Abstract group 864.4176: $\SL(2,3).S_3^2$

• Label: 864.4176
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $22$
• Trans deg.: $48$
• Rank: $2$

Group information

Description:	$\SL(2,3).S_3^2$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$D_6\wr C_2\times S_4$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$4$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	19	80	132	224	72	192	144	864
Conjugacy classes	1	3	8	4	10	4	5	4	39
Divisions	1	3	7	4	9	2	5	2	33
Autjugacy classes	1	2	5	3	6	1	3	1	22

Dimension	1	2	3	4	6	8	12	16
Irr. complex chars.	4	10	4	11	4	5	1	0	39
Irr. rational chars.	4	6	4	7	4	6	1	1	33

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$48$
Rank:	$2$
Inequivalent generating pairs:	$9$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	6	8	12

Constructions

Presentation:	${\langle a, b, c, d \mid b^{6}=d^{12}=1, a^{2}=d^{6}, c^{6}=d^{6}, b^{a}= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,2,3,6)(4,11,9,8)(5,13,10,7)(12,16,14,15)(17,18)(19,22)(20,21), (1,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 37 & 6 \\ 12 & 19 \end{array}\right), \left(\begin{array}{rr} 1 & 35 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 41 & 36 \\ 9 & 11 \end{array}\right), \left(\begin{array}{rr} 31 & 24 \\ 6 & 25 \end{array}\right), \left(\begin{array}{rr} 25 & 12 \\ 0 & 37 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 21 & 22 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/42\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6.S_4)$ $\,\rtimes\,$ $S_3$ (2)	$C_3$ $\,\rtimes\,$ $(D_6.S_4)$ (2)	$(C_3\times C_6.S_4)$ $\,\rtimes\,$ $C_2$ (2)	$C_3^2$ $\,\rtimes\,$ $(C_2^2.S_4)$	all 5
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:S_3)$ . $S_4$	$C_6$ . $(S_3\times S_4)$ (2)	$\SL(2,3)$ . $S_3^2$	$(C_{12}.D_6)$ . $S_3$	all 12

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 1758 subgroups in 200 conjugacy classes, 26 normal (12 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4:S_3^2$
Commutator:	$G' \simeq$ $C_3^2\times \SL(2,3)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_4:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_3^2$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $\SL(2,3).S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 864: $\SL(2,3).S_3^2$

Order 432: $C_3\times C_6.S_4$ x 2, $C_3:S_3\times \SL(2,3)$

Order 288: $D_6.S_4$ x 2, $Q_8.S_3^2$

Order 216: $C_3^2\times \SL(2,3)$, $C_6.S_3^2$

Order 144: $C_6.S_4$ x 4, $S_3\times \SL(2,3)$ x 3, $C_3^2:Q_{16}$ x 2, $C_3^2:Q_{16}$ x 2, $C_6.S_4$ x 2, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$

Order 108: $C_3^2:C_{12}$ x 2, $C_3^2:D_6$

Order 96: $S_3\times Q_{16}$ x 2, $C_2^2.S_4$

Order 72: $C_3\times \SL(2,3)$ x 6, $C_3^2:Q_8$ x 2, $C_6.D_6$ x 2, $C_3:C_{24}$ x 2, $Q_8\times C_3^2$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 2, $C_3^2\times C_6$

Order 48: $S_3\times Q_8$ x 5, $C_2.S_4$ x 4, $C_3:Q_{16}$ x 4, $C_3:Q_{16}$ x 2, $S_3\times C_8$ x 2, $C_3\times Q_{16}$ x 2, $C_2\times \SL(2,3)$

Order 36: $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 3, $C_6\times S_3$ x 3, $C_3\times C_{12}$, $C_6:S_3$

Order 32: $C_2\times Q_{16}$

Order 27: $C_3^3$

Order 24: $C_3:Q_8$ x 7, $C_4\times S_3$ x 5, $C_3\times Q_8$ x 5, $\SL(2,3)$ x 4, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_6:C_4$

Order 18: $C_3\times C_6$ x 7, $C_3\times S_3$ x 6, $C_3:S_3$ x 2

Order 16: $Q_{16}$ x 4, $C_2\times Q_8$ x 2, $C_2\times C_8$

Order 12: $C_3:C_4$ x 9, $C_{12}$ x 5, $D_6$ x 3, $C_2\times C_6$

Order 9: $C_3^2$ x 7

Order 8: $Q_8$ x 5, $C_2\times C_4$ x 2, $C_8$ x 2

Order 6: $C_6$ x 9, $S_3$ x 6

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

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $\SL(2,3).S_3^2$

Order 432: $C_3:S_3\times \SL(2,3)$, $C_3\times C_6.S_4$

Order 288: $D_6.S_4$, $Q_8.S_3^2$

Order 216: $C_3^2\times \SL(2,3)$, $C_6.S_3^2$

Order 144: $S_3\times \SL(2,3)$ x 2, $C_6.S_4$ x 2, $C_3^2:Q_{16}$, $C_3^2:Q_{16}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6.S_4$

