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<div class="ChapSects"><a href="chap1.html#X7C16EA1580AC7586">1 <span class="Heading">The Small Groups Library</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X8389AD927B74BA4A">1.1 <span class="Heading">Overview</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1.html#X7ECCCA82839EA283">1.2 <span class="Heading">Function Reference</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8398F2577B719D99">1.2-1 SmallGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X781EA70A7902B22C">1.2-2 SmallGroupsAvailable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7BB133CB7AA8F465">1.2-3 AllSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X875EB1167FF6BA82">1.2-4 OneSmallGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C587F2A82BEAD19">1.2-5 NumberSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X872991747D5CFD35">1.2-6 NumberSmallGroupsAvailable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7B5A1FD47C722EB2">1.2-7 SelectSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X83044B9D7E3BDF35">1.2-8 IdSmallGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7C0C616180DE5875">1.2-9 IdGroupsAvailable</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X85352440869327EC">1.2-10 IdsOfAllSmallGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X8162304487D0C3E2">1.2-11 IdGap3SolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X833DB8AB80B76D26">1.2-12 SmallGroupsInformation</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X850CC04E7855FF68">1.2-13 UnloadSmallGroupsData</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap1.html#X7CE8AEAF8133285D">1.2-14 SMALL_GROUPS_OLD_ORDER</a></span>
</div></div>
</div>
<h3>1 <span class="Heading">The Small Groups Library</span></h3>
<p><a id="X8389AD927B74BA4A" name="X8389AD927B74BA4A"></a></p>
<h4>1.1 <span class="Heading">Overview</span></h4>
<p>The Small Groups library gives access to all groups of certain <q>small</q> orders. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given. Currently, the library contains the following groups:</p>
<ul>
<li><p>those of order at most 2000 except 1024 (<span class="SimpleMath">423 164 062</span> groups);</p>
</li>
<li><p>those of cubefree order at most 50 000 (<span class="SimpleMath">395 703</span> groups);</p>
</li>
<li><p>those of order <span class="SimpleMath">p^7</span> for the primes <span class="SimpleMath">p = 3,5,7,11</span> (<span class="SimpleMath">907 489</span> groups);</p>
</li>
<li><p>those of order <span class="SimpleMath">p^n</span> for <span class="SimpleMath">n ≤ 6</span> and all primes <span class="SimpleMath">p</span></p>
</li>
<li><p>those of order <span class="SimpleMath">q^n ⋅ p</span> for <span class="SimpleMath">q^n</span> dividing <span class="SimpleMath">2^8</span>, <span class="SimpleMath">3^6</span>, <span class="SimpleMath">5^5</span> or <span class="SimpleMath">7^4</span> and all primes <span class="SimpleMath">p</span> with <span class="SimpleMath">p ≠ q</span>;</p>
</li>
<li><p>those of squarefree order;</p>
</li>
<li><p>those whose order factorises into at most 3 primes.</p>
</li>
</ul>
<p>The first three items in this list cover an explicit range of orders; the last four provide access to infinite families of groups having orders of certain types.</p>
<p>The library also has an identification function: it returns the library number of a given group. This function determines library numbers using invariants of groups. The function is available for all orders in the library except for the orders 512 and 1536 and except for the orders <span class="SimpleMath">p^5</span>, <span class="SimpleMath">p^6</span> and <span class="SimpleMath">p^7</span> above 2000.</p>
<p>The library is organised in 11 layers. Each layer contains the groups of certain orders and their corresponding group identification routines. It is possible to install the first <span class="SimpleMath">n</span> layers of the group library and the first <span class="SimpleMath">m</span> layers of the group identification for each <span class="SimpleMath">1 ≤ m ≤ n ≤ 11</span>. This might be useful to save disk space. There is an extensive <code class="file">README</code> file for the Small Groups library available in the <code class="code">small</code> directory of the <strong class="pkg">GAP</strong> distribution containing detailed information on the layers. A brief description of the layers is given here:</p>
<dl>
<dt><strong class="Mark">(1)</strong></dt>
