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<div class="ChapSects"><a href="chap2_mj.html#X78C56BAA804A56A1">2 <span class="Heading">Irreducible Matrix Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X82FD673384BF353B">2.1 <span class="Heading">Irreducible Solvable Matrix Groups</span></a>
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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7DF4B4D683A727E8">2.1-1 IrreducibleSolvableGroupMS</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X836AEF4A7E494724">2.1-2 NumberIrreducibleSolvableGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X7DAC64F17C8B49A2">2.1-3 AllIrreducibleSolvableGroups</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X844E60B87FC48D1B">2.1-4 OneIrreducibleSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X81B11EE77EFA745E">2.1-5 PrimitiveIndexIrreducibleSolvableGroup</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap2_mj.html#X816FF4DD8267B4A7">2.1-6 IrreducibleSolvableGroup</a></span>
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<h3>2 <span class="Heading">Irreducible Matrix Groups</span></h3>
<p><a id="X82FD673384BF353B" name="X82FD673384BF353B"></a></p>
<h4>2.1 <span class="Heading">Irreducible Solvable Matrix Groups</span></h4>
<p><a id="X7DF4B4D683A727E8" name="X7DF4B4D683A727E8"></a></p>
<h5>2.1-1 IrreducibleSolvableGroupMS</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleSolvableGroupMS</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a representative of the <var class="Arg">i</var>-th conjugacy class of irreducible solvable subgroup of GL(<var class="Arg">n</var>, <var class="Arg">p</var>), where <var class="Arg">n</var> is an integer <span class="SimpleMath">\(> 1\)</span>, <var class="Arg">p</var> is a prime, and <span class="SimpleMath">\(\textit{p}^{\textit{n}} < 256\)</span>.</p>
<p>The numbering of the representatives should be considered arbitrary. However, it is guaranteed that the <var class="Arg">i</var>-th group on this list will lie in the same conjugacy class in all future versions of <strong class="pkg">GAP</strong>, unless two (or more) groups on the list are discovered to be duplicates, in which case <code class="func">IrreducibleSolvableGroupMS</code> will return <code class="keyw">fail</code> for all but one of the duplicates.</p>
<p>For values of <var class="Arg">n</var>, <var class="Arg">p</var>, and <var class="Arg">i</var> admissible to <code class="func">IrreducibleSolvableGroup</code> (<a href="chap2_mj.html#X816FF4DD8267B4A7"><span class="RefLink">2.1-6</span></a>), <code class="func">IrreducibleSolvableGroupMS</code> returns a representative of the same conjugacy class of subgroups of GL(<var class="Arg">n</var>, <var class="Arg">p</var>) as <code class="func">IrreducibleSolvableGroup</code> (<a href="chap2_mj.html#X816FF4DD8267B4A7"><span class="RefLink">2.1-6</span></a>). Note that it currently adds two more groups (missing from the original list by Mark Short) for <var class="Arg">n</var> <span class="SimpleMath">\(= 2\)</span>, <var class="Arg">p</var> <span class="SimpleMath">\(= 13\)</span>.</p>
<p><a id="X836AEF4A7E494724" name="X836AEF4A7E494724"></a></p>
<h5>2.1-2 NumberIrreducibleSolvableGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberIrreducibleSolvableGroups</code>( <var class="Arg">n</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns the number of conjugacy classes of irreducible solvable subgroup of GL(<var class="Arg">n</var>, <var class="Arg">p</var>).</p>
<p><a id="X7DAC64F17C8B49A2" name="X7DAC64F17C8B49A2"></a></p>
<h5>2.1-3 AllIrreducibleSolvableGroups</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllIrreducibleSolvableGroups</code>( <var class="Arg">func1</var>, <var class="Arg">val1</var>, <var class="Arg">func2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns a list of conjugacy class representatives <span class="SimpleMath">\(G\)</span> of matrix groups over a prime field such that <span class="SimpleMath">\(f(G) = v\)</span> or <span class="SimpleMath">\(f(G) \in v\)</span>, for all pairs <span class="SimpleMath">\((f,v)\)</span> in (<var class="Arg">func1</var>, <var class="Arg">val1</var>), (<var class="Arg">func2</var>, <var class="Arg">val2</var>), <span class="SimpleMath">\(\ldots\)</span>. The following possibilities for the functions <span class="SimpleMath">\(f\)</span> are particularly efficient, because the values can be read off the information in the data base: <code class="code">DegreeOfMatrixGroup</code> (or <code class="func">Dimension</code> (<a href="../../../doc/ref/chap57_mj.html#X7E6926C6850E7C4E"><span class="RefLink">Reference: Dimension</span></a>) or <code class="func">DimensionOfMatrixGroup</code> (<a href="../../../doc/ref/chap44_mj.html#X7E55258C783C50CA"><span class="RefLink">Reference: DimensionOfMatrixGroup</span></a>)) for the linear degree, <code class="func">Characteristic</code> (<a href="../../../doc/ref/chap31_mj.html#X81278E53800BF64D"><span class="RefLink">Reference: Characteristic</span></a>) for the field characteristic, <code class="func">Size</code> (<a href="../../../doc/ref/chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">Reference: Size</span></a>), <code class="code">IsPrimitiveMatrixGroup</code> (or <code class="code">IsLinearlyPrimitive</code>), and <code class="code">MinimalBlockDimension</code>>.</p>
