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<div class="ChapSects"><a href="chap3.html#X87EAF8E578D95793">3 <span class="Heading">The User Interface of the <strong class="pkg">AtlasRep</strong> Package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X87D26B13819A8209">3.1 <span class="Heading">Accessing vs. Constructing Representations</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X81BF52FC7B8C08D4">3.2 <span class="Heading">Group Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X795DB7E486E0817D">3.3 <span class="Heading">Standard Generators Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X861CD545803B97E8">3.4 <span class="Heading">Class Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X850EEDEE831EE039">3.4-1 <span class="Heading">Definition of <strong class="pkg">ATLAS</strong> Class Names</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X79DACFFA7E2D1A99">3.5-1 DisplayAtlasInfo</a></span>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X841478AB7CD06D44">3.5-6 OneAtlasGeneratingSetInfo</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X790D5F8C7E8E6947">3.6 <span class="Heading"><strong class="pkg">Browse</strong> Applications Provided by <strong class="pkg">AtlasRep</strong></span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F31A7CB841FE63F">3.6-1 BrowseMinimalDegrees</a></span>
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<h3>3 <span class="Heading">The User Interface of the <strong class="pkg">AtlasRep</strong> Package</span></h3>
<p>The <em>user interface</em> is the part of the <strong class="pkg">GAP</strong> interface that allows one to display information about the current contents of the database and to access individual data (perhaps by downloading them, see Section <a href="chap4.html#X7C3293A98577EE68"><span class="RefLink">4.2-1</span></a>). The corresponding functions are described in this chapter. See Section <a href="chap2.html#X87ACE06E82B68589"><span class="RefLink">2.4</span></a> for some small examples how to use the functions of the interface.</p>
<p>Data extensions of the <strong class="pkg">AtlasRep</strong> package are regarded as another part of the <strong class="pkg">GAP</strong> interface, they are described in Chapter <a href="chap5.html#X7B0718A178BB10CA"><span class="RefLink">5</span></a>. Finally, the low level part of the interface is described in Chapter <a href="chap7.html#X7F77634D817156B3"><span class="RefLink">7</span></a>.</p>
<p><a id="X87D26B13819A8209" name="X87D26B13819A8209"></a></p>
<h4>3.1 <span class="Heading">Accessing vs. Constructing Representations</span></h4>
<p>Note that <em>accessing</em> the data means in particular that it is <em>not</em> the aim of this package to <em>construct</em> representations from known ones. For example, if at least one permutation representation for a group <span class="SimpleMath">G</span> is stored but no matrix representation in a positive characteristic <span class="SimpleMath">p</span>, say, then <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) returns <code class="keyw">fail</code> when it is asked for a description of an available set of matrix generators for <span class="SimpleMath">G</span> in characteristic <span class="SimpleMath">p</span>, although such a representation can be obtained by reduction modulo <span class="SimpleMath">p</span> of an integral matrix representation, which in turn can be constructed from any permutation representation.</p>
<p><a id="X81BF52FC7B8C08D4" name="X81BF52FC7B8C08D4"></a></p>
<h4>3.2 <span class="Heading">Group Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p><a id="sect:groupnames"/> When you access data via the <strong class="pkg">AtlasRep</strong> package, you specify the group in question by an admissible <em>name</em>. Thus it is essential to know these names, which are called <em>the <strong class="pkg">GAP</strong> names</em> of the group in the following.</p>
<p>For a group <span class="SimpleMath">G</span>, say, whose character table is available in <strong class="pkg">GAP</strong>'s Character Table Library (see <a href="chapBib.html#biBCTblLib">[Bre22]</a>), the admissible names of <span class="SimpleMath">G</span> are the admissible names of this character table. One such name is the <code class="func">Identifier</code> (<a href="../../../doc/ref/chap71.html#X79C40EE97890202F"><span class="RefLink">Reference: Identifier for character tables</span></a>) value of the character table, see <a href="../../../pkg/ctbllib/doc/chap3.html#X818A9DE5799A4809"><span class="RefLink">CTblLib: Admissible Names for Character Tables in CTblLib</span></a>. This name is usually very similar to the name used in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>. For example, <code class="code">"M22"</code> is a <strong class="pkg">GAP</strong> name of the Mathieu group <span class="SimpleMath">M_22</span>, <code class="code">"12_1.U4(3).2_1"</code> is a <strong class="pkg">GAP</strong> name of <span class="SimpleMath">12_1.U_4(3).2_1</span>, the two names <code class="code">"S5"</code> and <code class="code">"A5.2"</code> are <strong class="pkg">GAP</strong> names of the symmetric group <span class="SimpleMath">S_5</span>, and the two names <code class="code">"F3+"</code> and <code class="code">"Fi24'"</code> are <strong class="pkg">GAP</strong> names of the simple Fischer group <span class="SimpleMath">Fi_24^'</span>.</p>
<p>When a <strong class="pkg">GAP</strong> name is required as an input of a package function, this input is case insensitive. For example, both <code class="code">"A5"</code> and <code class="code">"a5"</code> are valid arguments of <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>).</p>
<p>Internally, for example as part of filenames (see Section <a href="chap7.html#X7A86627B80980F61"><span class="RefLink">7.6</span></a>), the package uses names that may differ from the <strong class="pkg">GAP</strong> names; these names are called <em><strong class="pkg">ATLAS</strong>-file names</em>. For example, <code class="code">"A5"</code>, <code class="code">"TE62"</code>, and <code class="code">"F24"</code> are <strong class="pkg">ATLAS</strong>-file names. Of these, only <code class="code">"A5"</code> is also a <strong class="pkg">GAP</strong> name, but the other two are not; corresponding <strong class="pkg">GAP</strong> names are <code class="code">"2E6(2)"</code> and <code class="code">"Fi24'"</code>, respectively.</p>
<p><a id="X795DB7E486E0817D" name="X795DB7E486E0817D"></a></p>
<h4>3.3 <span class="Heading">Standard Generators Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p>For the general definition of <em>standard generators</em> of a group, see <a href="chapBib.html#biBWil96">[Wil96]</a>.</p>
<p>Several <em>different</em> standard generators may be defined for a group, the definitions for each group that occurs in the <strong class="pkg">ATLAS</strong> of Group Representations can be found at</p>
<p><span class="URL"><a href="http://atlas.math.rwth-aachen.de/Atlas/v3">http://atlas.math.rwth-aachen.de/Atlas/v3</a></span>.</p>
<p>When one specifies the standardization, the <span class="SimpleMath">i</span>-th set of standard generators is denoted by the number <span class="SimpleMath">i</span>. Note that when more than one set of standard generators is defined for a group, one must be careful to use <em>compatible standardization</em>. For example, the straight line programs, straight line decisions and black box programs in the database refer to a specific standardization of their inputs. That is, a straight line program for computing generators of a certain subgroup of a group <span class="SimpleMath">G</span> is defined only for a specific set of standard generators of <span class="SimpleMath">G</span>, and applying the program to matrix or permutation generators of <span class="SimpleMath">G</span> but w. r. t. a different standardization may yield unpredictable results. Therefore the results returned by the functions described in this chapter contain information about the standardizations they refer to.</p>
<p><a id="X861CD545803B97E8" name="X861CD545803B97E8"></a></p>
<h4>3.4 <span class="Heading">Class Names Used in the <strong class="pkg">AtlasRep</strong> Package</span></h4>
<p>For each straight line program (see <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>)) that is used to compute lists of class representatives, it is essential to describe the classes in which these elements lie. Therefore, in these cases the records returned by the function <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>) contain a component <code class="code">outputs</code> with value a list of <em>class names</em>.</p>
<p>Currently we define these class names only for simple groups and certain extensions of simple groups, see Section <a href="chap3.html#X850EEDEE831EE039"><span class="RefLink">3.4-1</span></a>. The function <code class="func">AtlasClassNames</code> (<a href="chap3.html#X78166D1D7D18EFBF"><span class="RefLink">3.4-2</span></a>) can be used to compute the list of class names from the character table in the <strong class="pkg">GAP</strong> Library.</p>
<p><a id="X850EEDEE831EE039" name="X850EEDEE831EE039"></a></p>
<h5>3.4-1 <span class="Heading">Definition of <strong class="pkg">ATLAS</strong> Class Names</span></h5>
<p>For the definition of class names of an almost simple group, we assume that the ordinary character tables of all nontrivial normal subgroups are shown in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>.</p>
<p>Each class name is a string consisting of the element order of the class in question followed by a combination of capital letters, digits, and the characters <code class="code">'</code> and <code class="code">-</code> (starting with a capital letter). For example, <code class="code">1A</code>, <code class="code">12A1</code>, and <code class="code">3B'</code> denote the class that contains the identity element, a class of element order <span class="SimpleMath">12</span>, and a class of element order <span class="SimpleMath">3</span>, respectively.</p>
<ol>
<li><p>For the table of a <em>simple</em> group, the class names are the same as returned by the two argument version of the <strong class="pkg">GAP</strong> function <code class="func">ClassNames</code> (<a href="../../../doc/ref/chap71.html#X804CFD597C795801"><span class="RefLink">Reference: ClassNames</span></a>), cf. <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 5]</a>: The classes are arranged w. r. t. increasing element order and for each element order w. r. t. decreasing centralizer order, the conjugacy classes that contain elements of order <span class="SimpleMath">n</span> are named <span class="SimpleMath">n</span><code class="code">A</code>, <span class="SimpleMath">n</span><code class="code">B</code>, <span class="SimpleMath">n</span><code class="code">C</code>, <span class="SimpleMath">...</span>; the alphabet used here is potentially infinite, and reads <code class="code">A</code>, <code class="code">B</code>, <code class="code">C</code>, <span class="SimpleMath">...</span>, <code class="code">Z</code>, <code class="code">A1</code>, <code class="code">B1</code>, <span class="SimpleMath">...</span>, <code class="code">A2</code>, <code class="code">B2</code>, <span class="SimpleMath">...</span>.</p>
<p>For example, the classes of the alternating group <span class="SimpleMath">A_5</span> have the names <code class="code">1A</code>, <code class="code">2A</code>, <code class="code">3A</code>, <code class="code">5A</code>, and <code class="code">5B</code>.</p>
</li>
<li><p>Next we consider the case of an <em>upward extension</em> <span class="SimpleMath">G.A</span> of a simple group <span class="SimpleMath">G</span> by a <em>cyclic</em> group of order <span class="SimpleMath">A</span>. The <strong class="pkg">ATLAS</strong> defines class names for each element <span class="SimpleMath">g</span> of <span class="SimpleMath">G.A</span> only w. r. t. the group <span class="SimpleMath">G.a</span>, say, that is generated by <span class="SimpleMath">G</span> and <span class="SimpleMath">g</span>; namely, there is a power of <span class="SimpleMath">g</span> (with the exponent coprime to the order of <span class="SimpleMath">g</span>) for which the class has a name of the same form as the class names for simple groups, and the name of the class of <span class="SimpleMath">g</span> w. r. t. <span class="SimpleMath">G.a</span> is then obtained from this name by appending a suitable number of dashes <code class="code">'</code>. So dashed class names refer exactly to those classes that are not printed in the <strong class="pkg">ATLAS</strong>.</p>
<p>For example, those classes of the symmetric group <span class="SimpleMath">S_5</span> that do not lie in <span class="SimpleMath">A_5</span> have the names <code class="code">2B</code>, <code class="code">4A</code>, and <code class="code">6A</code>. The outer classes of the group <span class="SimpleMath">L_2(8).3</span> have the names <code class="code">3B</code>, <code class="code">6A</code>, <code class="code">9D</code>, and <code class="code">3B'</code>, <code class="code">6A'</code>, <code class="code">9D'</code>. The outer elements of order <span class="SimpleMath">5</span> in the group <span class="SimpleMath">Sz(32).5</span> lie in the classes with names <code class="code">5B</code>, <code class="code">5B'</code>, <code class="code">5B''</code>, and <code class="code">5B'''</code>.</p>
<p>In the group <span class="SimpleMath">G.A</span>, the class of <span class="SimpleMath">g</span> may fuse with other classes. The name of the class of <span class="SimpleMath">g</span> in <span class="SimpleMath">G.A</span> is obtained from the names of the involved classes of <span class="SimpleMath">G.a</span> by concatenating their names after removing the element order part from all of them except the first one.</p>
<p>For example, the elements of order <span class="SimpleMath">9</span> in the group <span class="SimpleMath">L_2(27).6</span> are contained in the subgroup <span class="SimpleMath">L_2(27).3</span> but not in <span class="SimpleMath">L_2(27)</span>. In <span class="SimpleMath">L_2(27).3</span>, they lie in the classes <code class="code">9A</code>, <code class="code">9A'</code>, <code class="code">9B</code>, and <code class="code">9B'</code>; in <span class="SimpleMath">L_2(27).6</span>, these classes fuse to <code class="code">9AB</code> and <code class="code">9A'B'</code>.</p>
</li>
