Abstract group 96.3: $C_2^3.A_4$

• Label: 96.3
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $12$
• Trans deg.: $12$
• Rank: $2$

Group information

Description:	$C_2^3.A_4$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^2.\GL(2,\mathbb{Z}/4)$, of order \(384\)\(\medspace = 2^{7} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$
Derived length:	$3$

This group is nonabelian and solvable.

Group statistics

Order	1	2	3	4	6
Elements	1	7	32	24	32	96
Conjugacy classes	1	3	2	4	2	12
Divisions	1	3	1	2	1	8
Autjugacy classes	1	3	1	1	1	7

Dimension	1	2	3	4	6
Irr. complex chars.	3	3	5	0	1	12
Irr. rational chars.	1	2	1	1	3	8

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$12$
Rank:	$2$
Inequivalent generating pairs:	$16$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{3}=b^{4}=c^{2}=d^{4}=[a,c]=[b,c]=[c,d]=1, b^{a}=b^{3}cd^{3}, d^{a}=b, d^{b}=cd \rangle$
Permutation group:	Degree $12$ $\langle(1,9,5)(2,10,6)(3,11,7)(4,12,8), (3,4)(5,8,6,7)(9,11)(10,12), (1,4,2,3) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrrr} 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \\ 0 & 0 & -1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{6}(\Z)$
	$\left\langle \left(\begin{array}{rrrrr} 0 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 1 & 1 & 1 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{5}(\F_{2})$
	$\left\langle \left(\begin{array}{rr} 5 & 4 \\ 4 & 5 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 3 & 4 \\ 2 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 2 \\ 2 & 1 \end{array}\right), \left(\begin{array}{rr} 4 & 3 \\ 1 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/8\Z)$
Transitive group:	12T57	24T179	24T180	32T417	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2.C_4^2)$ $\,\rtimes\,$ $C_3$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^3$ . $A_4$	$C_2^2$ . $\SL(2,3)$	$C_2$ . $(C_4^2:C_3)$		more information

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{8}\Z)$.

Homology

Abelianization:	$C_{3} $
Schur multiplier:	$C_{2}$
Commutator length:	$2$

Subgroups

There are 91 subgroups in 21 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

Order 32: $C_2.C_4^2$

Order 24: $C_2\times A_4$

Order 16: $C_2^2\times C_4$

Order 12: $A_4$

Order 8: $C_2\times C_4$ x 4, $C_2^3$

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: $C_2^3.A_4$

Order 32: $C_2.C_4^2$

Order 24: $C_2\times A_4$

Order 16: $C_2^2\times C_4$

Order 12: $A_4$

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

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $C_2^3.A_4$ ($C_1$)

Order 32: $C_2.C_4^2$ ($C_3$)

Order 8: $C_2^3$ ($A_4$)

Order 4: $C_2^2$ ($\SL(2,3)$)

Order 2: $C_2$ ($C_4^2:C_3$)

Order 1: $C_1$ ($C_2^3.A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $C_2^3.A_4$ ($C_1$)

Order 32: $C_2.C_4^2$ ($C_3$)

Order 8: $C_2^3$ ($A_4$)

Order 4: $C_2^2$ ($\SL(2,3)$)

Order 2: $C_2$ ($C_4^2:C_3$)

Order 1: $C_1$ ($C_2^3.A_4$)

Series

Derived series	$C_2^3.A_4$	$\rhd$	$C_2.C_4^2$	$\rhd$	$C_2$	$\rhd$	$C_1$
Chief series	$C_2^3.A_4$	$\rhd$	$C_2.C_4^2$	$\rhd$	$C_2^3$	$\rhd$	$C_2$	$\rhd$	$C_1$
Lower central series	$C_2^3.A_4$	$\rhd$	$C_2.C_4^2$
Upper central series	$C_1$	$\lhd$	$C_2$

Supergroups

This group is a maximal subgroup of 34 larger groups in the database.

This group is a maximal quotient of 28 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	3A1	3A-1	4A1	4A-1	4B1	4B-1	6A1	6A-1
	Size	1	1	3	3	16	16	6	6	6	6	16	16
	2 P	1A	1A	1A	1A	3A-1	3A1	2C	2C	2C	2C	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	4A-1	4A1	4B-1	4B1	2A	2A
	Type
96.3.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.3.1b1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
96.3.1b2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
96.3.2a	S	$2$	$-2$	$2$	$-2$	$-1$	$-1$	$0$	$0$	$0$	$0$	$1$	$1$
96.3.2b1	C	$2$	$-2$	$2$	$-2$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	${\zeta }_3^{-1}$	${\zeta }_3$
96.3.2b2	C	$2$	$-2$	$2$	$-2$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$	$0$	$0$	${\zeta }_3$	${\zeta }_3^{-1}$
96.3.3a	R	$3$	$3$	$3$	$3$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$0$	$0$
96.3.3b1	C	$3$	$3$	$-1$	$-1$	$0$	$0$	$-1-2i$	$-1+2i$	$1$	$1$	$0$	$0$
96.3.3b2	C	$3$	$3$	$-1$	$-1$	$0$	$0$	$-1+2i$	$-1-2i$	$1$	$1$	$0$	$0$
96.3.3c1	C	$3$	$3$	$-1$	$-1$	$0$	$0$	$1$	$1$	$-1-2i$	$-1+2i$	$0$	$0$
96.3.3c2	C	$3$	$3$	$-1$	$-1$	$0$	$0$	$1$	$1$	$-1+2i$	$-1-2i$	$0$	$0$
96.3.6a	R	$6$	$-6$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Rational character table

		1A	2A	2B	2C	3A	4A	4B	6A
	Size	1	1	3	3	32	12	12	32
	2 P	1A	1A	1A	1A	3A	2C	2C	3A
	3 P	1A	2A	2B	2C	1A	4A	4B	2A
	Schur
96.3.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.3.1b		$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$-1$
96.3.2a	2	$2$	$-2$	$2$	$-2$	$-1$	$0$	$0$	$1$
96.3.2b		$4$	$-4$	$4$	$-4$	$1$	$0$	$0$	$-1$
96.3.3a		$3$	$3$	$3$	$3$	$0$	$-1$	$-1$	$0$
96.3.3b		$6$	$6$	$-2$	$-2$	$0$	$-2$	$2$	$0$
96.3.3c		$6$	$6$	$-2$	$-2$	$0$	$2$	$-2$	$0$
96.3.6a		$6$	$-6$	$-2$	$2$	$0$	$0$	$0$	$0$

Additional information

For most small groups the commutator length is 1. The group 96.3 is one of the two smallest groups whose commutator length is greater than 1. The other, also of order 96, is 96.203. Both have commutator length 2.