<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; WedderburnDecomposition</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list of simple algebras.</p>
<p>The input <var class="Arg">FG</var> should be a group algebra of a finite group <span class="SimpleMath">\(G\)</span> over the field <span class="SimpleMath">\(F\)</span>, where <span class="SimpleMath">\(F\)</span> is either an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals) or a finite field of characteristic coprime with the order of <span class="SimpleMath">\(G\)</span>.</p>
<p>The function returns the list of all <em>Wedderburn components</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of the group algebra <var class="Arg">FG</var>. If <span class="SimpleMath">\(F\)</span> is an abelian number field then each Wedderburn component is given as a matrix algebra of a <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). If <span class="SimpleMath">\(F\)</span> is afinite field then the Wedderburn components are given as matrix algebras over finite fields.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( GroupRing( GF(5), DihedralGroup(16) ) );</span>
[ ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 1, 1 ] ),
( GF(5)^[ 1, 1 ] ), ( GF(5)^[ 2, 2 ] ), ( GF(5^2)^[ 2, 2 ] ) ]
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( GroupRing( Rationals, DihedralGroup(16) ) );</span>
[ Rationals, Rationals, Rationals, Rationals, ( Rationals^[ 2, 2 ] ),
<crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( GroupRing( CF(5), DihedralGroup(16) ) );</span>
[ CF(5), CF(5), CF(5), CF(5), ( CF(5)^[ 2, 2 ] ),
<crossed product with center NF(40,[ 1, 31 ]) over AsField( NF(40,
[ 1, 31 ]), CF(40) ) of a group of size 2> ]
</pre></div>
<p>The previous examples show that if <span class="SimpleMath">\(D_{16}\)</span> denotes the dihedral group of order <span class="SimpleMath">\(16\)</span> then the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(\mathbb F_5 D_{16}\)</span>, <span class="SimpleMath">\(ℚ D_{16}\)</span> and <span class="SimpleMath">\(ℚ (\xi_5) D_{16}\)</span> are respectively</p>
<p>where <span class="SimpleMath">\((K(\xi_8)/K,t)\)</span> is a <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>) with the centre <span class="SimpleMath">\(K=NF(8,[ 1, 7 ])= ℚ (\sqrt{2})\)</span>, <span class="SimpleMath">\((F(\xi_{40})/F,t) = ℚ (\sqrt{2},\xi_5)\)</span> is a cyclotomic algebra with centre <span class="SimpleMath">\(F=NF(40,[ 1, 31 ])\)</span> and <span class="SimpleMath">\(\xi_n\)</span> denotes a <span class="SimpleMath">\(n\)</span>-th root of unity.</p>
<p>Two more examples:</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( GroupRing( Rationals, SmallGroup(48,15) ) );</span>
[ Rationals, Rationals, Rationals, Rationals,
<crossed product with center Rationals over CF(3) of a group of size 2>,
<crossed product with center Rationals over GaussianRationals of a group of \
size 2>, <crossed product with center Rationals over CF(3) of a group of size 2>, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), CF(8) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ),
<crossed product with center Rationals over CF(12) of a group of size 4> ]
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition( GroupRing( CF(3), SmallGroup(48,15) ) );</span>
[ CF(3), CF(3), CF(3), CF(3), ( CF(3)^[ 2, 2 ] ),
<crossed product with center CF(3) over AsField( CF(3), CF( 12) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ),
<crossed product with center NF(24,[ 1, 7 ]) over AsField( NF(24,
[ 1, 7 ]), CF(24) ) of a group of size 2>, ( CF(3)^[ 2, 2 ] ),
( CF(3)^[ 2, 2 ] ), ( <crossed product with center CF(3) over AsField( CF( 3), CF(12) ) of a group of size 2>^[ 2, 2 ] ) ]
</pre></div>
<p>In some cases, in characteristic zero, some entries of the output of <code class="func">WedderburnDecomposition</code> do not provide full matrix algebras over a <em>cyclotomic algebra</em> (<ahref="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>), but "fractional matrix algebras". That entry is not an algebra that can be used as a <strong class="pkg">GAP</strong> object. Instead it is a pair formed by a rational giving the "size" of the matrices and a crossed product. See <a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a> for a theoretical explanation of this phenomenon. In this case a warning message is displayed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing(Rationals,SmallGroup(240,89));</span>
<algebra-with-one over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecomposition(QG);</span>
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!