Order 108: $C_3^2:D_6$, $C_3^2:C_{12}$

Order 96: $C_2^2.S_4$, $S_3\times Q_{16}$

Order 72: $C_3\times \SL(2,3)$ x 4, $Q_8\times C_3^2$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_3^2:Q_8$, $C_6.D_6$, $C_6.D_6$, $C_3:C_{24}$

Order 54: $C_3^2\times C_6$, $C_3^2:C_6$

Order 48: $S_3\times Q_8$ x 3, $C_2.S_4$ x 2, $C_3:Q_{16}$ x 2, $C_3:Q_{16}$, $S_3\times C_8$, $C_2\times \SL(2,3)$, $C_3\times Q_{16}$

Order 36: $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_6\times S_3$ x 2, $C_3\times C_{12}$, $C_6:S_3$

Order 32: $C_2\times Q_{16}$

Order 27: $C_3^3$

Order 24: $C_3:Q_8$ x 4, $C_4\times S_3$ x 3, $\SL(2,3)$ x 3, $C_3\times Q_8$ x 3, $C_6:C_4$, $C_{24}$, $C_3:C_8$

Order 18: $C_3\times C_6$ x 5, $C_3\times S_3$ x 2, $C_3:S_3$

Order 16: $Q_{16}$ x 2, $C_2\times Q_8$ x 2, $C_2\times C_8$

Order 12: $C_3:C_4$ x 5, $C_{12}$ x 3, $D_6$ x 2, $C_2\times C_6$

Order 9: $C_3^2$ x 5

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 2, $C_8$

Order 6: $C_6$ x 6, $S_3$ x 2

Order 4: $C_4$ x 3, $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $\SL(2,3).S_3^2$ ($C_1$)

Order 432: $C_3\times C_6.S_4$ ($C_2$) x 2, $C_3:S_3\times \SL(2,3)$ ($C_2$)

Order 216: $C_3^2\times \SL(2,3)$ ($C_2^2$)

Order 144: $C_6.S_4$ ($S_3$) x 2, $C_{12}.D_6$ ($S_3$)

Order 72: $C_3\times \SL(2,3)$ ($D_6$) x 2, $Q_8\times C_3^2$ ($D_6$)

Order 36: $C_6:S_3$ ($S_4$)

Order 24: $C_3\times Q_8$ ($S_3^2$) x 2, $\SL(2,3)$ ($S_3^2$)

Order 18: $C_3:S_3$ ($C_2.S_4$) x 2, $C_3\times C_6$ ($C_2\times S_4$)

Order 9: $C_3^2$ ($C_2^2.S_4$)

Order 8: $Q_8$ ($C_3:S_3^2$)

Order 6: $C_6$ ($S_3\times S_4$) x 2

Order 3: $C_3$ ($D_6.S_4$) x 2

Order 2: $C_2$ ($A_4:S_3^2$)

Order 1: $C_1$ ($\SL(2,3).S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $\SL(2,3).S_3^2$ ($C_1$)

Order 432: $C_3:S_3\times \SL(2,3)$ ($C_2$), $C_3\times C_6.S_4$ ($C_2$)

Order 216: $C_3^2\times \SL(2,3)$ ($C_2^2$)

Order 144: $C_{12}.D_6$ ($S_3$), $C_6.S_4$ ($S_3$)

Order 72: $Q_8\times C_3^2$ ($D_6$), $C_3\times \SL(2,3)$ ($D_6$)

Order 36: $C_6:S_3$ ($S_4$)

Order 24: $\SL(2,3)$ ($S_3^2$), $C_3\times Q_8$ ($S_3^2$)

Order 18: $C_3\times C_6$ ($C_2\times S_4$), $C_3:S_3$ ($C_2.S_4$)

Order 9: $C_3^2$ ($C_2^2.S_4$)

Order 8: $Q_8$ ($C_3:S_3^2$)

Order 6: $C_6$ ($S_3\times S_4$)

Order 3: $C_3$ ($D_6.S_4$)

Order 2: $C_2$ ($A_4:S_3^2$)

Order 1: $C_1$ ($\SL(2,3).S_3^2$)

Series

Derived series

$\SL(2,3).S_3^2$

$\rhd$

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

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$\SL(2,3).S_3^2$

$\rhd$

$C_3\times C_6.S_4$

$\rhd$

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

$\rhd$

$Q_8\times C_3^2$

$\rhd$

$C_3\times Q_8$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$\SL(2,3).S_3^2$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 16 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $39 \times 39$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $33 \times 33$ rational character table.