<li><p>Now we define class names for <em>general upward extensions</em> <span class="SimpleMath">G.A</span> of a simple group <span class="SimpleMath">G</span>. Each element <span class="SimpleMath">g</span> of such a group lies in an upward extension <span class="SimpleMath">G.a</span> by a cyclic group, and the class names w. r. t. <span class="SimpleMath">G.a</span> are already defined. The name of the class of <span class="SimpleMath">g</span> in <span class="SimpleMath">G.A</span> is obtained by concatenating the names of the classes in the orbit of <span class="SimpleMath">G.A</span> on the classes of cyclic upward extensions of <span class="SimpleMath">G</span>, after ordering the names lexicographically and removing the element order part from all of them except the first one. An <em>exception</em> is the situation where dashed and non-dashed class names appear in an orbit; in this case, the dashed names are omitted.</p>
<p>For example, the classes <code class="code">21A</code> and <code class="code">21B</code> of the group <span class="SimpleMath">U_3(5).3</span> fuse in <span class="SimpleMath">U_3(5).S_3</span> to the class <code class="code">21AB</code>, and the class <code class="code">2B</code> of <span class="SimpleMath">U_3(5).2</span> fuses with the involution classes <code class="code">2B'</code>, <code class="code">2B''</code> in the groups <span class="SimpleMath">U_3(5).2^'</span> and <span class="SimpleMath">U_3(5).2^{''}</span> to the class <code class="code">2B</code> of <span class="SimpleMath">U_3(5).S_3</span>.</p>
<p>It may happen that some names in the <code class="code">outputs</code> component of a record returned by <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>) do not uniquely determine the classes of the corresponding elements. For example, the (algebraically conjugate) classes <code class="code">39A</code> and <code class="code">39B</code> of the group <span class="SimpleMath">Co_1</span> have not been distinguished yet. In such cases, the names used contain a minus sign <code class="code">-</code>, and mean <q>one of the classes in the range described by the name before and the name after the minus sign</q>; the element order part of the name does not appear after the minus sign. So the name <code class="code">39A-B</code> for the group <span class="SimpleMath">Co_1</span> means <code class="code">39A</code> or <code class="code">39B</code>, and the name <code class="code">20A-B'''</code> for the group <span class="SimpleMath">Sz(32).5</span> means one of the classes of element order <span class="SimpleMath">20</span> in this group (these classes lie outside the simple group <span class="SimpleMath">Sz</span>).</p>
</li>
<li><p>For a <em>downward extension</em> <span class="SimpleMath">m.G.A</span> of an almost simple group <span class="SimpleMath">G.A</span> by a cyclic group of order <span class="SimpleMath">m</span>, let <span class="SimpleMath">π</span> denote the natural epimorphism from <span class="SimpleMath">m.G.A</span> onto <span class="SimpleMath">G.A</span>. Each class name of <span class="SimpleMath">m.G.A</span> has the form <code class="code">nX_0</code>, <code class="code">nX_1</code> etc., where <code class="code">nX</code> is the class name of the image under <span class="SimpleMath">π</span>, and the indices <code class="code">0</code>, <code class="code">1</code> etc. are chosen according to the position of the class in the lifting order rows for <span class="SimpleMath">G</span>, see <a href="chapBib.html#biBCCN85">[CCN+85, Chapter 7, Section 7, and the example in Section 8]</a>).</p>
<p>For example, if <span class="SimpleMath">m = 6</span> then <code class="code">1A_1</code> and <code class="code">1A_5</code> denote the classes containing the generators of the kernel of <span class="SimpleMath">π</span>, that is, central elements of order <span class="SimpleMath">6</span>.</p>
</li>
</ol>
<p><a id="X78166D1D7D18EFBF" name="X78166D1D7D18EFBF"></a></p>
<h5>3.4-2 AtlasClassNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasClassNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of class names.</p>
<p>Let <var class="Arg">tbl</var> be the ordinary or modular character table of a group <span class="SimpleMath">G</span>, say, that is almost simple or a downward extension of an almost simple group and such that <var class="Arg">tbl</var> is an <strong class="pkg">ATLAS</strong> table from the <strong class="pkg">GAP</strong> Character Table Library, according to its <code class="func">InfoText</code> (<a href="../../../doc/ref/chap12.html#X871562FD7F982C12"><span class="RefLink">Reference: InfoText</span></a>) value. Then <code class="func">AtlasClassNames</code> returns the list of class names for <span class="SimpleMath">G</span>, as defined in Section <a href="chap3.html#X850EEDEE831EE039"><span class="RefLink">3.4-1</span></a>. The ordering of class names is the same as the ordering of the columns of <var class="Arg">tbl</var>.</p>
<p>(The function may work also for character tables that are not <strong class="pkg">ATLAS</strong> tables, but then clearly the class names returned are somewhat arbitrary.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "L3(4).3" ) );</span>
[ "1A", "2A", "3A", "4ABC", "5A", "5B", "7A", "7B", "3B", "3B'",
"3C", "3C'", "6B", "6B'", "15A", "15A'", "15B", "15B'", "21A",
"21A'", "21B", "21B'" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "U3(5).2" ) );</span>
[ "1A", "2A", "3A", "4A", "5A", "5B", "5CD", "6A", "7AB", "8AB",
"10A", "2B", "4B", "6D", "8C", "10B", "12B", "20A", "20B" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "L2(27).6" ) );</span>
[ "1A", "2A", "3AB", "7ABC", "13ABC", "13DEF", "14ABC", "2B", "4A",
"26ABC", "26DEF", "28ABC", "28DEF", "3C", "3C'", "6A", "6A'",
"9AB", "9A'B'", "6B", "6B'", "12A", "12A'" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "L3(4).3.2_2" ) );</span>
[ "1A", "2A", "3A", "4ABC", "5AB", "7A", "7B", "3B", "3C", "6B",
"15A", "15B", "21A", "21B", "2C", "4E", "6E", "8D", "14A", "14B" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "3.A6" ) );</span>
[ "1A_0", "1A_1", "1A_2", "2A_0", "2A_1", "2A_2", "3A_0", "3B_0",
"4A_0", "4A_1", "4A_2", "5A_0", "5A_1", "5A_2", "5B_0", "5B_1",
"5B_2" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasClassNames( CharacterTable( "2.A5.2" ) );</span>
[ "1A_0", "1A_1", "2A_0", "3A_0", "3A_1", "5AB_0", "5AB_1", "2B_0",
"4A_0", "4A_1", "6A_0", "6A_1" ]
</pre></div>
<p><a id="X7B14A254870BA5A1" name="X7B14A254870BA5A1"></a></p>
<h5>3.4-3 AtlasCharacterNames</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasCharacterNames</code>( <var class="Arg">tbl</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a list of character names.</p>
<p>Let <var class="Arg">tbl</var> be the ordinary or modular character table of a simple group. <code class="func">AtlasCharacterNames</code> returns a list of strings, the <span class="SimpleMath">i</span>-th entry being the name of the <span class="SimpleMath">i</span>-th irreducible character of <var class="Arg">tbl</var>; this name consists of the degree of this character followed by distinguishing lowercase letters.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasCharacterNames( CharacterTable( "A5" ) ); </span>
[ "1a", "3a", "3b", "4a", "5a" ]
</pre></div>
<p><a id="X7CC88B2287A72204" name="X7CC88B2287A72204"></a></p>
<h4>3.5 <span class="Heading">Accessing Data via <strong class="pkg">AtlasRep</strong></span></h4>
<p>The examples shown in this section refer to the situation that no extensions have been notified, and to a perhaps outdated table of contents. That is, the current version of the database may contain more information than is shown here.</p>
<p><a id="X79DACFFA7E2D1A99" name="X79DACFFA7E2D1A99"></a></p>
<h5>3.5-1 DisplayAtlasInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasInfo</code>( [<var class="Arg">listofnames</var>][,] [<var class="Arg">std</var>][,] [<var class="Arg">"contents"</var>, <var class="Arg">sources</var>][,] [<var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DisplayAtlasInfo</code>( <var class="Arg">gapname</var>[, <var class="Arg">std</var>][, <var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function lists the information available via the <strong class="pkg">AtlasRep</strong> package, for the given input.</p>
<p>There are essentially three ways of calling this function.</p>
<ul>
<li><p>If there is no argument or if the first argument is a list <var class="Arg">listofnames</var> of strings that are <strong class="pkg">GAP</strong> names of groups, <code class="func">DisplayAtlasInfo</code> shows an overview of the known information.</p>
</li>
<li><p>If the first argument is a string <var class="Arg">gapname</var> that is a <strong class="pkg">GAP</strong> name of a group, <code class="func">DisplayAtlasInfo</code> shows an overview of the information that is available for this group.</p>
</li>
<li><p>If the string <code class="code">"contents"</code> is the only argument then the function shows which parts of the database are available; these are at least the <code class="code">"core"</code> part, which means the data from the <strong class="pkg">ATLAS</strong> of Group Representations, and the <code class="code">"internal"</code> part, which means the data that are distributed with the <strong class="pkg">AtlasRep</strong> package. Other parts can become available by calls to <code class="func">AtlasOfGroupRepresentationsNotifyData</code> (<a href="chap5.html#X81B5FA0578257653"><span class="RefLink">5.1-1</span></a>). Note that the shown numbers of locally available files depend on what has already been downloaded.</p>
</li>
</ul>
<p>In each case, the information will be printed to the screen or will be fed into a pager, see Section <a href="chap4.html#X81F055037F9D3068"><span class="RefLink">4.2-11</span></a>. An interactive alternative to <code class="func">DisplayAtlasInfo</code> is the function <code class="func">BrowseAtlasInfo</code> (<a href="../../../pkg/browse/doc/chap6.html#X8411AF928194C5AB"><span class="RefLink">Browse: BrowseAtlasInfo</span></a>), see <a href="chapBib.html#biBBrowse">[BL18]</a>.</p>
<p>The following paragraphs describe the structure of the output in the two cases. Examples can be found in Section <a href="chap3.html#X7CE4FF2380DB47F2"><span class="RefLink">3.5-2</span></a>.</p>
<p>Called without arguments, <code class="func">DisplayAtlasInfo</code> shows a general overview for all groups. If some information is available for the group <span class="SimpleMath">G</span>, say, then one line is shown for <span class="SimpleMath">G</span>, with the following columns.</p>
<dl>
<dt><strong class="Mark"><code class="code">group</code></strong></dt>
<dd><p>the <strong class="pkg">GAP</strong> name of <span class="SimpleMath">G</span> (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>), if applicable followed by a string (by default a star <code class="code">*</code>) indicating that at least one column refers to data not belonging to the core part of the database (see Section <a href="chap4.html#X862C660878D422FA"><span class="RefLink">4.2-12</span></a>).</p>
</dd>
<dt><strong class="Mark"><code class="code">#</code></strong></dt>
<dd><p>the number of faithful representations stored for <span class="SimpleMath">G</span> that satisfy the additional conditions given (see below),</p>
</dd>
<dt><strong class="Mark"><code class="code">maxes</code></strong></dt>
<dd><p>the number of available straight line programs for computing generators of maximal subgroups of <span class="SimpleMath">G</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">cl</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program for computing representatives of conjugacy classes of elements of <span class="SimpleMath">G</span> is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">cyc</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program for computing representatives of classes of maximally cyclic subgroups of <span class="SimpleMath">G</span> is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">out</code></strong></dt>
<dd><p>descriptions of outer automorphisms of <span class="SimpleMath">G</span> for which at least one program is stored,</p>
</dd>
<dt><strong class="Mark"><code class="code">fnd</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available for finding standard generators,</p>
</dd>
<dt><strong class="Mark"><code class="code">chk</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available for checking whether a set of generators is a set of standard generators, and</p>
</dd>
<dt><strong class="Mark"><code class="code">prs</code></strong></dt>
<dd><p>a <code class="code">+</code> sign if at least one program is available that encodes a presentation.</p>
</dd>
</dl>
<p>Called with a list <var class="Arg">listofnames</var> of strings that are <strong class="pkg">GAP</strong> names of some groups, <code class="func">DisplayAtlasInfo</code> prints the overview described above but restricted to the groups in this list.</p>
<p>In addition to or instead of <var class="Arg">listofnames</var>, the string <code class="code">"contents"</code> and a description <span class="SimpleMath">sources</span> of the data may be given about which the overview is formed. See below for admissible values of <span class="SimpleMath">sources</span>.</p>
<p>Called with a string <var class="Arg">gapname</var> that is a <strong class="pkg">GAP</strong> name of a group, <code class="func">DisplayAtlasInfo</code> prints an overview of the information that is available for this group. One line is printed for each faithful representation, showing the number of this representation (which can be used in calls of <code class="func">AtlasGenerators</code> (<a href="chap3.html#X7D1CCCF8852DFF39"><span class="RefLink">3.5-3</span></a>)), and a string of one of the following forms; in both cases, <span class="SimpleMath">id</span> is a (possibly empty) string.</p>
<dl>
<dt><strong class="Mark"><code class="code">G <= Sym(</code><span class="SimpleMath">n</span><span class="SimpleMath">id</span><code class="code">)</code></strong></dt>
<dd><p>denotes a permutation representation of degree <span class="SimpleMath">n</span>, for example <code class="code">G <= Sym(40a)</code> and <code class="code">G <= Sym(40b)</code> denote two (nonequivalent) representations of degree <span class="SimpleMath">40</span>.</p>
</dd>
<dt><strong class="Mark"><code class="code">G <= GL(</code><span class="SimpleMath">n</span><span class="SimpleMath">id</span>,<span class="SimpleMath">descr</span><code class="code">)</code></strong></dt>