<dd><p>the groups whose order factorises into at most 3 primes.</p>
</dd>
<dt><strong class="Mark">(2)</strong></dt>
<dd><p>the remaining groups of order at most 1000 except 512 and 768.</p>
</dd>
<dt><strong class="Mark">(3)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">2^n ⋅ p</span> with <span class="SimpleMath">n ≤ 8</span> and <span class="SimpleMath">p</span> an odd prime.</p>
</dd>
<dt><strong class="Mark">(4)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">5^5</span>, <span class="SimpleMath">7^4</span> and of order <span class="SimpleMath">q^n ⋅ p</span> for <span class="SimpleMath">q^n</span> dividing <span class="SimpleMath">3^6</span>, <span class="SimpleMath">5^5</span> or <span class="SimpleMath">7^4</span> and <span class="SimpleMath">p ≠ q</span> a prime.</p>
</dd>
<dt><strong class="Mark">(5)</strong></dt>
<dd><p>the remaining groups of order at most 2000 except 1024, 1152, 1536 and 1920.</p>
</dd>
<dt><strong class="Mark">(6)</strong></dt>
<dd><p>the groups of orders 1152 and 1920.</p>
</dd>
<dt><strong class="Mark">(7)</strong></dt>
<dd><p>the groups of order 512.</p>
</dd>
<dt><strong class="Mark">(8)</strong></dt>
<dd><p>the groups of order 1536.</p>
</dd>
<dt><strong class="Mark">(9)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">p^n</span> for <span class="SimpleMath">4 ≤ n ≤ 6</span>.</p>
</dd>
<dt><strong class="Mark">(10)</strong></dt>
<dd><p>the remaining groups of cubefree order at most 50 000 and of squarefree order.</p>
</dd>
<dt><strong class="Mark">(11)</strong></dt>
<dd><p>the remaining groups of order <span class="SimpleMath">p^7</span> for <span class="SimpleMath">p = 3,5,7,11</span>.</p>
</dd>
</dl>
<p>The data in this library has been carefully checked and cross-checked. It is believed to be reliable. However, no absolute guarantees are given and users should, as always, make their own checks in critical cases.</p>
<p>The data occupies about 30 MB (storing over 400 million groups in about 200 megabits). The group identification occupies about 47 MB of which 18 MB is used for the groups in layer (6). More information on the Small Groups library can be found on <span class="URL"><a href="http://www.icm.tu-bs.de/ag_algebra/software/small/">http://www.icm.tu-bs.de/ag_algebra/software/small/</a></span></p>
<p>This library has been constructed by Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. A survey on this topic and an account of the history of group constructions can be found in <a href="chapBib.html#biBBEO01">[BEO02]</a>. Further detailed information on the construction of this library is available in <a href="chapBib.html#biBNew77">[New77]</a>, <a href="chapBib.html#biBOBr90">[O'B90]</a>, <a href="chapBib.html#biBOBr91">[O'B91]</a>, <a href="chapBib.html#biBBescheEick98">[BE99a]</a>, <a href="chapBib.html#biBBescheEick1000">[BE99b]</a>, <a href="chapBib.html#biBBescheEick768">[BE01]</a>, <a href="chapBib.html#biBBEO00">[BEO01]</a>, <a href="chapBib.html#biBEOB99">[EO99a]</a>, <a href="chapBib.html#biBEOB98">[EO99b]</a>, <a href="chapBib.html#biBNOV04">[NOV04]</a>, <a href="chapBib.html#biBGir03">[Gir03]</a>, <a href="chapBib.html#biBDEi05">[DE05]</a>, <a href="chapBib.html#biBOV05">[OV05]</a>. The Small Groups library incorporates the <strong class="pkg">GAP</strong> 3 libraries <code class="code">TwoGroup</code> and <code class="code">ThreeGroup</code>. The data from these libraries was directly included into the Small Groups library, and the ordering there was preserved. The Small Groups library replaces the Gap 3 library of solvable groups of order at most 100. However, both the organisation and data descriptions of these groups has changed in the Small Groups library.</p>
<p>As of version 1.4 of this library, the arrangement of groups is the same as in Magma, Version 2.23. In earlier releases of this library, the arrangement in orders <span class="SimpleMath">p^7</span>, <span class="SimpleMath">p=3,5,7,11</span> disagreed. An attempt to fix this was instated on version 1.1 of this library, but a wrong permutation was used. If you would like to refer to index numbers for these orders in older versions of the library, see <code class="func">SMALL_GROUPS_OLD_ORDER</code> (<a href="chap1.html#X7CE8AEAF8133285D"><span class="RefLink">1.2-14</span></a>)). The arrangement of all other orders has always agreed and has remained stable.</p>
<p>In version 1.5, the number of groups of order 1024 was corrected. For more information, refer to <a href="chapBib.html#biBBurrell2021">[Bur21]</a>.</p>
