Abstract group 96.65: $A_4:C_8$

• Label: 96.65
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$A_4:C_8$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^2\times S_4$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	7	8	8	8	48	16	96
Conjugacy classes	1	3	1	4	1	8	2	20
Divisions	1	3	1	2	1	2	1	11
Autjugacy classes	1	3	1	2	1	2	1	11

Dimension	1	2	3	4	6	12
Irr. complex chars.	8	4	8	0	0	0	20
Irr. rational chars.	2	3	2	2	1	1	11

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$36$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	12
Arbitrary	3	5	7

Constructions

Presentation:	$\langle a, b, c, d \mid a^{8}=b^{3}=c^{2}=d^{2}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=cd, c^{b}=d, d^{b}=cd \rangle$
Permutation group:	Degree $12$ $\langle(2,3)(5,6,7,9,8,10,11,12), (5,7,8,11)(6,9,10,12), (5,8)(6,10)(7,11)(9,12), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{lll}\alpha & \alpha^{4} & \alpha^{3} \\ \alpha^{2} & \alpha & \alpha^{2} \\ \alpha & \alpha^{2} & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{4} & 0 & 0 \\ \alpha^{3} & \alpha^{4} & \alpha^{6} \\ \alpha^{4} & 0 & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{6} & \alpha^{4} & \alpha \\ 0 & \alpha^{7} & \alpha^{2} \\ 0 & \alpha^{7} & \alpha^{3} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{6} & 0 & 0 \\ 0 & \alpha^{6} & 0 \\ 0 & 0 & \alpha^{6} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{4} & 0 & 0 \\ 0 & \alpha^{4} & 0 \\ 0 & 0 & \alpha^{4} \\ \end{array}\right), \left(\begin{array}{lll}\alpha^{5} & 0 & 0 \\ \alpha^{2} & \alpha^{5} & \alpha \\ \alpha^{6} & 0 & \alpha \\ \end{array}\right) \right\rangle \subseteq \GL_{3}(\F_{9}) = \GL_{3}(\F_{3}[\alpha]/(\alpha^{2} + 2 \alpha + 2))$
	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 6 & 15 \\ 1 & 2 \end{array}\right), \left(\begin{array}{rr} 3 & 5 \\ 7 & 12 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 9 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
Transitive group:	24T89	32T413			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_4$ $\,\rtimes\,$ $C_8$	$C_2^2$ $\,\rtimes\,$ $(C_3:C_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $S_4$	$(C_4\times A_4)$ . $C_2$	$(C_2\times A_4)$ . $C_4$	$C_2$ . $(A_4:C_4)$	all 6

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{16}\Z)$.

Homology

Abelianization:	$C_{8} $
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 68 subgroups in 26 conjugacy classes, 10 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_8$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $A_4:C_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 96: $A_4:C_8$

Order 48: $C_4\times A_4$

Order 32: $C_2^2:C_8$

Order 24: $C_2\times A_4$, $C_3:C_8$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

Order 12: $A_4$, $C_{12}$

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

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $A_4:C_8$

Order 48: $C_4\times A_4$

Order 32: $C_2^2:C_8$

Order 24: $C_2\times A_4$, $C_3:C_8$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

Order 12: $A_4$, $C_{12}$

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

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $A_4:C_8$ ($C_1$)

Order 48: $C_4\times A_4$ ($C_2$)

Order 24: $C_2\times A_4$ ($C_4$)

Order 16: $C_2^2\times C_4$ ($S_3$)

Order 12: $A_4$ ($C_8$)

Order 8: $C_2^3$ ($C_3:C_4$)

Order 4: $C_2^2$ ($C_3:C_8$), $C_4$ ($S_4$)

Order 2: $C_2$ ($A_4:C_4$)

Order 1: $C_1$ ($A_4:C_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $A_4:C_8$ ($C_1$)

Order 48: $C_4\times A_4$ ($C_2$)

Order 24: $C_2\times A_4$ ($C_4$)

Order 16: $C_2^2\times C_4$ ($S_3$)

Order 12: $A_4$ ($C_8$)

Order 8: $C_2^3$ ($C_3:C_4$)

Order 4: $C_2^2$ ($C_3:C_8$), $C_4$ ($S_4$)

Order 2: $C_2$ ($A_4:C_4$)

Order 1: $C_1$ ($A_4:C_8$)

Series

Derived series

$A_4:C_8$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$A_4:C_8$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$A_4:C_8$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 42 larger groups in the database.

This group is a maximal quotient of 32 larger groups in the database.

Character theory

Complex character table

See the $20 \times 20$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	2B	2C	3A	4A	4B	6A	8A	8B	12A
	Size	1	1	3	3	8	2	6	8	24	24	16
	2 P	1A	1A	1A	1A	3A	2A	2A	3A	4A	4B	6A
	3 P	1A	2A	2B	2C	1A	4A	4B	2A	8A	8B	4A
	Schur
96.65.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.65.1b		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$
96.65.1c		$2$	$2$	$2$	$2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$-2$
96.65.1d		$4$	$-4$	$4$	$-4$	$4$	$0$	$0$	$-4$	$0$	$0$	$0$
96.65.2a		$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$-1$	$0$	$0$	$-1$
96.65.2b	2	$2$	$2$	$2$	$2$	$-1$	$-2$	$-2$	$-1$	$0$	$0$	$1$
96.65.2c		$4$	$-4$	$4$	$-4$	$-2$	$0$	$0$	$2$	$0$	$0$	$0$
96.65.3a		$3$	$3$	$-1$	$-1$	$0$	$3$	$-1$	$0$	$-1$	$1$	$0$
96.65.3b		$3$	$3$	$-1$	$-1$	$0$	$3$	$-1$	$0$	$1$	$-1$	$0$
96.65.3c		$6$	$6$	$-2$	$-2$	$0$	$-6$	$2$	$0$	$0$	$0$	$0$
96.65.3d		$12$	$-12$	$-4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$