<dd><p>denotes a matrix representation of dimension <span class="SimpleMath">n</span> over a coefficient ring described by <span class="SimpleMath">descr</span>, which can be a prime power, <code class="code">ℤ</code> (denoting the ring of integers), a description of an algebraic extension field, <code class="code">ℂ</code> (denoting an unspecified algebraic extension field), or <code class="code">ℤ/</code><span class="SimpleMath">m</span><code class="code">ℤ</code> for an integer <span class="SimpleMath">m</span> (denoting the ring of residues mod <span class="SimpleMath">m</span>); for example, <code class="code">G <= GL(2a,4)</code> and <code class="code">G <= GL(2b,4)</code> denote two (nonequivalent) representations of dimension <span class="SimpleMath">2</span> over the field with four elements.</p>
</dd>
</dl>
<p>Below the representations, the programs available for <var class="Arg">gapname</var> are listed. In each row of the overview, the entry in the first column is followed by a string (by default a star <code class="code">*</code>) if the row refers to data not belonging to the core part of the database (see Section <a href="chap4.html#X862C660878D422FA"><span class="RefLink">4.2-12</span></a>).</p>
<p>The following optional arguments can be used to restrict the overviews.</p>
<dl>
<dt><strong class="Mark"><var class="Arg">std</var></strong></dt>
<dd><p>must be a positive integer or a list of positive integers; if it is given then only those representations are considered that refer to the <var class="Arg">std</var>-th set of standard generators or the <span class="SimpleMath">i</span>-th set of standard generators, for <span class="SimpleMath">i</span> in <var class="Arg">std</var> (see Section <a href="chap3.html#X795DB7E486E0817D"><span class="RefLink">3.3</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">"contents"</code> and <span class="SimpleMath">sources</span></strong></dt>
<dd><p>for a string or a list of strings <span class="SimpleMath">sources</span>, restrict the data about which the overview is formed; if <span class="SimpleMath">sources</span> is the string <code class="code">"core"</code> then only data from the <strong class="pkg">ATLAS</strong> of Group Representations are considered, if <span class="SimpleMath">sources</span> is a string that denotes a data extension in the sense of a <code class="code">dirid</code> argument of <code class="func">AtlasOfGroupRepresentationsNotifyData</code> (<a href="chap5.html#X81B5FA0578257653"><span class="RefLink">5.1-1</span></a>) then only the data that belong to this data extension are considered; also a list of such strings may be given, then the union of these data is considered,</p>
</dd>
<dt><strong class="Mark"><code class="code">Identifier</code> and <span class="SimpleMath">id</span></strong></dt>
<dd><p>restrict to representations with <code class="code">id</code> component in the list <span class="SimpleMath">id</span> (note that this component is itself a list, entering this list is not admissible), or satisfying the function <span class="SimpleMath">id</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">IsPermGroup</code> and <code class="keyw">true</code> (or <code class="keyw">false</code>)</strong></dt>
<dd><p>restrict to permutation representations (or to representations that are not permutation representations),</p>
</dd>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a positive integer, a list of positive integers, or a property <span class="SimpleMath">n</span>, restrict to permutation representations of degree equal to <span class="SimpleMath">n</span>, or in the list <span class="SimpleMath">n</span>, or satisfying the function <span class="SimpleMath">n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code> and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>restrict to faithful permutation representations of minimal degree (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsTransitive</code> and a boolean value</strong></dt>
<dd><p>restrict to transitive or intransitive permutation representations where this information is available (if the value <code class="keyw">true</code> or <code class="keyw">false</code> is given), or to representations for which this information is not available (if the value <code class="keyw">fail</code> is given),</p>
</dd>
<dt><strong class="Mark"><code class="code">IsPrimitive</code> and a boolean value</strong></dt>
<dd><p>restrict to primitive or imprimitive permutation representations where this information is available (if the value <code class="keyw">true</code> or <code class="keyw">false</code> is given), or to representations for which this information is not available (if the value <code class="keyw">fail</code> is given),</p>
</dd>
<dt><strong class="Mark"><code class="code">Transitivity</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a nonnegative integer, a list of nonnegative integers, or a property <span class="SimpleMath">n</span>, restrict to permutation representations for which the information is available that the transitivity is equal to <span class="SimpleMath">n</span>, or is in the list <span class="SimpleMath">n</span>, or satisfies the function <span class="SimpleMath">n</span>; if <span class="SimpleMath">n</span> is <code class="keyw">fail</code> then restrict to all permutation representations for which this information is not available,</p>
</dd>
<dt><strong class="Mark"><code class="code">RankAction</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a nonnegative integer, a list of nonnegative integers, or a property <span class="SimpleMath">n</span>, restrict to permutation representations for which the information is available that the rank is equal to <span class="SimpleMath">n</span>, or is in the list <span class="SimpleMath">n</span>, or satisfies the function <span class="SimpleMath">n</span>; if <span class="SimpleMath">n</span> is <code class="keyw">fail</code> then restrict to all permutation representations for which this information is not available,</p>
</dd>
<dt><strong class="Mark"><code class="code">IsMatrixGroup</code> and <code class="keyw">true</code> (or <code class="keyw">false</code>)</strong></dt>
<dd><p>restrict to matrix representations (or to representations that are not matrix representations),</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code> and <span class="SimpleMath">p</span></strong></dt>
<dd><p>for a prime integer, a list of prime integers, or a property <span class="SimpleMath">p</span>, restrict to matrix representations over fields of characteristic equal to <span class="SimpleMath">p</span>, or in the list <span class="SimpleMath">p</span>, or satisfying the function <span class="SimpleMath">p</span> (representations over residue class rings that are not fields can be addressed by entering <code class="keyw">fail</code> as the value of <span class="SimpleMath">p</span>),</p>
</dd>
<dt><strong class="Mark"><code class="code">Dimension</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a positive integer, a list of positive integers, or a property <span class="SimpleMath">n</span>, restrict to matrix representations of dimension equal to <span class="SimpleMath">n</span>, or in the list <span class="SimpleMath">n</span>, or satisfying the function <span class="SimpleMath">n</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code>, <span class="SimpleMath">p</span>, <code class="code">Dimension</code>,
and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>for a prime integer <span class="SimpleMath">p</span>, restrict to faithful matrix representations over fields of characteristic <span class="SimpleMath">p</span> that have minimal dimension (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">Ring</code> and <span class="SimpleMath">R</span></strong></dt>
<dd><p>for a ring or a property <span class="SimpleMath">R</span>, restrict to matrix representations for which the information is available that the ring spanned by the matrix entries is contained in this ring or satisfies this property (note that the representation might be defined over a proper subring); if <span class="SimpleMath">R</span> is <code class="keyw">fail</code> then restrict to all matrix representations for which this information is not available,</p>
</dd>
<dt><strong class="Mark"><code class="code">Ring</code>, <span class="SimpleMath">R</span>, <code class="code">Dimension</code>,
and the string <code class="code">"minimal"</code></strong></dt>
<dd><p>for a ring <span class="SimpleMath">R</span>, restrict to faithful matrix representations over this ring that have minimal dimension (if this information is available),</p>
</dd>
<dt><strong class="Mark"><code class="code">Character</code> and <span class="SimpleMath">chi</span></strong></dt>
<dd><p>for a class function or a list of class functions <span class="SimpleMath">chi</span>, restrict to representations with these characters (note that the underlying characteristic of the class function, see Section <a href="../../../doc/ref/chap71.html#X7F58A82F7D88000A"><span class="RefLink">Reference: UnderlyingCharacteristic</span></a>, determines the characteristic of the representation),</p>
</dd>
<dt><strong class="Mark"><code class="code">Character</code> and <span class="SimpleMath">name</span></strong></dt>
<dd><p>for a string <span class="SimpleMath">name</span>, restrict to representations for which the character is known to have this name, according to the information shown by <code class="func">DisplayAtlasInfo</code>; if the characteristic is not specified then it defaults to zero,</p>
</dd>
<dt><strong class="Mark"><code class="code">Character</code> and <span class="SimpleMath">n</span></strong></dt>
<dd><p>for a positive integer <span class="SimpleMath">n</span>, restrict to representations for which the character is known to be the <span class="SimpleMath">n</span>-th irreducible character in <strong class="pkg">GAP</strong>'s library character table of the group in question; if the characteristic is not specified then it defaults to zero,</p>
</dd>
<dt><strong class="Mark"><code class="code">IsStraightLineProgram</code> and <code class="keyw">true</code></strong></dt>
<dd><p>restrict to straight line programs, straight line decisions (see Section <a href="chap6.html#X8121E9567A7137C9"><span class="RefLink">6.1</span></a>), and black box programs (see Section <a href="chap6.html#X7BE856BC785A9E8F"><span class="RefLink">6.2</span></a>), and</p>
</dd>
<dt><strong class="Mark"><code class="code">IsStraightLineProgram</code> and <code class="keyw">false</code></strong></dt>
<dd><p>restrict to representations.</p>
</dd>
</dl>
<p>Note that the above conditions refer only to the information that is available without accessing the representations. For example, if it is not stored in the table of contents whether a permutation representation is primitive then this representation does not match an <code class="code">IsPrimitive</code> condition in <code class="func">DisplayAtlasInfo</code>.</p>
<p>If <q>minimality</q> information is requested and no available representation matches this condition then either no minimal representation is available or the information about the minimality is missing. See <code class="func">MinimalRepresentationInfo</code> (<a href="chap6.html#X7DC66D8282B2BB7F"><span class="RefLink">6.3-1</span></a>) for checking whether the minimality information is available for the group in question. Note that in the cases where the string <code class="code">"minimal"</code> occurs as an argument, <code class="func">MinimalRepresentationInfo</code> (<a href="chap6.html#X7DC66D8282B2BB7F"><span class="RefLink">6.3-1</span></a>) is called with third argument <code class="code">"lookup"</code>; this is because the stored information was precomputed just for the groups in the <strong class="pkg">ATLAS</strong> of Group Representations, so trying to compute non-stored minimality information (using other available databases) will hardly be successful.</p>
<p>The representations are ordered as follows. Permutation representations come first (ordered according to their degrees), followed by matrix representations over finite fields (ordered first according to the field size and second according to the dimension), matrix representations over the integers, and then matrix representations over algebraic extension fields (both kinds ordered according to the dimension), the last representations are matrix representations over residue class rings (ordered first according to the modulus and second according to the dimension).</p>
<p>The maximal subgroups are ordered according to decreasing group order. For an extension <span class="SimpleMath">G.p</span> of a simple group <span class="SimpleMath">G</span> by an outer automorphism of prime order <span class="SimpleMath">p</span>, this means that <span class="SimpleMath">G</span> is the first maximal subgroup and then come the extensions of the maximal subgroups of <span class="SimpleMath">G</span> and the novelties; so the <span class="SimpleMath">n</span>-th maximal subgroup of <span class="SimpleMath">G</span> and the <span class="SimpleMath">n</span>-th maximal subgroup of <span class="SimpleMath">G.p</span> are in general not related. (This coincides with the numbering used for the <code class="func">Maxes</code> (<a href="../../../pkg/ctbllib/doc/chap3.html#X8150E63F7DBDF252"><span class="RefLink">CTblLib: Maxes</span></a>) attribute for character tables.)</p>
<p><a id="X7CE4FF2380DB47F2" name="X7CE4FF2380DB47F2"></a></p>
<h5>3.5-2 <span class="Heading">Examples for DisplayAtlasInfo</span></h5>
<p>Here are some examples how <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>) can be called, and how its output can be interpreted.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "contents" );</span>
- AtlasRepAccessRemoteFiles: false
- AtlasRepDataDirectory: /home/you/gap/pkg/atlasrep/
ID | address, version, files
---------+------------------------------------------------
core | http://atlas.math.rwth-aachen.de/Atlas/,
| version 2019-04-08,
| 10586 files locally available.
---------+------------------------------------------------
internal | atlasrep/datapkg,
| version 2019-05-06,
| 276 files locally available.
---------+------------------------------------------------
mfer | http://www.math.rwth-aachen.de/~mfer/datagens/,
| version 2015-10-06,
| 34 files locally available.
---------+------------------------------------------------
ctblocks | ctblocks/atlas/,
| version 2019-04-08,
| 121 files locally available.