[ Rationals, Rationals, <crossed product with center Rationals over CF( 5) of a group of size 4>, ( Rationals^[ 4, 4 ] ), ( Rationals^[ 4, 4 ] ),
( Rationals^[ 5, 5 ] ), ( Rationals^[ 5, 5 ] ), ( Rationals^[ 6, 6 ] ),
<crossed product with center NF(12,[ 1, 11 ]) over AsField( NF(12,
[ 1, 11 ]), NF(60,[ 1, 11 ]) ) of a group of size 4>,
[ 3/2, <crossed product with center NF(8,[ 1, 7 ]) over AsField( NF(8,
[ 1, 7 ]), NF(40,[ 1, 31 ]) ) of a group of size 4> ] ]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; WedderburnDecompositionInfo</code>( <var class="Arg">FG</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Returns: A list with each entry a numerical description of a <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>).</p>
<p>The input <var class="Arg">FG</var> should be a group algebra of a finite group <span class="SimpleMath">\(G\)</span> over the field <span class="SimpleMath">\(F\)</span>, where <span class="SimpleMath">\(F\)</span> is either an abelian number field (i.e. a subfield of a finite cyclotomic extension of the rationals) or a finite field of characteristic coprime to the order of <span class="SimpleMath">\(G\)</span>.</p>
<p>This function is a numerical counterpart of <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>).</p>
<p>It returns a list formed by lists of lengths 2, 4 or 5.</p>
<p>The lists of length 2 are of the form</p>
<p class="center">\[
[n,F],
\]</p>
<p>where <span class="SimpleMath">\(n\)</span> is a positive integer and <span class="SimpleMath">\(F\)</span> is a field. It represents the <span class="SimpleMath">\(n\times n\)</span> matrix algebra <span class="SimpleMath">\(M_n(F)\)</span> over the field <span class="SimpleMath">\(F\)</span>.</p>
<p>where <span class="SimpleMath">\(F\)</span> is a field and <span class="SimpleMath">\(n,k,d,\alpha,\beta\)</span> are non-negative integers, satisfying the conditions mentioned in Section <a href="chap9_mj.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a>. It represents the <span class="SimpleMath">\(n\times n\)</span> matrix algebra <span class="SimpleMath">\(M_n(A)\)</span> over the cyclic algebra</p>
<p>where <span class="SimpleMath">\(F\)</span> is a field and <span class="SimpleMath">\(n,k,d_i,\alpha_i,\beta_i,\gamma_{i,j}\)</span> are non-negative integers. It represents the <span class="SimpleMath">\(n\times n\)</span> matrix algebra <span class="SimpleMath">\(M_n(A)\)</span> over the <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>)</p>
<p>where <span class="SimpleMath">\(\xi_k\)</span> is a primitive <span class="SimpleMath">\(k\)</span>-th root of unity (see <a href="chap9_mj.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a>).</p>
<p>The interpretation of the previous example gives rise to the following <em>Wedderburn decompositions</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>), where <span class="SimpleMath">\(D_{16}\)</span> is the dihedral group of order 16 and <span class="SimpleMath">\(\xi_5\)</span> is a primitive <span class="SimpleMath">\(5\)</span>-th root of unity.</p>
<p>In the previous example we computed the Wedderburn decomposition of the rational group algebra <span class="SimpleMath">\(ℚ Q_{16}\)</span> of the quaternion group of order <span class="SimpleMath">\(16\)</span> and the rational group algebra <span class="SimpleMath">\(ℚ S_{4}\)</span> of the symmetric group on four letters. For the two group algebras we used both <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) and <code class="func">WedderburnDecompositionInfo</code>.</p>
<p>The output of <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) shows that</p>
<p>where <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> are <em>crossed products</em> (<a href="chap9_mj.html#X7FB21779832CE1CB"><span class="RefLink">9.6</span></a>) with coefficients in the cyclotomic fields <span class="SimpleMath">\(ℚ (\xi_8)\)</span> and <span class="SimpleMath">\(ℚ (\xi_3)\)</span> respectively. This output can be used as a <strongclass="pkg">GAP</strong> object, but it does not give clear information on the structure of the algebras <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span>.</p>
<p>The numerical information displayed by <code class="func">WedderburnDecompositionInfo</code> means that</p>
<p>Both <span class="SimpleMath">\(A\)</span> and <span class="SimpleMath">\(B\)</span> are quaternion algebras over its centre which is <span class="SimpleMath">\(ℚ (\xi+\xi^{-1})\)</span> and the former is equal to <span class="SimpleMath">\(ℚ (\sqrt{2})\)</span> and <span class="SimpleMath">\(ℚ\)</span> respectively.</p>
<p>In <span class="SimpleMath">\(B\)</span>, one has <span class="SimpleMath">\((g+1)(g-1)=0\)</span>, while <span class="SimpleMath">\(g\)</span> is neither <span class="SimpleMath">\(1\)</span> nor <span class="SimpleMath">\(-1\)</span>. This shows that <span class="SimpleMath">\(B=M_2( ℚ )\)</span>. However the relation <span class="SimpleMath">\(g^2=-1\)</span> in <span class="SimpleMath">\(A\)</span> shows that</p>