<p><a id="X7ECCCA82839EA283" name="X7ECCCA82839EA283"></a></p>
<h4>1.2 <span class="Heading">Function Reference</span></h4>
<p><a id="X8398F2577B719D99" name="X8398F2577B719D99"></a></p>
<h5>1.2-1 SmallGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGroup</code>( <var class="Arg">order</var>, <var class="Arg">i</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">‣ SmallGroup</code>( <var class="Arg">pair</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the <var class="Arg">i</var>-th group of order <var class="Arg">order</var> in the catalogue. If the group is solvable, it will be given as a PcGroup; otherwise it will be given as a permutation group. If the groups of order <var class="Arg">order</var> are not installed, the function reports an error and enters a break loop.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( 60, 4 );</span>
<pc group of size 60 with 4 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( G );</span>
"C60"
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( 60, 5 );</span>
Group([ (1,2,3,4,5), (1,2,3) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( G );</span>
"A5"
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( 768, 1000000 );</span>
<pc group of size 768 with 9 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := SmallGroup( [768, 1000000] );</span>
<pc group of size 768 with 9 generators>
</pre></div>
<p><a id="X781EA70A7902B22C" name="X781EA70A7902B22C"></a></p>
<h5>1.2-2 SmallGroupsAvailable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGroupsAvailable</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code">true</code> if the library of groups of order <var class="Arg">order</var> is installed, and <code class="code">false</code> otherwise.</p>
<p><a id="X7BB133CB7AA8F465" name="X7BB133CB7AA8F465"></a></p>
<h5>1.2-3 AllSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllSmallGroups</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns all groups with certain properties as specified by <var class="Arg">arg</var>. If <var class="Arg">arg</var> is a number <span class="SimpleMath">n</span>, then this function returns all groups of order <span class="SimpleMath">n</span>. However, the function can also take several arguments which then must be organized in pairs <code class="code">function</code> and <code class="code">value</code>. In this case the first function must be <code class="func">Size</code> (<a href="../../../doc/ref/chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">Reference: Size</span></a>) and the first value an order or a range of orders. If value is a list then it is considered a list of possible function values to include. The function returns those groups of the specified orders having those properties specified by the remaining functions and their values.</p>
<p>Precomputed information is stored for the properties <code class="func">IsAbelian</code> (<a href="../../../doc/ref/chap35_mj.html#X830A4A4C795FBC2D"><span class="RefLink">Reference: IsAbelian</span></a>), <code class="func">IsNilpotentGroup</code> (<a href="../../../doc/ref/chap39_mj.html#X87D062608719F2CD"><span class="RefLink">Reference: IsNilpotentGroup</span></a>), <code class="func">IsSupersolvableGroup</code> (<a href="../../../doc/ref/chap39_mj.html#X7AADF2E88501B9FF"><span class="RefLink">Reference: IsSupersolvableGroup</span></a>), <code class="func">IsSolvableGroup</code> (<a href="../../../doc/ref/chap39_mj.html#X809C78D5877D31DF"><span class="RefLink">Reference: IsSolvableGroup</span></a>), <code class="func">RankPGroup</code> (<a href="../../../doc/ref/chap39_mj.html#X840A4F937ABF15E1"><span class="RefLink">Reference: RankPGroup</span></a>), <code class="func">PClassPGroup</code> (<a href="../../../doc/ref/chap39_mj.html#X863434AD7DDE514B"><span class="RefLink">Reference: PClassPGroup</span></a>), <code class="func">LGLength</code> (<a href="../../../doc/ref/chap45_mj.html#X7C3912F77B12C8B6"><span class="RefLink">Reference: LGLength</span></a>), <code class="code">FrattinifactorSize</code> and <code class="code">FrattinifactorId</code> for the groups of order at most <span class="SimpleMath">2000</span> which have more than three prime factors, except those of order <span class="SimpleMath">512</span>, <span class="SimpleMath">768</span>, <span class="SimpleMath">1024</span>, <span class="SimpleMath">1152</span>, <span class="SimpleMath">1536</span>, <span class="SimpleMath">1920</span> and those of order <span class="SimpleMath">p^n ⋅ q > 1000</span> with <span class="SimpleMath">n > 2</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallGroups( 6 );</span>