</pre></div>
<p>Note: The above output does not fit to the rest of the manual examples, since data extensions except <code class="code">internal</code> have been removed at the beginning of Chapter <a href="chap2.html#X8171B3798425E183"><span class="RefLink">2</span></a>.</p>
<p>The output tells us that two data extensions have been notified in addition to the core data from the <strong class="pkg">ATLAS</strong> of Group Representations and the (local) internal data distributed with the <strong class="pkg">AtlasRep</strong> package. The files of the extension <code class="code">mfer</code> must be downloaded before they can be read (but note that the access to remote files is disabled), and the files of the extension <code class="code">ctblocks</code> are locally available in the <code class="file">ctblocks/atlas</code> subdirectory of the <strong class="pkg">GAP</strong> package directory. This table (in particular the numbers of locally available files) depends on your installation of the package and how many files you have already downloaded.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( [ "M11", "A5" ] );</span>
group | # | maxes | cl | cyc | out | fnd | chk | prs
------+----+-------+----+-----+-----+-----+-----+----
M11 | 42 | 5 | + | + | | + | + | +
A5* | 18 | 3 | + | | | | + | +
</pre></div>
<p>The above output means that the database provides <span class="SimpleMath">42</span> representations of the Mathieu group <span class="SimpleMath">M_11</span>, straight line programs for computing generators of representatives of all five classes of maximal subgroups, for computing representatives of the conjugacy classes of elements and of generators of maximally cyclic subgroups, contains no straight line program for applying outer automorphisms (well, in fact <span class="SimpleMath">M_11</span> admits no nontrivial outer automorphism), and contains straight line decisions that check a set of generators or a set of group elements for being a set of standard generators. Analogously, <span class="SimpleMath">18</span> representations of the alternating group <span class="SimpleMath">A_5</span> are available, straight line programs for computing generators of representatives of all three classes of maximal subgroups, and no straight line programs for computing representatives of the conjugacy classes of elements, of generators of maximally cyclic subgroups, and no for computing images under outer automorphisms; straight line decisions for checking the standardization of generators or group elements are available. The star <code class="code">*</code> in the first column of the row for <span class="SimpleMath">A_5</span> means that some of the available data do not belong to the core part of the database (see Section <a href="chap4.html#X862C660878D422FA"><span class="RefLink">4.2-12</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( [ "M11", "A5" ], NrMovedPoints, 11 );</span>
group | # | maxes | cl | cyc | out | fnd | chk | prs
------+---+-------+----+-----+-----+-----+-----+----
M11 | 1 | 5 | + | + | | + | + | +
</pre></div>
<p>The given conditions restrict the overview to permutation representations on <span class="SimpleMath">11</span> points. The rows for all those groups are omitted for which no such representation is available, and the numbers of those representations are shown that satisfy the given conditions. In the above example, we see that no representation on <span class="SimpleMath">11</span> points is available for <span class="SimpleMath">A_5</span>, and exactly one such representation is available for <span class="SimpleMath">M_11</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", IsPermGroup, true );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)
3: G <= Sym(10) rank 3, on cosets of S3 (3rd max.)
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", NrMovedPoints, [ 4 .. 9 ] );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
2: G <= Sym(6) 2-trans., on cosets of D10 (2nd max.)
</pre></div>
<p>The first three representations stored for <span class="SimpleMath">A_5</span> are (in fact primitive) permutation representations.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", Dimension, [ 1 .. 3 ] );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
8: G <= GL(2a,4) character 2a
9: G <= GL(2b,4) character 2b
10: G <= GL(3,5) character 3a
12: G <= GL(3a,9) character 3a
13: G <= GL(3b,9) character 3b
17: G <= GL(3a,Field([Sqrt(5)])) character 3a
18: G <= GL(3b,Field([Sqrt(5)])) character 3b
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", Characteristic, 0 );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
14: G <= GL(4,Z) character 4a
15: G <= GL(5,Z) character 5a
16: G <= GL(6,Z) character 3ab
17: G <= GL(3a,Field([Sqrt(5)])) character 3a
18: G <= GL(3b,Field([Sqrt(5)])) character 3b
</pre></div>
<p>The representations with number between <span class="SimpleMath">4</span> and <span class="SimpleMath">13</span> are (in fact irreducible) matrix representations over various finite fields, those with numbers <span class="SimpleMath">14</span> to <span class="SimpleMath">16</span> are integral matrix representations, and the last two are matrix representations over the field generated by <span class="SimpleMath">sqrt{5}</span> over the rational number field.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", Identifier, "a" );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
4: G <= GL(4a,2) character 4a
8: G <= GL(2a,4) character 2a
12: G <= GL(3a,9) character 3a
17: G <= GL(3a,Field([Sqrt(5)])) character 3a
</pre></div>
<p>Each of the representations with the numbers <span class="SimpleMath">4, 8, 12</span>, and <span class="SimpleMath">17</span> is labeled with the distinguishing letter <code class="code">a</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", NrMovedPoints, IsPrimeInt );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
1: G <= Sym(5) 3-trans., on cosets of A4 (1st max.)
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", Characteristic, IsOddInt );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
6: G <= GL(4,3) character 4a
7: G <= GL(6,3) character 3ab
10: G <= GL(3,5) character 3a
11: G <= GL(5,5) character 5a
12: G <= GL(3a,9) character 3a
13: G <= GL(3b,9) character 3b
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", Dimension, IsPrimeInt );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
8: G <= GL(2a,4) character 2a
9: G <= GL(2b,4) character 2b
10: G <= GL(3,5) character 3a
11: G <= GL(5,5) character 5a
12: G <= GL(3a,9) character 3a
13: G <= GL(3b,9) character 3b
15: G <= GL(5,Z) character 5a
17: G <= GL(3a,Field([Sqrt(5)])) character 3a
18: G <= GL(3b,Field([Sqrt(5)])) character 3b
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", Ring, IsFinite and IsPrimeField );</span>
Representations for G = A5: (all refer to std. generators 1)
---------------------------
4: G <= GL(4a,2) character 4a
5: G <= GL(4b,2) character 2ab
6: G <= GL(4,3) character 4a
7: G <= GL(6,3) character 3ab
10: G <= GL(3,5) character 3a
11: G <= GL(5,5) character 5a
</pre></div>
<p>The above examples show how the output can be restricted using a property (a unary function that returns either <code class="keyw">true</code> or <code class="keyw">false</code>) that follows <code class="func">NrMovedPoints</code> (<a href="../../../doc/ref/chap42.html#X85E7B1E28430F49E"><span class="RefLink">Reference: NrMovedPoints for a permutation</span></a>), <code class="func">Characteristic</code> (<a href="../../../doc/ref/chap31.html#X81278E53800BF64D"><span class="RefLink">Reference: Characteristic</span></a>), <code class="func">Dimension</code> (<a href="../../../doc/ref/chap57.html#X7E6926C6850E7C4E"><span class="RefLink">Reference: Dimension</span></a>), or <code class="func">Ring</code> (<a href="../../../doc/ref/chap56.html#X820B172A860A5B1A"><span class="RefLink">Reference: Ring</span></a>) in the argument list of <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DisplayAtlasInfo( "A5", IsStraightLineProgram, true );</span>
Programs for G = A5: (all refer to std. generators 1)
--------------------
- class repres.*
- presentation
- maxes (all 3):
1: A4
2: D10
3: S3
- std. gen. checker:
(check)
(pres)
</pre></div>
<p>Straight line programs are available for computing generators of representatives of the three classes of maximal subgroups of <span class="SimpleMath">A_5</span>, and a straight line decision for checking whether given generators are in fact standard generators is available as well as a presentation in terms of standard generators, see <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>). A straight line program for computing conjugacy class representatives is available, and the star <code class="code">*</code> says that this program does not belong to the core part of the database (see Section <a href="chap4.html#X862C660878D422FA"><span class="RefLink">4.2-12</span></a>).</p>
<p><a id="X7D1CCCF8852DFF39" name="X7D1CCCF8852DFF39"></a></p>
<h5>3.5-3 AtlasGenerators</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasGenerators</code>( <var class="Arg">gapname</var>, <var class="Arg">repnr</var>[, <var class="Arg">maxnr</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasGenerators</code>( <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record containing generators for a representation, or <code class="keyw">fail</code>.</p>
<p>In the first form, <var class="Arg">gapname</var> must be a string denoting a <strong class="pkg">GAP</strong> name (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>) of a group, and <var class="Arg">repnr</var> a positive integer. If at least <var class="Arg">repnr</var> representations for the group with <strong class="pkg">GAP</strong> name <var class="Arg">gapname</var> are available then <code class="func">AtlasGenerators</code>, when called with <var class="Arg">gapname</var> and <var class="Arg">repnr</var>, returns an immutable record describing the <var class="Arg">repnr</var>-th representation; otherwise <code class="keyw">fail</code> is returned. If a third argument <var class="Arg">maxnr</var>, a positive integer, is given then an immutable record describing the restriction of the <var class="Arg">repnr</var>-th representation to the <var class="Arg">maxnr</var>-th maximal subgroup is returned.</p>
<p>The result record has at least the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">contents</code></strong></dt>
<dd><p>the identifier of the part of the database to which the generators belong, for example <code class="code">"core"</code> or <code class="code">"internal"</code>,</p>
</dd>
<dt><strong class="Mark"><code class="code">generators</code></strong></dt>
<dd><p>a list of generators for the group,</p>
</dd>
<dt><strong class="Mark"><code class="code">groupname</code></strong></dt>
<dd><p>the <strong class="pkg">GAP</strong> name of the group (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>a <strong class="pkg">GAP</strong> object (a list of filenames plus additional information) that uniquely determines the representation, see Section <a href="chap7.html#X7CCA3DE97E756F01"><span class="RefLink">7.7</span></a>; the value can be used as <code class="code">identifier</code> argument of <code class="func">AtlasGenerators</code>.</p>
</dd>
<dt><strong class="Mark"><code class="code">repname</code></strong></dt>
<dd><p>a string that is an initial part of the filenames of the generators.</p>
</dd>
<dt><strong class="Mark"><code class="code">repnr</code></strong></dt>
<dd><p>the number of the representation in the current session, equal to the argument <var class="Arg">repnr</var> if this is given.</p>
</dd>
<dt><strong class="Mark"><code class="code">standardization</code></strong></dt>
<dd><p>the positive integer denoting the underlying standard generators,</p>
</dd>
<dt><strong class="Mark"><code class="code">type</code></strong></dt>
<dd><p>a string that describes the type of the representation (<code class="code">"perm"</code> for a permutation representation, <code class="code">"matff"</code> for a matrix representation over a finite field, <code class="code">"matint"</code> for a matrix representation over the ring of integers, <code class="code">"matalg"</code> for a matrix representation over an algebraic number field).</p>
</dd>
</dl>
<p>Additionally, the following <em>describing components</em> may be available if they are known, and depending on the data type of the representation.</p>
<dl>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>the group order,</p>
</dd>
<dt><strong class="Mark"><code class="code">id</code></strong></dt>
<dd><p>the distinguishing string as described for <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">charactername</code></strong></dt>
<dd><p>a string that describes the character of the representation,</p>
</dd>
<dt><strong class="Mark"><code class="code">constituents</code></strong></dt>
<dd><p>a list of positive integers denoting the positions of the irreducible constituents of the character of the representation,</p>
</dd>
<dt><strong class="Mark"><code class="code">p</code> (for permutation representations)</strong></dt>
<dd><p>for the number of moved points,</p>
</dd>
<dt><strong class="Mark"><code class="code">dim</code> (for matrix representations)</strong></dt>
<dd><p>the dimension of the matrices,</p>
</dd>
<dt><strong class="Mark"><code class="code">ring</code> (for matrix representations)</strong></dt>
<dd><p>the ring generated by the matrix entries,</p>
</dd>
<dt><strong class="Mark"><code class="code">transitivity</code> (for permutation representations)</strong></dt>
<dd><p>a nonnegative integer, see <code class="func">Transitivity</code> (<a href="../../../doc/ref/chap41.html#X8295D733796B7A37"><span class="RefLink">Reference: Transitivity</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">orbits</code> (for intransitive permutation representations)</strong></dt>
<dd><p>the sorted list of orbit lengths on the set of moved points,</p>
</dd>
<dt><strong class="Mark"><code class="code">rankAction</code> (for transitive permutation representations)</strong></dt>
<dd><p>the number of orbits of the point stabilizer on the set of moved points, see <code class="func">RankAction</code> (<a href="../../../doc/ref/chap41.html#X8166A6A17C8D6E73"><span class="RefLink">Reference: RankAction</span></a>),</p>
</dd>
<dt><strong class="Mark"><code class="code">stabilizer</code> (for transitive permutation representations)</strong></dt>
<dd><p>a string that describes the structure of the point stabilizers,</p>
</dd>
<dt><strong class="Mark"><code class="code">isPrimitive</code> (for transitive permutation representations)</strong></dt>
<dd><p><code class="keyw">true</code> if the point stabilizers are maximal subgroups, and <code class="keyw">false</code> otherwise,</p>
</dd>
<dt><strong class="Mark"><code class="code">maxnr</code> (for primitive permutation representations)</strong></dt>
<dd><p>the number of the class of maximal subgroups that contains the point stabilizers, w. r. t. the <code class="func">Maxes</code> (<a href="../../../pkg/ctbllib/doc/chap3.html#X8150E63F7DBDF252"><span class="RefLink">CTblLib: Maxes</span></a>) list.</p>
</dd>
</dl>
<p>It should be noted that the number <var class="Arg">repnr</var> refers to the number shown by <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>) <em>in the current session</em>; it may be that after the addition of new representations (for example after loading a package that provides some), <var class="Arg">repnr</var> refers to another representation.</p>