<p>and so <span class="SimpleMath">\(A\)</span> is a division algebra with centre <span class="SimpleMath">\(ℚ (\sqrt{2})\)</span>, which is a subalgebra of the algebra of Hamiltonian quaternions. This could be deduced also using well known methods on cyclic algebras (see e.g. <a href="chapBib_mj.html#biBR">[Rei03]</a>).</p>
<p>The next example shows the output of <code class="code">WedderburnDecompositionInfo</code> for <span class="SimpleMath">\(ℚ G\)</span> and <span class="SimpleMath">\(ℚ (\xi_3) G\)</span>, where <span class="SimpleMath">\(G=SmallGroup(48,15)\)</span>. The user can compare it with the output of <code class="func">WedderburnDecomposition</code> (<a href="chap2_mj.html#X7F1779ED8777F3E7"><span class="RefLink">2.1-1</span></a>) for the same group in the previous section. Notice that the last entry of the <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(ℚ G\)</span> is not given as a matrix algebra of a cyclic algebra. However, the corresponding entry of <span class="SimpleMath">\(ℚ (\xi_3) G\)</span> is a matrix algebra of a cyclic algebra.</p>
<p>In some cases some of the first entries of the output of <code class="func">WedderburnDecompositionInfo</code> are not integers and so the correspoding <em>Wedderburn components</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) are given as "fractional matrix algebras" of <em>cyclotomic algebras</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>). See <a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a> for a theoretical explanation of this phenomenon. In that case a warning message will be displayed during the first call of <code class="code">WedderburnDecompositionInfo</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">QG:=GroupRing(Rationals,SmallGroup(240,89));</span>
<algebra-with-one over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">WedderburnDecompositionInfo(QG);</span>
Wedderga: Warning!!!
Some of the Wedderburn components displayed are FRACTIONAL MATRIX ALGEBRAS!!!
<p>The interpretation of the output in the previous example gives rise to the following <em>Wedderburn decomposition</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of <span class="SimpleMath">\(ℚ G\)</span> for <span class="SimpleMath">\(G\)</span> the small group <span class="SimpleMath">\([240,89]\)</span>:</p>
<p class="center">\[
ℚ G = 2 ℚ \oplus 2 M_4( ℚ ) \oplus 2 M_5( ℚ ) \oplus M_6( ℚ ) \oplus A \oplus B \oplus C
\]</p>
<p><span class="SimpleMath">\(B\)</span> is an algebra of degree <span class="SimpleMath">\((4*2)/2 = 4 \)</span> which is <em>Brauer equivalent</em> (<a href="chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to</p>
<p>and <span class="SimpleMath">\(C\)</span> is an algebra of degree <span class="SimpleMath">\((4*2)*3/4 = 6 \)</span> which is <em>Brauer equivalent</em> (<a href="chap9_mj.html#X7A24D5407F72C633"><span class="RefLink">9.5</span></a>) to</p>
<p>The precise description of <span class="SimpleMath">\(B\)</span> and <span class="SimpleMath">\(C\)</span> requires the usage of "ad hoc" arguments.</p>
<p>The first input <var class="Arg">FG</var> should be a <em>semisimple group algebra</em> (<a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) over a finite group <span class="SimpleMath">\(G\)</span> and the second input should be an irreducible character of <span class="SimpleMath">\(G\)</span>.</p>
<p>The output is a matrix algebra of a <em>cyclotomic algebras</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>) which is isomorphic to the unique <em>Wedderburn component</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) <span class="SimpleMath">\(A\)</span> of <var class="Arg">FG</var> such that <span class="SimpleMath">\(\chi(A)\ne 0\)</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A5 := AlternatingGroup(5);</span>
Alt( [ 1 .. 5 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">chi := First(Irr( A5 ), chi -> Degree(chi) = 3);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SimpleAlgebraByCharacter( GroupRing( Rationals, A5 ), chi );</span>
( NF(5,[ 1, 4 ])^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">SimpleAlgebraByCharacter( GroupRing( GF(7), A5 ), chi );</span>
( GF(7^2)^[ 3, 3 ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">G:=SmallGroup(128,100); </span>
<pc group of size 128 with 7 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">chi4:=Filtered(Irr(G),x->Degree(x)=4);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">List(chi4,x->SimpleAlgebraByCharacter(GroupRing(Rationals,G),x));</span>
[ ( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ),
( <crossed product with center NF(8,[ 1, 3 ]) over AsField( NF(8,
[ 1, 3 ]), CF(8) ) of a group of size 2>^[ 2, 2 ] ) ]
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SimpleAlgebraByCharacterInfo</code>( <var class="Arg">FG</var>, <var class="Arg">chi</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Returns: The numerical description of the output of <code class="func">SimpleAlgebraByCharacter</code> (<a href="chap2_mj.html#X8349114C83161C2D"><span class="RefLink">2.2-1</span></a>).</p>
<p>The first input <var class="Arg">FG</var> is a <em>semisimple group algebra</em> (<a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) over a finite group <span class="SimpleMath">\(G\)</span> and the second input is an irreducible character of <span class="SimpleMath">\(G\)</span>.</p>
