Abstract group 576.8440: $C_2\times C_6\times \GL(2,3)$

• Label: 576.8440
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_2\times C_6\times \GL(2,3)$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^4.S_4^2$, of order \(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	55	26	24	278	48	48	96	576
Conjugacy classes	1	11	5	4	43	8	8	16	96
Divisions	1	11	3	4	25	4	4	4	56
Autjugacy classes	1	3	3	2	7	1	2	1	20

Dimension	1	2	3	4	6	8
Irr. complex chars.	24	36	24	12	0	0	96
Irr. rational chars.	8	12	8	12	8	8	56

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$5460$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{6}=e^{12}=[a,b]=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(3,4)(5,6), (1,2)(3,5)(4,6)(7,8)(9,10)(11,12), (1,2)(3,5)(4,6)(7,8) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 10 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 5 & 6 \\ 6 & 11 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 6 \\ 6 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_3$ $\, \times\, $ $\GL(2,3)$
Semidirect product:	$(C_6\times Q_8)$ $\,\rtimes\,$ $D_6$	$Q_8$ $\,\rtimes\,$ $(C_6\times D_6)$	$(C_6.C_2^4)$ $\,\rtimes\,$ $S_3$	$(Q_8:C_6^2)$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^2\times C_6)$ . $S_4$	$C_2^3$ . $(C_3\times S_4)$	$C_6$ . $(C_2^2\times S_4)$	$C_2^2$ . $(C_6\times S_4)$	all 6
Aut. group:	$\Aut(C_3\times C_{84})$

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{12}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1888 subgroups in 447 conjugacy classes, 74 normal (16 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_6$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $\SL(2,3)$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times C_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_6.C_2^4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2\times C_6\times \GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 576: $C_2\times C_6\times \GL(2,3)$

Order 288: $C_6\times \GL(2,3)$ x 6, $Q_8:C_6^2$

Order 192: $C_2^2\times \GL(2,3)$, $C_{24}:C_2^3$

Order 144: $C_3\times \GL(2,3)$ x 4, $C_6\times \SL(2,3)$ x 3, $C_6^2:C_2^2$

Order 96: $C_6\times \SD_{16}$ x 12, $C_2\times \GL(2,3)$ x 6, $C_2^2\times \SL(2,3)$ x 2, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_2^2\times C_{24}$

Order 72: $C_6\times D_6$ x 14, $C_2\times C_6^2$, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 48: $C_3\times \SD_{16}$ x 16, $C_6\times D_4$ x 9, $C_6\times Q_8$ x 6, $C_2\times \SL(2,3)$ x 6, $C_2\times C_{24}$ x 6, $\GL(2,3)$ x 4, $C_2^3\times C_6$, $C_2^2\times D_6$, $C_2^2\times C_{12}$

Order 36: $C_6\times S_3$ x 28, $C_6^2$ x 7

Order 32: $C_2\times \SD_{16}$ x 12, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 24: $C_2\times D_6$ x 14, $C_2^2\times C_6$ x 13, $C_3\times D_4$ x 10, $C_2\times C_{12}$ x 6, $C_3\times Q_8$ x 5, $C_{24}$ x 4, $\SL(2,3)$ x 2

Order 18: $C_3\times S_3$ x 8, $C_3\times C_6$ x 7

Order 16: $\SD_{16}$ x 16, $C_2\times D_4$ x 9, $C_2\times C_8$ x 6, $C_2\times Q_8$ x 6, $C_2^4$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 37, $D_6$ x 28, $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $C_2^3$ x 11, $D_4$ x 10, $C_2\times C_4$ x 6, $Q_8$ x 5, $C_8$ x 4

Order 6: $C_6$ x 25, $S_3$ x 8

Order 4: $C_2^2$ x 23, $C_4$ x 4

Order 3: $C_3$ x 3

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_2\times C_6\times \GL(2,3)$

Order 288: $Q_8:C_6^2$, $C_6\times \GL(2,3)$

Order 192: $C_2^2\times \GL(2,3)$, $C_{24}:C_2^3$

Order 144: $C_6^2:C_2^2$, $C_6\times \SL(2,3)$, $C_3\times \GL(2,3)$

Order 96: $C_2^2\times \SL(2,3)$ x 2, $C_6\times \SD_{16}$ x 2, $C_6.C_2^4$, $C_{12}:C_2^3$, $C_2\times \GL(2,3)$, $C_2^2\times C_{24}$

Order 72: $C_6\times D_6$ x 2, $C_2\times C_6^2$, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 48: $C_6\times Q_8$ x 2, $C_6\times D_4$ x 2, $C_2\times \SL(2,3)$ x 2, $C_3\times \SD_{16}$ x 2, $C_2^3\times C_6$, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $\GL(2,3)$, $C_2\times C_{24}$

Order 36: $C_6^2$ x 2, $C_6\times S_3$ x 2

Order 32: $C_2\times \SD_{16}$ x 2, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 24: $C_2^2\times C_6$ x 5, $C_3\times Q_8$ x 3, $C_2\times C_{12}$ x 2, $\SL(2,3)$ x 2, $C_2\times D_6$ x 2, $C_3\times D_4$ x 2, $C_{24}$