<p>The alternative form of <code class="func">AtlasGenerators</code>, with only argument <var class="Arg">identifier</var>, can be used to fetch the result record with <code class="code">identifier</code> value equal to <var class="Arg">identifier</var>. The purpose of this variant is to access the <em>same</em> representation also in <em>different</em> <strong class="pkg">GAP</strong> sessions.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens1:= AtlasGenerators( "A5", 1 );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", generators := [ (1,2)(3,4), (1,3,5) ],
groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens8:= AtlasGenerators( "A5", 8 );</span>
rec( charactername := "2a", constituents := [ 2 ], contents := "core",
dim := 2,
generators := [ [ [ Z(2)^0, 0*Z(2) ], [ Z(2^2), Z(2)^0 ] ],
[ [ 0*Z(2), Z(2)^0 ], [ Z(2)^0, Z(2)^0 ] ] ], groupname := "A5",
id := "a",
identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1,
4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2),
size := 60, standardization := 1, type := "matff" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens17:= AtlasGenerators( "A5", 17 );</span>
rec( charactername := "3a", constituents := [ 2 ], contents := "core",
dim := 3,
generators :=
[ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ]
], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ],
groupname := "A5", id := "a",
identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
type := "matalg" )
</pre></div>
<p>Each of the above pairs of elements generates a group isomorphic to <span class="SimpleMath">A_5</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens1max2:= AtlasGenerators( "A5", 1, 2 );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", generators := [ (1,2)(3,4), (2,3)(4,5) ],
groupname := "D10", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5, 2 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 10, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">id:= gens1max2.identifier;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens1max2 = AtlasGenerators( id );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">max2:= Group( gens1max2.generators );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( max2 );</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">IdGroup( max2 ) = IdGroup( DihedralGroup( 10 ) );</span>
true
</pre></div>
<p>The elements stored in <code class="code">gens1max2.generators</code> describe the restriction of the first representation of <span class="SimpleMath">A_5</span> to a group in the second class of maximal subgroups of <span class="SimpleMath">A_5</span> according to the list in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a>; this subgroup is isomorphic to the dihedral group <span class="SimpleMath">D_10</span>.</p>
<p><a id="X801F2E657C8A79ED" name="X801F2E657C8A79ED"></a></p>
<h5>3.5-4 AtlasProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasProgram</code>( <var class="Arg">gapname</var>[, <var class="Arg">std</var>][, <var class="Arg">"contents"</var>, <var class="Arg">sources</var>][, <var class="Arg">"version"</var>, <var class="Arg">vers</var>], <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasProgram</code>( <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record containing a program, or <code class="keyw">fail</code>.</p>
<p>In the first form, <var class="Arg">gapname</var> must be a string denoting a <strong class="pkg">GAP</strong> name (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>) of a group <span class="SimpleMath">G</span>, say. If the database contains a straight line program (see Section <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB"><span class="RefLink">Reference: Straight Line Programs</span></a>) or straight line decision (see Section <a href="chap6.html#X8121E9567A7137C9"><span class="RefLink">6.1</span></a>) or black box program (see Section <a href="chap6.html#X7BE856BC785A9E8F"><span class="RefLink">6.2</span></a>) as described by the arguments indicated by <var class="Arg">...</var> (see below) then <code class="func">AtlasProgram</code> returns an immutable record containing this program. Otherwise <code class="keyw">fail</code> is returned.</p>
<p>If the optional argument <var class="Arg">std</var> is given, only those straight line programs/decisions are considered that take generators from the <var class="Arg">std</var>-th set of standard generators of <span class="SimpleMath">G</span> as input, see Section <a href="chap3.html#X795DB7E486E0817D"><span class="RefLink">3.3</span></a>.</p>
<p>If the optional arguments <code class="code">"contents"</code> and <var class="Arg">sources</var> are given then the latter must be either a string or a list of strings, with the same meaning as described for <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>).</p>
<p>If the optional arguments <code class="code">"version"</code> and <var class="Arg">vers</var> are given then the latter must be either a number or a list of numbers, and only those straight line programs/decisions are considered whose version number fits to <var class="Arg">vers</var>.</p>
<p>The result record has at least the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">groupname</code></strong></dt>
<dd><p>the string <var class="Arg">gapname</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">identifier</code></strong></dt>
<dd><p>a <strong class="pkg">GAP</strong> object (a list of filenames plus additional information) that uniquely determines the program; the value can be used as <var class="Arg">identifier</var> argument of <code class="func">AtlasProgram</code> (see below),</p>
</dd>
<dt><strong class="Mark"><code class="code">program</code></strong></dt>
<dd><p>the required straight line program/decision, or black box program,</p>
</dd>
<dt><strong class="Mark"><code class="code">standardization</code></strong></dt>
<dd><p>the positive integer denoting the underlying standard generators of <span class="SimpleMath">G</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">version</code></strong></dt>
<dd><p>the substring of the filename of the program that denotes the version of the program.</p>
</dd>
</dl>
<p>If the program computes generators of the restriction to a maximal subgroup then also the following components are present.</p>
<dl>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>the order of the maximal subgroup,</p>
</dd>
<dt><strong class="Mark"><code class="code">subgroupname</code></strong></dt>
<dd><p>a string denoting a name of the maximal subgroup.</p>
</dd>
</dl>
<p>In the first form, the arguments indicated by <var class="Arg">...</var> must be as follows.</p>
<dl>
<dt><strong class="Mark">(the string <code class="code">"maxes"</code> and) a positive integer <span class="SimpleMath">maxnr</span>
</strong></dt>
<dd><p>the required program computes generators of the <span class="SimpleMath">maxnr</span>-th maximal subgroup of the group with <strong class="pkg">GAP</strong> name <span class="SimpleMath">gapname</span>.</p>
<p>In this case, the result record of <code class="func">AtlasProgram</code> also may contain a component <code class="code">size</code>, whose value is the order of the maximal subgroup in question.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"maxes"</code>
and two positive integers <span class="SimpleMath">maxnr</span> and <span class="SimpleMath">std2</span></strong></dt>
<dd><p>the required program computes standard generators of the <span class="SimpleMath">maxnr</span>-th maximal subgroup of the group with <strong class="pkg">GAP</strong> name <span class="SimpleMath">gapname</span>, w. r. t. the standardization <span class="SimpleMath">std2</span>.</p>
<p>A prescribed <code class="code">"version"</code> parameter refers to the straight line program for computing the restriction, not to the program for standardizing the result of the restriction.</p>
<p>The meaning of the component <code class="code">size</code> in the result, if present, is the same as in the previous case.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"maxstd"</code> and three positive integers
<span class="SimpleMath">maxnr</span>, <span class="SimpleMath">vers</span>, <span class="SimpleMath">substd</span></strong></dt>
<dd><p>the required program computes standard generators of the <span class="SimpleMath">maxnr</span>-th maximal subgroup of the group with <strong class="pkg">GAP</strong> name <span class="SimpleMath">gapname</span> w. r. t. standardization <span class="SimpleMath">substd</span>; in this case, the inputs of the program are <em>not</em> standard generators of the group with <strong class="pkg">GAP</strong> name <span class="SimpleMath">gapname</span> but the outputs of the straight line program with version <span class="SimpleMath">vers</span> for computing generators of its <span class="SimpleMath">maxnr</span>-th maximal subgroup.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"kernel"</code> and a string <span class="SimpleMath">factname</span></strong></dt>
<dd><p>the required program computes generators of the kernel of an epimorphism from <span class="SimpleMath">G</span> to a group with <strong class="pkg">GAP</strong> name <span class="SimpleMath">factname</span>.</p>
</dd>
<dt><strong class="Mark">one of the strings <code class="code">"classes"</code> or <code class="code">"cyclic"</code></strong></dt>
<dd><p>the required program computes representatives of conjugacy classes of elements or representatives of generators of maximally cyclic subgroups of <span class="SimpleMath">G</span>, respectively.</p>
<p>See <a href="chapBib.html#biBBSW01">[BSWW01]</a> and <a href="chapBib.html#biBSWW00">[SWW00]</a> for the background concerning these straight line programs. In these cases, the result record of <code class="func">AtlasProgram</code> also contains a component <code class="code">outputs</code>, whose value is a list of class names of the outputs, as described in Section <a href="chap3.html#X861CD545803B97E8"><span class="RefLink">3.4</span></a>.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"cyc2ccl"</code> (and the string <span class="SimpleMath">vers</span>)</strong></dt>
<dd><p>the required program computes representatives of conjugacy classes of elements from representatives of generators of maximally cyclic subgroups of <span class="SimpleMath">G</span>. Thus the inputs are the outputs of the program of type <code class="code">"cyclic"</code> whose version is <span class="SimpleMath">vers</span>.</p>
</dd>
<dt><strong class="Mark">the strings <code class="code">"cyc2ccl"</code>, <span class="SimpleMath">vers1</span>, <code class="code">"version"</code>, <span class="SimpleMath">vers2</span></strong></dt>
<dd><p>the required program computes representatives of conjugacy classes of elements from representatives of generators of maximally cyclic subgroups of <span class="SimpleMath">G</span>, where the inputs are the outputs of the program of type <code class="code">"cyclic"</code> whose version is <span class="SimpleMath">vers1</span> and the required program itself has version <span class="SimpleMath">vers2</span>.</p>
</dd>
<dt><strong class="Mark">the strings <code class="code">"automorphism"</code> and <span class="SimpleMath">autname</span></strong></dt>
<dd><p>the required program computes images of standard generators under the outer automorphism of <span class="SimpleMath">G</span> that is given by this string.</p>
<p>Note that a value <code class="code">"2"</code> of <span class="SimpleMath">autname</span> means that the square of the automorphism is an inner automorphism of <span class="SimpleMath">G</span> (not necessarily the identity mapping) but the automorphism itself is not.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"check"</code></strong></dt>
<dd><p>the required result is a straight line decision that takes a list of generators for <span class="SimpleMath">G</span> and returns <code class="keyw">true</code> if these generators are standard generators of <span class="SimpleMath">G</span> w. r. t. the standardization <var class="Arg">std</var>, and <code class="keyw">false</code> otherwise.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"presentation"</code></strong></dt>
<dd><p>the required result is a straight line decision that takes a list of group elements and returns <code class="keyw">true</code> if these elements are standard generators of <span class="SimpleMath">G</span> w. r. t. the standardization <var class="Arg">std</var>, and <code class="keyw">false</code> otherwise.</p>
<p>See <code class="func">StraightLineProgramFromStraightLineDecision</code> (<a href="chap6.html#X7EA613C57DDC67D5"><span class="RefLink">6.1-9</span></a>) for an example how to derive defining relators for <span class="SimpleMath">G</span> in terms of the standard generators from such a straight line decision.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"find"</code></strong></dt>
<dd><p>the required result is a black box program that takes <span class="SimpleMath">G</span> and returns a list of standard generators of <span class="SimpleMath">G</span>, w. r. t. the standardization <var class="Arg">std</var>.</p>
</dd>
<dt><strong class="Mark">the string <code class="code">"restandardize"</code> and an integer <span class="SimpleMath">std2</span></strong></dt>
<dd><p>the required result is a straight line program that computes standard generators of <span class="SimpleMath">G</span> w. r. t. the <span class="SimpleMath">std2</span>-th set of standard generators of <span class="SimpleMath">G</span>; in this case, the argument <var class="Arg">std</var> must be given.</p>
</dd>
<dt><strong class="Mark">the strings <code class="code">"other"</code> and <span class="SimpleMath">descr</span></strong></dt>
<dd><p>the required program is described by <span class="SimpleMath">descr</span>.</p>
</dd>
</dl>
<p>The second form of <code class="func">AtlasProgram</code>, with only argument the list <var class="Arg">identifier</var>, can be used to fetch the result record with <code class="code">identifier</code> value equal to <var class="Arg">identifier</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">prog:= AtlasProgram( "A5", 2 );</span>
rec( groupname := "A5", identifier := [ "A5", "A5G1-max2W1", 1 ],
program := <straight line program>, size := 10,
standardization := 1, subgroupname := "D10", version := "1" )
<span class="GAPprompt">gap></span> <span class="GAPinput">StringOfResultOfStraightLineProgram( prog.program, [ "a", "b" ] );</span>
"[ a, bbab ]"
<span class="GAPprompt">gap></span> <span class="GAPinput">gens1:= AtlasGenerators( "A5", 1 );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", generators := [ (1,2)(3,4), (1,3,5) ],
groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">maxgens:= ResultOfStraightLineProgram( prog.program,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> gens1.generators );</span>
[ (1,2)(3,4), (2,3)(4,5) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">maxgens = gens1max2.generators;</span>
true
</pre></div>