<p>The output is the numerical description <a href="chap9_mj.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a> of the <em>cyclotomic algebra</em> (<a href="chap9_mj.html#X8099A8C784255672"><span class="RefLink">9.11</span></a>) which is isomorphic to the unique <em>Wedderburn component</em> (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) <span class="SimpleMath">\(A\)</span> of <var class="Arg">FG</var> such that <span class="SimpleMath">\(\chi(A)\ne 0\)</span>.</p>
<p>See <a href="chap9_mj.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a> for the interpretation of the numerical information given by the output.</p>
<p>In the three-argument version the input must be formed by a <em>semisimple rational group algebra</em> <var class="Arg">QG</var> (see <a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) and two subgroups <var class="Arg">K</var> and <var class="Arg">H</var> of <span class="SimpleMath">\(G\)</span> which forma <em>strong Shoda pair</em> (<a href="chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">\(G\)</span>.</p>
<p>The three-argument version returns the Wedderburn component (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of the rational group algebra <var class="Arg">QG</var> realized by the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>).</p>
<p>In the four-argument version the first argument is a semisimple finite group algebra <var class="Arg">FG</var>, <var class="Arg">(K,H)</var> is astrong Shoda pair of <span class="SimpleMath">\(G\)</span> and the fourth input data is either a generating <span class="SimpleMath">\(q\)</span>-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or a representative of a generating <span class="SimpleMath">\(q\)</span>-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> (see <a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>).</p>
<p>The four-argument version returns the Wedderburn component (<a href="chap9_mj.html#X84BB4A6081EAE905"><span class="RefLink">9.3</span></a>) of the finite group algebra <var class="Arg">FG</var> realized by the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>) and the cyclotomic class <var class="Arg">C</var> (or the cyclotomic class containing <var class="Arg">C</var>).</p>
<p>The versions ending in NC do not check if (<var class="Arg">K</var>,<var class="Arg">H</var>) is astrong Shoda pair of <span class="SimpleMath">\(G\)</span>. In the four-argument version it is also not checked whether <var class="Arg">C</var> is either a generating <span class="SimpleMath">\(q\)</span>-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or an integer coprime to the index of <var class="Arg">H</var> in <var class="Arg">K</var>.</p>
<p>In the three-argument version the input must be formed by a <em>semisimple rational group algebra</em> (<a href="chap9_mj.html#X7FDD93FB79ADCC91"><span class="RefLink">9.2</span></a>) <var class="Arg">QG</var> and two subgroups <var class="Arg">K</var> and <var class="Arg">H</var> of <span class="SimpleMath">\(G\)</span> which forma <em>strong Shoda pair</em> (<a href="chap9_mj.html#X7E3479527BAE5B9E"><span class="RefLink">9.15</span></a>) of <span class="SimpleMath">\(G\)</span>. It returns the numerical information describing the Wedderburn component (<a href="chap9_mj.html#X84A142407B7565E0"><span class="RefLink">9.12</span></a>) of the rational group algebra <var class="Arg">QG</var> realized by a the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>).</p>
<p>In the four-argument version the first input is a semisimple finite group algebra <var class="Arg">FG</var>, <var class="Arg">(K,H)</var> is astrong Shoda pair of <span class="SimpleMath">\(G\)</span> and the fourth input data is either a generating <span class="SimpleMath">\(q\)</span>-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or a representative of a generating <span class="SimpleMath">\(q\)</span>-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> (<a href="chap9_mj.html#X800D8C5087D79DC8"><span class="RefLink">9.19</span></a>). It returns a pair of positive integers <span class="SimpleMath">\([n,r]\)</span> which represent the <span class="SimpleMath">\(n\times n\)</span> matrix algebra over the field of order <span class="SimpleMath">\(r\)</span> which is isomorphic to the Wedderburn component of <var class="Arg">FG</var> realized by a the strong Shoda pair (<var class="Arg">K</var>,<var class="Arg">H</var>) and the cyclotomic class <var class="Arg">C</var> (or the cyclotomic class containing the integer <var class="Arg">C</var>).</p>
<p>The versions ending in NC do not check if (<var class="Arg">K</var>,<var class="Arg">H</var>) is astrong Shoda pair of <span class="SimpleMath">\(G\)</span>. In the four-argument version it is also not checked whether <var class="Arg">C</var> is either a generating <span class="SimpleMath">\(q\)</span>-cyclotomic class modulo the index of <var class="Arg">H</var> in <var class="Arg">K</var> or an integer coprime with the index of <var class="Arg">H</var> in <var class="Arg">K</var>.</p>