Order 18: $C_3\times C_6$ x 2, $C_3\times S_3$

Order 16: $\SD_{16}$ x 2, $C_2\times Q_8$ x 2, $C_2\times D_4$ x 2, $C_2\times C_8$, $C_2^4$, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 8, $D_6$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 3, $Q_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $C_8$

Order 6: $C_6$ x 7, $S_3$

Order 4: $C_2^2$ x 4, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_2\times C_6\times \GL(2,3)$ ($C_1$)

Order 288: $C_6\times \GL(2,3)$ ($C_2$) x 6, $Q_8:C_6^2$ ($C_2$)

Order 192: $C_2^2\times \GL(2,3)$ ($C_3$)

Order 144: $C_3\times \GL(2,3)$ ($C_2^2$) x 4, $C_6\times \SL(2,3)$ ($C_2^2$) x 3

Order 96: $C_2\times \GL(2,3)$ ($C_6$) x 6, $C_6.C_2^4$ ($S_3$), $C_2^2\times \SL(2,3)$ ($C_6$)

Order 72: $C_3\times \SL(2,3)$ ($C_2^3$)

Order 48: $\GL(2,3)$ ($C_2\times C_6$) x 4, $C_6\times Q_8$ ($D_6$) x 3, $C_2\times \SL(2,3)$ ($C_2\times C_6$) x 3

Order 32: $C_2^2\times Q_8$ ($C_3\times S_3$)

Order 24: $\SL(2,3)$ ($C_2^2\times C_6$), $C_2^2\times C_6$ ($S_4$), $C_3\times Q_8$ ($C_2\times D_6$)

Order 16: $C_2\times Q_8$ ($C_6\times S_3$) x 3

Order 12: $C_2\times C_6$ ($\GL(2,3)$) x 4, $C_2\times C_6$ ($C_2\times S_4$) x 3

Order 8: $C_2^3$ ($C_3\times S_4$), $Q_8$ ($C_6\times D_6$)

Order 6: $C_6$ ($C_2\times \GL(2,3)$) x 6, $C_6$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_3\times \GL(2,3)$) x 4, $C_2^2$ ($C_6\times S_4$) x 3

Order 3: $C_3$ ($C_2^2\times \GL(2,3)$)

Order 2: $C_2$ ($C_6\times \GL(2,3)$) x 6, $C_2$ ($C_2\times C_6\times S_4$)

Order 1: $C_1$ ($C_2\times C_6\times \GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_2\times C_6\times \GL(2,3)$ ($C_1$)

Order 288: $Q_8:C_6^2$ ($C_2$), $C_6\times \GL(2,3)$ ($C_2$)

Order 192: $C_2^2\times \GL(2,3)$ ($C_3$)

Order 144: $C_6\times \SL(2,3)$ ($C_2^2$), $C_3\times \GL(2,3)$ ($C_2^2$)

Order 96: $C_6.C_2^4$ ($S_3$), $C_2^2\times \SL(2,3)$ ($C_6$), $C_2\times \GL(2,3)$ ($C_6$)

Order 72: $C_3\times \SL(2,3)$ ($C_2^3$)

Order 48: $C_6\times Q_8$ ($D_6$), $C_2\times \SL(2,3)$ ($C_2\times C_6$), $\GL(2,3)$ ($C_2\times C_6$)

Order 32: $C_2^2\times Q_8$ ($C_3\times S_3$)

Order 24: $\SL(2,3)$ ($C_2^2\times C_6$), $C_2^2\times C_6$ ($S_4$), $C_3\times Q_8$ ($C_2\times D_6$)

Order 16: $C_2\times Q_8$ ($C_6\times S_3$)

Order 12: $C_2\times C_6$ ($C_2\times S_4$), $C_2\times C_6$ ($\GL(2,3)$)

Order 8: $C_2^3$ ($C_3\times S_4$), $Q_8$ ($C_6\times D_6$)

Order 6: $C_6$ ($C_2^2\times S_4$), $C_6$ ($C_2\times \GL(2,3)$)

Order 4: $C_2^2$ ($C_6\times S_4$), $C_2^2$ ($C_3\times \GL(2,3)$)

Order 3: $C_3$ ($C_2^2\times \GL(2,3)$)

Order 2: $C_2$ ($C_6\times \GL(2,3)$), $C_2$ ($C_2\times C_6\times S_4$)

Order 1: $C_1$ ($C_2\times C_6\times \GL(2,3)$)

Series

Derived series

$C_2\times C_6\times \GL(2,3)$

$\rhd$

$\SL(2,3)$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_2\times C_6\times \GL(2,3)$

$\rhd$

$C_6\times \GL(2,3)$

$\rhd$

$C_6\times \SL(2,3)$

$\rhd$

$C_3\times \SL(2,3)$

$\rhd$

$C_3\times Q_8$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2\times C_6\times \GL(2,3)$

$\rhd$

$\SL(2,3)$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_6$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $96 \times 96$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $56 \times 56$ rational character table.