<p>The above example shows that for restricting representations given by standard generators to a maximal subgroup of <span class="SimpleMath">A_5</span>, we can also fetch and apply the appropriate straight line program. Such a program (see <a href="../../../doc/ref/chap37.html#X7DC99E4284093FBB"><span class="RefLink">Reference: Straight Line Programs</span></a>) takes standard generators of a group –in this example <span class="SimpleMath">A_5</span>– as its input, and returns a list of elements in this group –in this example generators of the <span class="SimpleMath">D_10</span> subgroup we had met above– which are computed essentially by evaluating structured words in terms of the standard generators.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">prog:= AtlasProgram( "J1", "cyclic" );</span>
rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],
outputs := [ "6A", "7A", "10B", "11A", "15B", "19A" ],
program := <straight line program>, standardization := 1,
version := "1" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( FreeGroup( "x", "y" ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ResultOfStraightLineProgram( prog.program, gens );</span>
[ (x*y)^2*((y*x)^2*y^2*x)^2*y^2, x*y, (x*(y*x*y)^2)^2*y,
(x*y*x*(y*x*y)^3*x*y^2)^2*x*y*x*(y*x*y)^2*y, x*y*x*(y*x*y)^2*y,
(x*y)^2*y ]
</pre></div>
<p>The above example shows how to fetch and use straight line programs for computing generators of representatives of maximally cyclic subgroups of a given group.</p>
<p><a id="X83DFD8967E6BC831" name="X83DFD8967E6BC831"></a></p>
<h5>3.5-5 AtlasProgramInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasProgramInfo</code>( <var class="Arg">gapname</var>[, <var class="Arg">std</var>][, <var class="Arg">"contents"</var>, <var class="Arg">sources</var>][, <var class="Arg">"version"</var>, <var class="Arg">vers</var>], <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record describing a program, or <code class="keyw">fail</code>.</p>
<p><code class="func">AtlasProgramInfo</code> takes the same arguments as <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>), and returns a similar result. The only difference is that the records returned by <code class="func">AtlasProgramInfo</code> have no components <code class="code">program</code> and <code class="code">outputs</code>. The idea is that one can use <code class="func">AtlasProgramInfo</code> for testing whether the program in question is available at all, but without downloading files. The <code class="code">identifier</code> component of the result of <code class="func">AtlasProgramInfo</code> can then be used to fetch the program with <code class="func">AtlasProgram</code> (<a href="chap3.html#X801F2E657C8A79ED"><span class="RefLink">3.5-4</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasProgramInfo( "J1", "cyclic" );</span>
rec( groupname := "J1", identifier := [ "J1", "J1G1-cycW1", 1 ],
standardization := 1, version := "1" )
</pre></div>
<p><a id="X841478AB7CD06D44" name="X841478AB7CD06D44"></a></p>
<h5>3.5-6 OneAtlasGeneratingSetInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneAtlasGeneratingSetInfo</code>( [<var class="Arg">gapname</var>][,] [<var class="Arg">std</var>][,] [<var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record describing a representation that satisfies the conditions, or <code class="keyw">fail</code>.</p>
<p>Let <var class="Arg">gapname</var> be a string denoting a <strong class="pkg">GAP</strong> name (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>) of a group <span class="SimpleMath">G</span>, say. If the database contains at least one representation for <span class="SimpleMath">G</span> with the required properties then <code class="func">OneAtlasGeneratingSetInfo</code> returns a record <span class="SimpleMath">r</span> whose components are the same as those of the records returned by <code class="func">AtlasGenerators</code> (<a href="chap3.html#X7D1CCCF8852DFF39"><span class="RefLink">3.5-3</span></a>), except that the component <code class="code">generators</code> is not contained, and an additional component <code class="code">givenRing</code> is present if <code class="code">Ring</code> is one of the arguments in the function call.</p>
<p>The information in <code class="code">givenRing</code> can be used later to construct the matrices over the prescribed ring. Note that this ring may be for example a domain constructed with <code class="func">AlgebraicExtension</code> (<a href="../../../doc/ref/chap67.html#X7CDA90537D2BAC8A"><span class="RefLink">Reference: AlgebraicExtension</span></a>) instead of a field of cyclotomics or of a finite field constructed with <code class="func">GF</code> (<a href="../../../doc/ref/chap59.html#X8592DBB086A8A9BE"><span class="RefLink">Reference: GF for field size</span></a>).</p>
<p>The component <code class="code">identifier</code> of <span class="SimpleMath">r</span> can be used as input for <code class="func">AtlasGenerators</code> (<a href="chap3.html#X7D1CCCF8852DFF39"><span class="RefLink">3.5-3</span></a>) in order to fetch the generators. If no representation satisfying the given conditions is available then <code class="keyw">fail</code> is returned.</p>
<p>If the argument <var class="Arg">std</var> is given then it must be a positive integer or a list of positive integers, denoting the sets of standard generators w. r. t. which the representation shall be given (see Section <a href="chap3.html#X795DB7E486E0817D"><span class="RefLink">3.3</span></a>).</p>
<p>The argument <var class="Arg">gapname</var> can be missing (then all available groups are considered), or a list of group names can be given instead.</p>
<p>Further restrictions can be entered as arguments, with the same meaning as described for <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>). The result of <code class="func">OneAtlasGeneratingSetInfo</code> describes the first generating set for <span class="SimpleMath">G</span> that matches the restrictions, in the ordering shown by <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>).</p>
<p>Note that even in the case that the user preference <code class="code">AtlasRepAccessRemoteFiles</code> has the value <code class="keyw">true</code> (see Section <a href="chap4.html#X7C3293A98577EE68"><span class="RefLink">4.2-1</span></a>), <code class="func">OneAtlasGeneratingSetInfo</code> does <em>not</em> attempt to <em>transfer</em> remote data files, just the table of contents is evaluated. So this function (as well as <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap3.html#X84C2D76482E60E42"><span class="RefLink">3.5-7</span></a>)) can be used to check for the availability of certain representations, and afterwards one can call <code class="func">AtlasGenerators</code> (<a href="chap3.html#X7D1CCCF8852DFF39"><span class="RefLink">3.5-3</span></a>) for those representations one wants to work with.</p>
<p>In the following example, we try to access information about permutation representations for the alternating group <span class="SimpleMath">A_5</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= OneAtlasGeneratingSetInfo( "A5" );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= AtlasGenerators( info.identifier );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", generators := [ (1,2)(3,4), (1,3,5) ],
groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">info = OneAtlasGeneratingSetInfo( "A5", IsPermGroup, true );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, "minimal" );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">info = OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, [ 1 .. 10 ] );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">OneAtlasGeneratingSetInfo( "A5", NrMovedPoints, 20 );</span>
fail
</pre></div>
<p>Note that a permutation representation of degree <span class="SimpleMath">20</span> could be obtained by taking twice the primitive representation on <span class="SimpleMath">10</span> points; however, the database does not store this imprimitive representation (cf. Section <a href="chap3.html#X87D26B13819A8209"><span class="RefLink">3.1</span></a>).</p>
<p>We continue this example. Next we access matrix representations of <span class="SimpleMath">A_5</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= OneAtlasGeneratingSetInfo( "A5", IsMatrixGroup, true );</span>
rec( charactername := "4a", constituents := [ 4 ], contents := "core",
dim := 4, groupname := "A5", id := "a",
identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1,
2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2),
size := 60, standardization := 1, type := "matff" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= AtlasGenerators( info.identifier );</span>
rec( charactername := "4a", constituents := [ 4 ], contents := "core",
dim := 4,
generators := [ <an immutable 4x4 matrix over GF2>,
<an immutable 4x4 matrix over GF2> ], groupname := "A5",
id := "a",
identifier := [ "A5", [ "A5G1-f2r4aB0.m1", "A5G1-f2r4aB0.m2" ], 1,
2 ], repname := "A5G1-f2r4aB0", repnr := 4, ring := GF(2),
size := 60, standardization := 1, type := "matff" )
<span class="GAPprompt">gap></span> <span class="GAPinput">info = OneAtlasGeneratingSetInfo( "A5", Dimension, 4 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">info = OneAtlasGeneratingSetInfo( "A5", Characteristic, 2 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">info2:= OneAtlasGeneratingSetInfo( "A5", Ring, GF(2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">info.identifier = info2.identifier; </span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 2 );</span>
rec( charactername := "2a", constituents := [ 2 ], contents := "core",
dim := 2, groupname := "A5", id := "a",
identifier := [ "A5", [ "A5G1-f4r2aB0.m1", "A5G1-f4r2aB0.m2" ], 1,
4 ], repname := "A5G1-f4r2aB0", repnr := 8, ring := GF(2^2),
size := 60, standardization := 1, type := "matff" )
<span class="GAPprompt">gap></span> <span class="GAPinput">OneAtlasGeneratingSetInfo( "A5", Characteristic, [2,5], Dimension, 1 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= OneAtlasGeneratingSetInfo( "A5", Characteristic, 0,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> Dimension, 4 );</span>
rec( charactername := "4a", constituents := [ 4 ], contents := "core",
dim := 4, groupname := "A5", id := "",
identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ],
repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60,
standardization := 1, type := "matint" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= AtlasGenerators( info.identifier );</span>
rec( charactername := "4a", constituents := [ 4 ], contents := "core",
dim := 4,
generators :=
[
[ [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ],
[ -1, -1, -1, -1 ] ],
[ [ 0, 1, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 0, 1, 0 ],
[ 1, 0, 0, 0 ] ] ], groupname := "A5", id := "",
identifier := [ "A5", "A5G1-Zr4B0.g", 1, 4 ],
repname := "A5G1-Zr4B0", repnr := 14, ring := Integers, size := 60,
standardization := 1, type := "matint" )
<span class="GAPprompt">gap></span> <span class="GAPinput">info = OneAtlasGeneratingSetInfo( "A5", Ring, Integers );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">info2:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(37) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">info = info2;</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">Difference( RecNames( info2 ), RecNames( info ) );</span>
[ "givenRing" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">info2.givenRing;</span>
CF(37)
<span class="GAPprompt">gap></span> <span class="GAPinput">OneAtlasGeneratingSetInfo( "A5", Ring, Integers mod 77 );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= OneAtlasGeneratingSetInfo( "A5", Ring, CF(5), Dimension, 3 );</span>
rec( charactername := "3a", constituents := [ 2 ], contents := "core",
dim := 3, givenRing := CF(5), groupname := "A5", id := "a",
identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
type := "matalg" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= AtlasGenerators( info );</span>
rec( charactername := "3a", constituents := [ 2 ], contents := "core",
dim := 3,
generators :=
[ [ [ -1, 0, 0 ], [ 0, -1, 0 ], [ -E(5)-E(5)^4, -E(5)-E(5)^4, 1 ]
], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ],
givenRing := CF(5), groupname := "A5", id := "a",
identifier := [ "A5", "A5G1-Ar3aB0.g", 1, 3 ],
polynomial := [ -1, 1, 1 ], repname := "A5G1-Ar3aB0", repnr := 17,
ring := NF(5,[ 1, 4 ]), size := 60, standardization := 1,
type := "matalg" )
<span class="GAPprompt">gap></span> <span class="GAPinput">gens2:= AtlasGenerators( info.identifier );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Difference( RecNames( gens ), RecNames( gens2 ) );</span>
[ "givenRing" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">OneAtlasGeneratingSetInfo( "A5", Ring, GF(17) );</span>
fail
</pre></div>
<p><a id="X84C2D76482E60E42" name="X84C2D76482E60E42"></a></p>
<h5>3.5-7 AllAtlasGeneratingSetInfos</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllAtlasGeneratingSetInfos</code>( [<var class="Arg">gapname</var>][,] [<var class="Arg">std</var>][,] [<var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the list of all records describing representations that satisfy the conditions.</p>
<p><code class="func">AllAtlasGeneratingSetInfos</code> is similar to <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>). The difference is that the list of <em>all</em> records describing the available representations with the given properties is returned instead of just one such component. In particular an empty list is returned if no such representation is available.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllAtlasGeneratingSetInfos( "A5", IsPermGroup, true );</span>
[ rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ]
, isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60,
stabilizer := "A4", standardization := 1, transitivity := 3,
type := "perm" ),
rec( charactername := "1a+5a", constituents := [ 1, 5 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p6B0.m1", "A5G1-p6B0.m2" ], 1, 6 ]
, isPrimitive := true, maxnr := 2, p := 6, rankAction := 2,
repname := "A5G1-p6B0", repnr := 2, size := 60,
stabilizer := "D10", standardization := 1, transitivity := 2,
type := "perm" ),
rec( charactername := "1a+4a+5a", constituents := [ 1, 4, 5 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p10B0.m1", "A5G1-p10B0.m2" ], 1,
10 ], isPrimitive := true, maxnr := 3, p := 10,
rankAction := 3, repname := "A5G1-p10B0", repnr := 3,
size := 60, stabilizer := "S3", standardization := 1,
transitivity := 1, type := "perm" ) ]
</pre></div>
<p>Note that a matrix representation in any characteristic can be obtained by reducing a permutation representation or an integral matrix representation; however, the database does not <em>store</em> such a representation (cf. Section <a href="chap3.html#X87D26B13819A8209"><span class="RefLink">3.1</span></a>).</p>
<p><a id="X80AABEE783363B70" name="X80AABEE783363B70"></a></p>
<h5>3.5-8 <span class="Heading">AtlasGroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasGroup</code>( [<var class="Arg">gapname</var>][,] [<var class="Arg">std</var>][,] [<var class="Arg">...</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasGroup</code>( <var class="Arg">identifier</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a group that satisfies the conditions, or <code class="keyw">fail</code>.</p>
