The conjugacy class of an element $g$ of a group $G$ is the set of all conjugates $g^x=x^{\left(-1\right)}·g·x$ for $x$ in $G$.

The ConjugacyClass( g, G ) command constructs the conjugacy class of an element g of a group G.

The group G must be an instance of a permutation group or a Cayley table group.

The conjugacy class of g is represented as an object cc for which the following methods are defined.

The ConjugacyClasses( G ) command computes all the conjugacy classes of a group (or character table) G, and returns them as a set. The group G must be one for which it is possible to compute the set of all elements of G.

The ClassNumber( G ) command computes the number of conjugacy classes of the group (or character table) G.

The ConjugateRank( G ) command returns the conjugate rank of a finite group (or character table) G, which is the number of distinct lengths of non-trivial (that is, non-central) conjugacy classes of G.

Note that the class number of a group is a searchable property for the SearchSmallGroups command.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{g1}≔\mathrm{SymmetricGroup}\left(6\right)$

$\mathrm{g1}≔{\mathbf{S}}_6$

$c≔\mathrm{ConjugacyClass}\left(\left[\left[1\,2\right]\,\left[3\,4\,5\right]\right]\,\mathrm{g1}\right)$

$c≔{\left(\left(1\,2\right)\left(3\,4\,5\right)\right)}^{{\mathbf{S}}_6}$

$\mathrm{numelems}\left(c\right)=\mathrm{binomial}\left(6\,3\right)\cdot 6$

$120=120$

$\mathrm{evalb}\left(\left\{\mathrm{seq}\left(x\,x\mathbf{in}c\right)\right\}=\mathrm{Elements}\left(c\right)\right)$

$\mathrm{true}$

$\mathrm{ops}≔\mathrm{Operations}\left(\mathrm{g1}\right)\:$

$\mathrm{singles}≔\varnothing \:$

$\mathrm{pairs}≔\varnothing \:$

$$\begin{gathered} \mathbf{for}x\mathbf{in}c\mathbf{do} \\ \mathrm{inverse}≔\mathrm{ops}:-\mathrm{`/`}\left(x\right); \\ \mathbf{if}\mathrm{inverse}\mathbf{in}\mathrm{singles}\mathbf{then} \\ \mathrm{singles}≔\mathrm{singles}\mathbf{minus}\left\{\mathrm{inverse}\right\}; \\ \mathrm{pairs}≔\left\{\mathrm{op}\left(\mathrm{pairs}\right)\,\left\{\mathrm{inverse}\,x\right\}\right\} \\ \mathbf{else} \\ \mathrm{singles}≔\left\{x\,\mathrm{op}\left(\mathrm{singles}\right)\right\} \\ \mathbf{end}\mathbf{if} \\ \mathbf{end}\mathbf{do}\: \end{gathered}$$

$\mathrm{singles}$

$\varnothing$

$\frac{\mathrm{numelems}\left(c\right)}{\mathrm{numelems}\left(\mathrm{pairs}\right)}$

$2$

$\mathrm{cs1}≔\mathrm{ConjugacyClasses}\left(\mathrm{g1}\right)$

$\mathrm{cs1}≔\left[{\left(\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\right)\left(3\,4\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\right)\left(3\,4\right)\left(5\,6\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\right)\left(4\,5\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\right)\left(4\,5\,6\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\,4\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\,4\right)\left(5\,6\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\,4\,5\right)\right)}^{{\mathbf{S}}_6}\,{\left(\left(1\,2\,3\,4\,5\,6\right)\right)}^{{\mathbf{S}}_6}\right]$

$\mathrm{numelems}\left(\mathrm{cs1}\right)=\mathrm{combinat}:-\mathrm{numbpart}\left(6\right)$

$11=11$

$M≔\mathrm{CayleyTable}\left(\mathrm{QuaternionGroup}\left(\right)\right)$

$M≔\left[\begin{array}{cccccccc}1& 2& 3& 4& 5& 6& 7& 8\\ 2& 3& 4& 1& 7& 5& 8& 6\\ 3& 4& 1& 2& 8& 7& 6& 5\\ 4& 1& 2& 3& 6& 8& 5& 7\\ 5& 6& 8& 7& 3& 4& 2& 1\\ 6& 8& 7& 5& 2& 3& 1& 4\\ 7& 5& 6& 8& 4& 1& 3& 2\\ 8& 7& 5& 6& 1& 2& 4& 3\end{array}\right]$

$\mathrm{g2}≔\mathrm{CayleyTableGroup}\left(M\right)$

$\mathrm{g2}≔\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}$

$\mathrm{cs2}≔\mathrm{ConjugacyClasses}\left(\mathrm{g2}\right)$

$\mathrm{cs2}≔\left[1^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\&comma;2^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\&comma;3^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\&comma;5^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\&comma;6^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\right]$

$\mathrm{map}\left(\mathrm{print}\&comma;\mathrm{map}\left(x\mapsto '\mathrm{abs}'\left(x\right)=\mathrm{numelems}\left(x\right)\&comma;\mathrm{cs2}\right)\right)\&colon;$

$\left|1^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\right|=1$

$\left|2^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\right|=2$

$\left|3^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\right|=1$

$\left|5^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\right|=2$

$\left|6^{\mathrm{<\; a\; Cayley\; table\; group\; with\; 8\; elements\; >}}\right|=2$

$\mathrm{cn3}≔\mathrm{SearchSmallGroups}\left('\mathrm{classnumber}'=3\right)$

$\mathrm{cn3}≔\left[3\&comma;1\right],\left[6\&comma;1\right]$

$\mathrm{map}\left(\mathrm{ClassNumber}@\mathrm{SmallGroup}\&comma;\left[\mathrm{cn3}\right]\right)$

$\left[3\&comma;3\right]$

$\mathrm{ClassNumber}\left(\mathrm{DirectProduct}\left(\mathrm{BabyMonster}\left(\right)\&comma;\mathrm{Symm}\left(1000\right)\&comma;\mathrm{ElementaryGroup}\left(19\&comma;30\right)\right)\right)$

$1020347182128879289726571449479665023113345049543549124072606701932879944$

$\mathrm{ConjugateRank}\left(\mathrm{Symm}\left(3\right)\right)$

$2$

The GroupTheory[ConjugacyClass], GroupTheory[ConjugacyClasses] and GroupTheory[ConjugateRank] commands were introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[ClassNumber] command was introduced in Maple 2018.

For more information on Maple 2018 changes, see Updates in Maple 2018.

The GroupTheory[ClassNumber] command was updated in Maple 2020.