[ <pc group of size 6 with 2 generators>,
<pc group of size 6 with 2 generators> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AllSmallGroups( 60, IsNilpotentGroup );</span>
[ <pc group of size 60 with 4 generators>,
<pc group of size 60 with 4 generators> ]
</pre></div>
<p><a id="X875EB1167FF6BA82" name="X875EB1167FF6BA82"></a></p>
<h5>1.2-4 OneSmallGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneSmallGroup</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns one group with certain properties as specified by <var class="Arg">arg</var>. The permitted arguments are those supported by <code class="func">AllSmallGroups</code> (<a href="chap1.html#X7BB133CB7AA8F465"><span class="RefLink">1.2-3</span></a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">G := OneSmallGroup( 6, IsAbelian );</span>
<pc group of size 6 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( G );</span>
"C6"
<span class="GAPprompt">gap></span> <span class="GAPinput">G := OneSmallGroup( 6, IsAbelian, false );</span>
<pc group of size 6 with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription( G );</span>
"S3"
<span class="GAPprompt">gap></span> <span class="GAPinput">G := OneSmallGroup( Size, [1..1000], IsSolvableGroup, false );</span>
Group([ (1,2,3,4,5), (1,2,3) ])
</pre></div>
<p><a id="X7C587F2A82BEAD19" name="X7C587F2A82BEAD19"></a></p>
<h5>1.2-5 NumberSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberSmallGroups</code>( <var class="Arg">order</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">‣ NrSmallGroups</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns the number of groups of order <var class="Arg">order</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 512 );</span>
10494213
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 2^8 * 23 );</span>
1083472
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 4096 );</span>
Error, the library of groups of size 4096 is not available
</pre></div>
<p><a id="X872991747D5CFD35" name="X872991747D5CFD35"></a></p>
<h5>1.2-6 NumberSmallGroupsAvailable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberSmallGroupsAvailable</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code">true</code> if the number of groups of order <var class="Arg">order</var> is known, and <code class="code">false</code> otherwise.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroupsAvailable( 100 );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 100 );</span>
16
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroupsAvailable( 4096 );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">NumberSmallGroups( 4096 );</span>
Error, the library of groups of size 4096 is not available
</pre></div>
<p><a id="X7B5A1FD47C722EB2" name="X7B5A1FD47C722EB2"></a></p>
<h5>1.2-7 SelectSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SelectSmallGroups</code>( <var class="Arg">argl</var>, <var class="Arg">all</var>, <var class="Arg">id</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>universal function for 'AllSmallGroups', 'OneSmallGroup' and 'IdsOfAllSmallGroups'.</p>
<p><a id="X83044B9D7E3BDF35" name="X83044B9D7E3BDF35"></a></p>
<h5>1.2-8 IdSmallGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdSmallGroup</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">‣ IdGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the library number of <var class="Arg">G</var>; that is, the function returns a pair <code class="code">[<var class="Arg">order</var>, <var class="Arg">i</var>]</code> where <var class="Arg">G</var> is isomorphic to <code class="code">SmallGroup( <var class="Arg">order</var>, <var class="Arg">i</var> )</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallGroup( GL( 2,3 ) );</span>
[ 48, 29 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdSmallGroup( Group( (1,2,3,4),(4,5) ) );</span>
[ 120, 34 ]
</pre></div>
<p><a id="X7C0C616180DE5875" name="X7C0C616180DE5875"></a></p>
<h5>1.2-9 IdGroupsAvailable</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdGroupsAvailable</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>returns <code class="code">true</code>, if the identification routines for groups of order <var class="Arg">order</var> are installed, otherwise returns <code class="code">false</code>.</p>
<p><a id="X85352440869327EC" name="X85352440869327EC"></a></p>