<p><code class="func">AtlasGroup</code> takes the same arguments as <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>), and returns the group generated by the <code class="code">generators</code> component of the record that is returned by <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) with these arguments; if <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) returns <code class="keyw">fail</code> then also <code class="func">AtlasGroup</code> returns <code class="keyw">fail</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasGroup( "A5" );</span>
Group([ (1,2)(3,4), (1,3,5) ])
</pre></div>
<p>Alternatively, it is possible to enter exactly one argument, a record <var class="Arg">identifier</var> as returned by <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) or <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap3.html#X84C2D76482E60E42"><span class="RefLink">3.5-7</span></a>), or the <code class="code">identifier</code> component of such a record.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= OneAtlasGeneratingSetInfo( "A5" );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasGroup( info );</span>
Group([ (1,2)(3,4), (1,3,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasGroup( info.identifier );</span>
Group([ (1,2)(3,4), (1,3,5) ])
</pre></div>
<p>In the groups returned by <code class="func">AtlasGroup</code>, the value of the attribute <code class="func">AtlasRepInfoRecord</code> (<a href="chap3.html#X87BC7D9C7BA2F27A"><span class="RefLink">3.5-10</span></a>) is set. This information is used for example by <code class="func">AtlasSubgroup</code> (<a href="chap3.html#X7A3E460C82B3D9A3"><span class="RefLink">3.5-9</span></a>) when this function is called with second argument a group created by <code class="func">AtlasGroup</code>.</p>
<p><a id="X7A3E460C82B3D9A3" name="X7A3E460C82B3D9A3"></a></p>
<h5>3.5-9 <span class="Heading">AtlasSubgroup</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasSubgroup</code>( <var class="Arg">gapname</var>[, <var class="Arg">std</var>][, <var class="Arg">...</var>], <var class="Arg">maxnr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasSubgroup</code>( <var class="Arg">identifier</var>, <var class="Arg">maxnr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasSubgroup</code>( <var class="Arg">G</var>, <var class="Arg">maxnr</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a group that satisfies the conditions, or <code class="keyw">fail</code>.</p>
<p>The arguments of <code class="func">AtlasSubgroup</code>, except the last argument <var class="Arg">maxnr</var>, are the same as for <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>). If the database provides a straight line program for restricting representations of the group with name <var class="Arg">gapname</var> (given w. r. t. the <var class="Arg">std</var>-th standard generators) to the <var class="Arg">maxnr</var>-th maximal subgroup and if a representation with the required properties is available, in the sense that calling <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>) with the same arguments except <var class="Arg">maxnr</var> yields a group, then <code class="func">AtlasSubgroup</code> returns the restriction of this representation to the <var class="Arg">maxnr</var>-th maximal subgroup.</p>
<p>In all other cases, <code class="keyw">fail</code> is returned.</p>
<p>Note that the conditions refer to the group and not to the subgroup. It may happen that in the restriction of a permutation representation to a subgroup, fewer points are moved, or that the restriction of a matrix representation turns out to be defined over a smaller ring. Here is an example.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasSubgroup( "A5", NrMovedPoints, 5, 1 );</span>
Group([ (1,5)(2,3), (1,3,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">NrMovedPoints( g );</span>
4
</pre></div>
<p>Alternatively, it is possible to enter exactly two arguments, the first being a record <var class="Arg">identifier</var> as returned by <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) or <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap3.html#X84C2D76482E60E42"><span class="RefLink">3.5-7</span></a>), or the <code class="code">identifier</code> component of such a record, or a group <var class="Arg">G</var> constructed with <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">info:= OneAtlasGeneratingSetInfo( "A5" );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasSubgroup( info, 1 );</span>
Group([ (1,5)(2,3), (1,3,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasSubgroup( info.identifier, 1 );</span>
Group([ (1,5)(2,3), (1,3,5) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasSubgroup( AtlasGroup( "A5" ), 1 );</span>
Group([ (1,5)(2,3), (1,3,5) ])
</pre></div>
<p><a id="X87BC7D9C7BA2F27A" name="X87BC7D9C7BA2F27A"></a></p>
<h5>3.5-10 AtlasRepInfoRecord</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasRepInfoRecord</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AtlasRepInfoRecord</code>( <var class="Arg">name</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: the record stored in the group <var class="Arg">G</var> when this was constructed with <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>), or a record with information about the group with name <var class="Arg">name</var>.</p>
<p>For a group <var class="Arg">G</var> that has been constructed with <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>), the value of this attribute is the info record that describes <var class="Arg">G</var>, in the sense that this record was the first argument of the call to <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>), or it is the result of the call to <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) with the conditions that were listed in the call to <code class="func">AtlasGroup</code> (<a href="chap3.html#X80AABEE783363B70"><span class="RefLink">3.5-8</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasRepInfoRecord( AtlasGroup( "A5" ) );</span>
rec( charactername := "1a+4a", constituents := [ 1, 4 ],
contents := "core", groupname := "A5", id := "",
identifier := [ "A5", [ "A5G1-p5B0.m1", "A5G1-p5B0.m2" ], 1, 5 ],
isPrimitive := true, maxnr := 1, p := 5, rankAction := 2,
repname := "A5G1-p5B0", repnr := 1, size := 60, stabilizer := "A4",
standardization := 1, transitivity := 3, type := "perm" )
</pre></div>
<p>For a string <var class="Arg">name</var> that is a <strong class="pkg">GAP</strong> name of a group <span class="SimpleMath">G</span>, say, <code class="func">AtlasRepInfoRecord</code> returns a record that contains information about <span class="SimpleMath">G</span> which is used by <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>). The following components may be bound in the record.</p>
<dl>
<dt><strong class="Mark"><code class="code">name</code></strong></dt>
<dd><p>the string <var class="Arg">name</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">nrMaxes</code></strong></dt>
<dd><p>the number of conjugacy classes of maximal subgroups of <span class="SimpleMath">G</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">size</code></strong></dt>
<dd><p>the order of <span class="SimpleMath">G</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">sizesMaxes</code></strong></dt>
<dd><p>a list which contains at position <span class="SimpleMath">i</span>, if bound, the order of a subgroup in the <span class="SimpleMath">i</span>-th class of maximal subgroups of <span class="SimpleMath">G</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">slpMaxes</code></strong></dt>
<dd><p>a list of length two; the first entry is a list of positions <span class="SimpleMath">i</span> such that a straight line program for computing the restriction of representations of <span class="SimpleMath">G</span> to a subgroup in the <span class="SimpleMath">i</span>-th class of maximal subgroups is available via <strong class="pkg">AtlasRep</strong>; the second entry is the corresponding list of standardizations of the generators of <span class="SimpleMath">G</span> for which these straight line programs are available,</p>
</dd>
<dt><strong class="Mark"><code class="code">structureMaxes</code></strong></dt>
<dd><p>a list which contains at position <span class="SimpleMath">i</span>, if bound, a string that describes the structure of the subgroups in the <span class="SimpleMath">i</span>-th class of maximal subgroups of <span class="SimpleMath">G</span>.</p>
</dd>
</dl>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasRepInfoRecord( "A5" );</span>
rec( name := "A5", nrMaxes := 3, size := 60,
sizesMaxes := [ 12, 10, 6 ],
slpMaxes := [ [ 1 .. 3 ], [ [ 1 ], [ 1 ], [ 1 ] ] ],
structureMaxes := [ "A4", "D10", "S3" ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">AtlasRepInfoRecord( "J5" );</span>
rec( )
</pre></div>
<p><a id="X87B012B080D01413" name="X87B012B080D01413"></a></p>
<h5>3.5-11 <span class="Heading">EvaluatePresentation</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EvaluatePresentation</code>( <var class="Arg">G</var>, <var class="Arg">gapname</var>[, <var class="Arg">std</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ EvaluatePresentation</code>( <var class="Arg">gens</var>, <var class="Arg">gapname</var>[, <var class="Arg">std</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a list of group elements or <code class="keyw">fail</code>.</p>
<p>The first argument must be either a group <var class="Arg">G</var> or a list <var class="Arg">gens</var> of group generators, and <var class="Arg">gapname</var> must be a string that is a <strong class="pkg">GAP</strong> name (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>) of a group <span class="SimpleMath">H</span>, say. The optional argument <var class="Arg">std</var>, if given, must be a positive integer that denotes a standardization of generators of <span class="SimpleMath">H</span>, the default is <span class="SimpleMath">1</span>.</p>
<p><code class="func">EvaluatePresentation</code> returns <code class="keyw">fail</code> if no presentation for <span class="SimpleMath">H</span> w. r. t. the standardization <var class="Arg">std</var> is stored in the database, and otherwise returns the list of results of evaluating the relators of a presentation for <span class="SimpleMath">H</span> at <var class="Arg">gens</var> or the <code class="func">GeneratorsOfGroup</code> (<a href="../../../doc/ref/chap39.html#X79C44528864044C5"><span class="RefLink">Reference: GeneratorsOfGroup</span></a>) value of <var class="Arg">G</var>, respectively. (An error is signalled if the number of generators is not equal to the number of inputs of the presentation.)</p>
<p>The result can be used as follows. Let <span class="SimpleMath">N</span> be the normal closure of the the result in <var class="Arg">G</var>. The factor group <var class="Arg">G</var><span class="SimpleMath">/N</span> is an epimorphic image of <span class="SimpleMath">H</span>. In particular, if all entries of the result have order <span class="SimpleMath">1</span> then <var class="Arg">G</var> itself is an epimorphic image of <span class="SimpleMath">H</span>. Moreover, an epimorphism is given by mapping the <var class="Arg">std</var>-th standard generators of <span class="SimpleMath">H</span> to the <span class="SimpleMath">N</span>-cosets of the given generators of <var class="Arg">G</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= MathieuGroup( 12 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( g );; # switch to 2 generators</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= Group( gens[1] * gens[3], gens[2] * gens[3] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">EvaluatePresentation( g, "J0" ); # no pres. for group "J0"</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">relimgs:= EvaluatePresentation( g, "M11" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( relimgs, Order ); # wrong group</span>
[ 3, 1, 5, 4, 10 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">relimgs:= EvaluatePresentation( g, "M12" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( relimgs, Order ); # generators are not standard</span>
[ 3, 4, 5, 4, 4 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasGroup( "M12" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">relimgs:= EvaluatePresentation( g, "M12", 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( relimgs, Order ); # right group, std. generators</span>
[ 1, 1, 1, 1, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasGroup( "2.M12" );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">relimgs:= EvaluatePresentation( g, "M12", 1 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List( relimgs, Order ); # std. generators for extension</span>
[ 1, 2, 1, 1, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( NormalClosure( g, SubgroupNC( g, relimgs ) ) );</span>
2
</pre></div>
<p><a id="X79F63403821C1E24" name="X79F63403821C1E24"></a></p>
<h5>3.5-12 <span class="Heading">StandardGeneratorsData</span></h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorsData</code>( <var class="Arg">G</var>, <var class="Arg">gapname</var>[, <var class="Arg">std</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ StandardGeneratorsData</code>( <var class="Arg">gens</var>, <var class="Arg">gapname</var>[, <var class="Arg">std</var>] )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: a record that describes standard generators of the group in question, or <code class="keyw">fail</code>, or the string <code class="code">"timeout"</code>.</p>
<p>The first argument must be either a group <var class="Arg">G</var> or a list <var class="Arg">gens</var> of group generators, and <var class="Arg">gapname</var> must be a string that is a <strong class="pkg">GAP</strong> name (see Section <a href="chap3.html#X81BF52FC7B8C08D4"><span class="RefLink">3.2</span></a>) of a group <span class="SimpleMath">H</span>, say. The optional argument <var class="Arg">std</var>, if given, must be a positive integer that denotes a standardization of generators of <span class="SimpleMath">H</span>, the default is <span class="SimpleMath">1</span>.</p>
<p>If the global option <code class="code">projective</code> is given then the group elements must be matrices over a finite field, and the group must be a central extension of the group <span class="SimpleMath">H</span> by a normal subgroup that consists of scalar matrices. In this case, all computations will be carried out modulo scalar matrices (in particular, element orders will be computed using <code class="func">ProjectiveOrder</code> (<a href="../../../doc/ref/chap24.html#X84A76F7A7B4166BC"><span class="RefLink">Reference: ProjectiveOrder</span></a>)), and the returned standard generators will belong to <span class="SimpleMath">H</span>.</p>
<p><code class="func">StandardGeneratorsData</code> returns</p>
<dl>
<dt><strong class="Mark"><code class="keyw">fail</code></strong></dt>
<dd><p>if no black box program for computing standard generators of <span class="SimpleMath">H</span> w. r. t. the standardization <var class="Arg">std</var> is stored in the database, or if the black box program returns <code class="keyw">fail</code> because a runtime error occurred or the program has proved that the given group or generators cannot generate a group isomorphic to <span class="SimpleMath">H</span>,</p>
</dd>
<dt><strong class="Mark"><code class="code">"timeout"</code></strong></dt>
<dd><p>if the black box program returns <code class="code">"timeout"</code>, typically because some elements of a given order were not found among a reasonable number of random elements, or</p>