<p><a id="X844E60B87FC48D1B" name="X844E60B87FC48D1B"></a></p>
<h5>2.1-4 OneIrreducibleSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ OneIrreducibleSolvableGroup</code>( <var class="Arg">func1</var>, <var class="Arg">val1</var>, <var class="Arg">func2</var>, <var class="Arg">val2</var>, <var class="Arg">...</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns one solvable subgroup <span class="SimpleMath">\(G\)</span> of a matrix group over a prime field such that <span class="SimpleMath">\(f(G) = v\)</span> or <span class="SimpleMath">\(f(G) \in v\)</span>, for all pairs <span class="SimpleMath">\((f,v)\)</span> in (<var class="Arg">func1</var>, <var class="Arg">val1</var>), (<var class="Arg">func2</var>, <var class="Arg">val2</var>), <span class="SimpleMath">\(\ldots\)</span>. The following possibilities for the functions <span class="SimpleMath">\(f\)</span> are particularly efficient, because the values can be read off the information in the data base: <code class="code">DegreeOfMatrixGroup</code> (or <code class="func">Dimension</code> (<a href="../../../doc/ref/chap57_mj.html#X7E6926C6850E7C4E"><span class="RefLink">Reference: Dimension</span></a>) or <code class="func">DimensionOfMatrixGroup</code> (<a href="../../../doc/ref/chap44_mj.html#X7E55258C783C50CA"><span class="RefLink">Reference: DimensionOfMatrixGroup</span></a>)) for the linear degree, <code class="func">Characteristic</code> (<a href="../../../doc/ref/chap31_mj.html#X81278E53800BF64D"><span class="RefLink">Reference: Characteristic</span></a>) for the field characteristic, <code class="func">Size</code> (<a href="../../../doc/ref/chap30_mj.html#X858ADA3B7A684421"><span class="RefLink">Reference: Size</span></a>), <code class="code">IsPrimitiveMatrixGroup</code> (or <code class="code">IsLinearlyPrimitive</code>), and <code class="code">MinimalBlockDimension</code>>.</p>
<p><a id="X81B11EE77EFA745E" name="X81B11EE77EFA745E"></a></p>
<h5>2.1-5 PrimitiveIndexIrreducibleSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimitiveIndexIrreducibleSolvableGroup</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This variable provides a way to get from irreducible solvable groups to primitive groups and vice versa. For the group <span class="SimpleMath">\(G\)</span> = <code class="code">IrreducibleSolvableGroup( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">k</var> )</code> and <span class="SimpleMath">\(d = p^n\)</span>, the entry <code class="code">PrimitiveIndexIrreducibleSolvableGroup[d][i]</code> gives the index number of the semidirect product <span class="SimpleMath">\(p^n:G\)</span> in the library of primitive groups.</p>
<p>Searching for an index in this list with <code class="func">Position</code> (<a href="../../../doc/ref/chap21_mj.html#X79975EC6783B4293"><span class="RefLink">Reference: Position</span></a>) gives the translation in the other direction.</p>
<p><a id="X816FF4DD8267B4A7" name="X816FF4DD8267B4A7"></a></p>
<h5>2.1-6 IrreducibleSolvableGroup</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IrreducibleSolvableGroup</code>( <var class="Arg">n</var>, <var class="Arg">p</var>, <var class="Arg">i</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function is obsolete, because for <var class="Arg">n</var> <span class="SimpleMath">\(= 2\)</span>, <var class="Arg">p</var> <span class="SimpleMath">\(= 13\)</span>, two groups were missing from the underlying database. It has been replaced by the function <code class="func">IrreducibleSolvableGroupMS</code> (<a href="chap2_mj.html#X7DF4B4D683A727E8"><span class="RefLink">2.1-1</span></a>). Please note that the latter function does not guarantee any ordering of the groups in the database. However, for values of <var class="Arg">n</var>, <var class="Arg">p</var>, and <var class="Arg">i</var> admissible to <code class="func">IrreducibleSolvableGroup</code>, <code class="func">IrreducibleSolvableGroupMS</code> (<a href="chap2_mj.html#X7DF4B4D683A727E8"><span class="RefLink">2.1-1</span></a>) returns a representative of the same conjugacy class of subgroups of GL(<var class="Arg">n</var>, <var class="Arg">p</var>) as <code class="func">IrreducibleSolvableGroup</code> did before.</p>
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