<h5>1.2-10 IdsOfAllSmallGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdsOfAllSmallGroups</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>similar to <code class="code">AllSmallGroups</code> but returns ids instead of groups. This may prevent workspace overflows, if a large number of groups are expected in the output.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdsOfAllSmallGroups( 60, IsNilpotentGroup );</span>
[ [ 60, 4 ], [ 60, 13 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdsOfAllSmallGroups( 60, IsNilpotentGroup, false );</span>
[ [ 60, 1 ], [ 60, 2 ], [ 60, 3 ], [ 60, 5 ], [ 60, 6 ], [ 60, 7 ],
[ 60, 8 ], [ 60, 9 ], [ 60, 10 ], [ 60, 11 ], [ 60, 12 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">IdsOfAllSmallGroups( Size, 60, IsSupersolvableGroup, true );</span>
[ [ 60, 1 ], [ 60, 2 ], [ 60, 3 ], [ 60, 4 ], [ 60, 6 ], [ 60, 7 ],
[ 60, 8 ], [ 60, 10 ], [ 60, 11 ], [ 60, 12 ], [ 60, 13 ] ]
</pre></div>
<p><a id="X8162304487D0C3E2" name="X8162304487D0C3E2"></a></p>
<h5>1.2-11 IdGap3SolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdGap3SolvableGroup</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">‣ Gap3CatalogueIdGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>returns the catalogue number of <var class="Arg">G</var> in the <strong class="pkg">GAP</strong> 3 catalogue of solvable groups; that is, the function returns a pair <code class="code">[<var class="Arg">order</var>, <var class="Arg">i</var>]</code> meaning that <var class="Arg">G</var> is isomorphic to the group <code class="code">SolvableGroup( <var class="Arg">order</var>, <var class="Arg">i</var> )</code> in <strong class="pkg">GAP</strong> 3.</p>
<p><a id="X833DB8AB80B76D26" name="X833DB8AB80B76D26"></a></p>
<h5>1.2-12 SmallGroupsInformation</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SmallGroupsInformation</code>( <var class="Arg">order</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>prints information on the groups of the specified order.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">SmallGroupsInformation( 32 );</span>
There are 51 groups of order 32.
They are sorted by their ranks.
1 is cyclic.
2 - 20 have rank 2.
21 - 44 have rank 3.
45 - 50 have rank 4.
51 is elementary abelian.
For the selection functions the values of the following attributes
are precomputed and stored:
IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and
FrattinifactorId.
This size belongs to layer 2 of the SmallGroups library.
IdSmallGroup is available for this size.
</pre></div>
<p><a id="X850CC04E7855FF68" name="X850CC04E7855FF68"></a></p>
<h5>1.2-13 UnloadSmallGroupsData</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnloadSmallGroupsData</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p><strong class="pkg">GAP</strong> loads all necessary data from the library automatically, but it does not delete the data from the workspace again. Usually, this will be not necessary, since the data is stored in a compressed format. However, if a large number of groups from the library have been loaded, then the user might wish to remove the data from the workspace and this can be done by the above function call.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">UnloadSmallGroupsData();</span>
</pre></div>
<p><a id="X7CE8AEAF8133285D" name="X7CE8AEAF8133285D"></a></p>
<h5>1.2-14 SMALL_GROUPS_OLD_ORDER</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SMALL_GROUPS_OLD_ORDER</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>If set to <code class="code">true</code>, then groups of order <span class="SimpleMath">3^7</span>, <span class="SimpleMath">5^7</span>, <span class="SimpleMath">7^7</span>, and <span class="SimpleMath">11^7</span> are ordered in the way they were ordered up to version 1.0 of the package. If this variable is set to <code class="code">false</code>, which is the default as of version 1.4, then a different ordering, that agrees with the one in Magma version 2.23, is used. The functions <code class="code">SMALLGP_PERM</code><span class="SimpleMath">x</span>, with <span class="SimpleMath">x=3,5,7,11</span>, give the old index position corresponding to a new index position. In releases 1.1-1.3 a misunderstood ordering, based on the old ordering and the permutations <span class="SimpleMath">(2,30083)(3,30084)(4,30085)(5,30086)</span>, <span class="SimpleMath">(2,104599)(3,104600)(4,104601)(5,104602)</span>, and <span class="SimpleMath">(2,721053)(3,721054)(4,721055)(5,721059)</span> respectively were used.</p>
<p> </p>
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