</dd>
<dt><strong class="Mark">a record containing standard generators</strong></dt>
<dd><p>otherwise.</p>
</dd>
</dl>
<p>When the result is not a record then either the group is not isomorphic to <span class="SimpleMath">H</span> (modulo scalars if applicable), or we were unlucky with choosing random elements.</p>
<p>When a record is returned <em>and</em> <var class="Arg">G</var> or the group generated by <var class="Arg">gens</var>, respectively, is isomorphic to <span class="SimpleMath">H</span> (or to a central extension of <span class="SimpleMath">H</span> by a group of scalar matrices if the global option <code class="code">projective</code> is given) then the result describes the desired standard generators.</p>
<p>If <var class="Arg">G</var> or the group generated by <var class="Arg">gens</var>, respectively, is <em>not</em> isomorphic to <span class="SimpleMath">H</span> then it may still happen that <code class="func">StandardGeneratorsData</code> returns a record. For a proof that the returned record describes the desired standard generators, one can use a presentation of <span class="SimpleMath">H</span> whose generators correspond to the <var class="Arg">std</var>-th standard generators, see <code class="func">EvaluatePresentation</code> (<a href="chap3.html#X87B012B080D01413"><span class="RefLink">3.5-11</span></a>).</p>
<p>A returned record has the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">gapname</code></strong></dt>
<dd><p>the string <var class="Arg">gapname</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">givengens</code></strong></dt>
<dd><p>the list of group generators from which standard generators were computed, either <var class="Arg">gens</var> or the <code class="func">GeneratorsOfGroup</code> (<a href="../../../doc/ref/chap39.html#X79C44528864044C5"><span class="RefLink">Reference: GeneratorsOfGroup</span></a>) value of <var class="Arg">G</var>,</p>
</dd>
<dt><strong class="Mark"><code class="code">stdgens</code></strong></dt>
<dd><p>a list of standard generators of the group,</p>
</dd>
<dt><strong class="Mark"><code class="code">givengenstostdgens</code></strong></dt>
<dd><p>a straight line program that takes <code class="code">givengens</code> as inputs, and returns <code class="code">stdgens</code>,</p>
</dd>
<dt><strong class="Mark"><code class="code">std</code></strong></dt>
<dd><p>the underlying standardization <var class="Arg">std</var>.</p>
</dd>
</dl>
<p>The first examples show three cases of failure, due to the unavailability of a suitable black box program or to a wrong choice of <var class="Arg">gapname</var>. (In the search for standard generators of <span class="SimpleMath">M_11</span> in the group <span class="SimpleMath">M_12</span>, one may or may not find an element whose order does not appear in <span class="SimpleMath">M_11</span>; in the first case, the result is <code class="keyw">fail</code>, whereas a record is returned in the second case. Both cases occur.)</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">StandardGeneratorsData( MathieuGroup( 11 ), "J0" );</span>
fail
<span class="GAPprompt">gap></span> <span class="GAPinput">StandardGeneratorsData( MathieuGroup( 11 ), "M12" );</span>
"timeout"
<span class="GAPprompt">gap></span> <span class="GAPinput">repeat</span>
<span class="GAPprompt">></span> <span class="GAPinput"> res:= StandardGeneratorsData( MathieuGroup( 12 ), "M11" );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> until res = fail;</span>
</pre></div>
<p>The next example shows a computation of standard generators for the Mathieu group <span class="SimpleMath">M_12</span>. Using a presentation of <span class="SimpleMath">M_12</span> w. r. t. these standard generators, we prove that the given group is isomorphic to <span class="SimpleMath">M_12</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfGroup( MathieuGroup( 12 ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">std:= 1;;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">res:= StandardGeneratorsData( gens, "M12", std );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Set( RecNames( res ) );</span>
[ "gapname", "givengens", "givengenstostdgens", "std", "stdgens" ]
<span class="GAPprompt">gap></span> <span class="GAPinput">gens = res.givengens;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ResultOfStraightLineProgram( res.givengenstostdgens, gens )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> = res.stdgens;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">evl:= EvaluatePresentation( res.stdgens, "M12", std );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( evl, IsOne );</span>
true
</pre></div>
<p>The next example shows the use of the global option <code class="code">projective</code>. We take an irreducible matrix representation of the double cover of the Mathieu group <span class="SimpleMath">M_12</span> (thus the center is represented by scalar matrices) and compute standard generators of the factor group <span class="SimpleMath">M_12</span>. Using a presentation of <span class="SimpleMath">M_12</span> w. r. t. these standard generators, we prove that the given group is modulo scalars isomorphic to <span class="SimpleMath">M_12</span>, and we get generators for the kernel.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">g:= AtlasGroup( "2.M12", IsMatrixGroup, Characteristic, IsPosInt );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens:= Permuted( GeneratorsOfGroup( g ), (1,2) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">res:= StandardGeneratorsData( gens, "M12", std : projective );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">gens = res.givengens;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">ResultOfStraightLineProgram( res.givengenstostdgens, gens )</span>
<span class="GAPprompt">></span> <span class="GAPinput"> = res.stdgens;</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">evl:= EvaluatePresentation( res.stdgens, "M12", std );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( evl, IsOne );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">ForAll( evl, x -> IsCentral( g, x ) );</span>
true
</pre></div>
<p><a id="X790D5F8C7E8E6947" name="X790D5F8C7E8E6947"></a></p>
<h4>3.6 <span class="Heading"><strong class="pkg">Browse</strong> Applications Provided by <strong class="pkg">AtlasRep</strong></span></h4>
<p>The functions <code class="func">BrowseMinimalDegrees</code> (<a href="chap3.html#X7F31A7CB841FE63F"><span class="RefLink">3.6-1</span></a>), <code class="func">BrowseBibliographySporadicSimple</code> (<a href="chap3.html#X84ED4FC182C28198"><span class="RefLink">3.6-2</span></a>), and <code class="func">BrowseAtlasInfo</code> (<a href="../../../pkg/browse/doc/chap6.html#X8411AF928194C5AB"><span class="RefLink">Browse: BrowseAtlasInfo</span></a>) (an alternative to <code class="func">DisplayAtlasInfo</code> (<a href="chap3.html#X79DACFFA7E2D1A99"><span class="RefLink">3.5-1</span></a>)) are available only if the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong> (see <a href="chapBib.html#biBBrowse">[BL18]</a>) is loaded.</p>
<p><a id="X7F31A7CB841FE63F" name="X7F31A7CB841FE63F"></a></p>
<h5>3.6-1 BrowseMinimalDegrees</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseMinimalDegrees</code>( [<var class="Arg">gapnames</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: the list of info records for the clicked representations.</p>
<p>If the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong> (see <a href="chapBib.html#biBBrowse">[BL18]</a>) is loaded then this function is available. It opens a browse table whose rows correspond to the groups for which <strong class="pkg">AtlasRep</strong> knows some information about minimal degrees, whose columns correspond to the characteristics that occur, and whose entries are the known minimal degrees.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">if IsBound( BrowseMinimalDegrees ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> down:= NCurses.keys.DOWN;; DOWN:= NCurses.keys.NPAGE;;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> right:= NCurses.keys.RIGHT;; END:= NCurses.keys.END;;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> enter:= NCurses.keys.ENTER;; nop:= [ 14, 14, 14 ];;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # just scroll in the table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> right, right, right ], "sedddrrrddd", nop, nop, "Q" ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseMinimalDegrees();;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # restrict the table to the groups with minimal ordinary degree 6</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseData.SetReplay( Concatenation( "scf6",</span>
<span class="GAPprompt">></span> <span class="GAPinput"> [ down, down, right, enter, enter ] , nop, nop, "Q" ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseMinimalDegrees();;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseData.SetReplay( false );</span>
<span class="GAPprompt">></span> <span class="GAPinput">fi;</span>
</pre></div>
<p>If an argument <var class="Arg">gapnames</var> is given then it must be a list of <strong class="pkg">GAP</strong> names of groups. The browse table is then restricted to the rows corresponding to these group names and to the columns that are relevant for these groups. A perhaps interesting example is the subtable with the data concerning sporadic simple groups and their covering groups, which has been published in <a href="chapBib.html#biBJan05">[Jan05]</a>. This table can be shown as follows.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">if IsBound( BrowseMinimalDegrees ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # just scroll in the table</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN, END ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "rrrrrrrrrrrrrr", nop, nop, "Q" ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseMinimalDegrees( BibliographySporadicSimple.groupNamesJan05 );;</span>
<span class="GAPprompt">></span> <span class="GAPinput">fi;</span>
</pre></div>
<p>The browse table does <em>not</em> contain rows for the groups <span class="SimpleMath">6.M_22</span>, <span class="SimpleMath">12.M_22</span>, <span class="SimpleMath">6.Fi_22</span>. Note that in spite of the title of <a href="chapBib.html#biBJan05">[Jan05]</a>, the entries in Table 1 of this paper are in fact the minimal degrees of faithful <em>irreducible</em> representations, and in the above three cases, these degrees are larger than the minimal degrees of faithful representations. The underlying data of the browse table is about the minimal faithful (but not necessarily irreducible) degrees.</p>
<p>The return value of <code class="func">BrowseMinimalDegrees</code> is the list of <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap3.html#X841478AB7CD06D44"><span class="RefLink">3.5-6</span></a>) values for those representations that have been <q>clicked</q> in visual mode.</p>
<p>The variant without arguments of this function is also available in the menu shown by <code class="func">BrowseGapData</code> (<a href="../../../pkg/browse/doc/chap6.html#X850C786C87A4877B"><span class="RefLink">Browse: BrowseGapData</span></a>).</p>
<p><a id="X84ED4FC182C28198" name="X84ED4FC182C28198"></a></p>
<h5>3.6-2 BrowseBibliographySporadicSimple</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BrowseBibliographySporadicSimple</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns: a record as returned by <code class="func">ParseBibXMLExtString</code> (<a href="../../../pkg/gapdoc/doc/chap7.html#X86BD29AE7A453721"><span class="RefLink">GAPDoc: ParseBibXMLextString</span></a>).</p>
<p>If the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong> (see <a href="chapBib.html#biBBrowse">[BL18]</a>) is loaded then this function is available. It opens a browse table whose rows correspond to the entries of the bibliographies in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> and in the <strong class="pkg">ATLAS</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a>.</p>
<p>The function is based on <code class="func">BrowseBibliography</code> (<a href="../../../pkg/browse/doc/chap6.html#X7F0FE4CC7F46ABF3"><span class="RefLink">Browse: BrowseBibliography</span></a>), see the documentation of this function for details, e.g., about the return value.</p>
<p>The returned record encodes the bibliography entries corresponding to those rows of the table that are <q>clicked</q> in visual mode, in the same format as the return value of <code class="func">ParseBibXMLExtString</code> (<a href="../../../pkg/gapdoc/doc/chap7.html#X86BD29AE7A453721"><span class="RefLink">GAPDoc: ParseBibXMLextString</span></a>), see the manual of the <strong class="pkg">GAP</strong> package <strong class="pkg">GAPDoc</strong> <a href="chapBib.html#biBGAPDoc">[LN18]</a> for details.</p>
<p><code class="func">BrowseBibliographySporadicSimple</code> can be called also via the menu shown by <code class="func">BrowseGapData</code> (<a href="../../../pkg/browse/doc/chap6.html#X850C786C87A4877B"><span class="RefLink">Browse: BrowseGapData</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">if IsBound( BrowseBibliographySporadicSimple ) then</span>
<span class="GAPprompt">></span> <span class="GAPinput"> enter:= NCurses.keys.ENTER;; nop:= [ 14, 14, 14 ];;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseData.SetReplay( Concatenation(</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # choose the application</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "/Bibliography of Sporadic Simple Groups", [ enter, enter ],</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # search in the title column for the Atlas of Finite Groups</span>
<span class="GAPprompt">></span> <span class="GAPinput"> "scr/Atlas of finite groups", [ enter,</span>
<span class="GAPprompt">></span> <span class="GAPinput"> # and quit</span>
<span class="GAPprompt">></span> <span class="GAPinput"> nop, nop, nop, nop ], "Q" ) );</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseGapData();;</span>
<span class="GAPprompt">></span> <span class="GAPinput"> BrowseData.SetReplay( false );</span>
<span class="GAPprompt">></span> <span class="GAPinput">fi;</span>
</pre></div>
<p>The bibliographies contained in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCN+85]</a> and in the <strong class="pkg">ATLAS</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> are available online in HTML format, see <span class="URL"><a href="http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/bibl/index.html">http://www.math.rwth-aachen.de/~Thomas.Breuer/atlasrep/bibl/index.html</a></span>.</p>
<p>The source data in BibXMLext format, which are used by <code class="func">BrowseBibliographySporadicSimple</code>, are distributed with the <strong class="pkg">AtlasRep</strong> package, in four files with suffix <code class="file">xml</code> in the package's <code class="file">bibl</code> directory. Note that each of the two books contains two bibliographies.</p>
<p>Details about the BibXMLext format, including information how to transform the data into other formats such as BibTeX, can be found in the <strong class="pkg">GAP</strong> package <strong class="pkg">GAPDoc</strong> (see <a href="chapBib.html#biBGAPDoc">[LN18]</a>).</p>
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