Abstract group $A_4:Q_8$

• Label: 96.185
• Order: \( 2^{5} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{6} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$A_4:Q_8$
Order:	\(96\)\(\medspace = 2^{5} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$D_4\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
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	12
Elements	1	7	8	56	8	16	96
Conjugacy classes	1	3	1	6	1	2	14
Divisions	1	3	1	6	1	1	13
Autjugacy classes	1	3	1	4	1	1	11

Dimension	1	2	3	4	6
Irr. complex chars.	4	5	4	0	1	14
Irr. rational chars.	4	3	4	1	1	13

Minimal Presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$9$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid b^{12}=c^{2}=d^{2}=[a,d]=[c,d]=1, a^{2}=b^{6}, b^{a}=b^{11}, c^{a}=cd, c^{b}=cd, d^{b}=c \rangle$
Permutation group:	Degree $12$ $\langle(2,3)(5,6,8,10)(7,11,12,9), (5,7,8,12)(6,9,10,11), (5,8)(6,10)(7,12)(9,11), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 8 \\ 3 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 5 \\ 7 & 12 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 9 & 8 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$
	$\left\langle \left(\begin{array}{rr} 7 & 0 \\ 6 & 7 \end{array}\right), \left(\begin{array}{rr} 0 & 5 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 9 \\ 9 & 10 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 11 & 4 \\ 1 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/12\Z)$
Transitive group:	24T87	24T131	32T410		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_4$ $\,\rtimes\,$ $Q_8$	$C_2^2$ $\,\rtimes\,$ $(C_3:Q_8)$			more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $S_4$	$C_2^3$ . $D_6$	$C_2$ . $(C_2\times S_4)$	$(A_4:C_4)$ . $C_2$ (2)	all 7

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

Homology

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

Subgroups

There are 126 subgroups in 42 conjugacy classes, 12 normal (10 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2\times S_4$
Commutator:	$G' \simeq$ $C_2\times A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_4$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $A_4:Q_8$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
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: $A_4:Q_8$

Order 48: $A_4:C_4$ x 2, $C_4\times A_4$

Order 32: $C_2^2:Q_8$

Order 24: $C_3:Q_8$, $C_2\times A_4$

Order 16: $C_4:C_4$ x 3, $C_2^2:C_4$ x 2, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $C_3:C_4$ x 2, $A_4$, $C_{12}$

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 2, $C_2^3$

Order 6: $C_6$

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 96: $A_4:Q_8$

Order 48: $C_4\times A_4$, $A_4:C_4$

Order 32: $C_2^2:Q_8$

Order 24: $C_3:Q_8$, $C_2\times A_4$

Order 16: $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $A_4$, $C_{12}$, $C_3:C_4$

Order 8: $C_2\times C_4$ x 4, $Q_8$ x 2, $C_2^3$

Order 6: $C_6$

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

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 96: $A_4:Q_8$ ($C_1$)

Order 48: $A_4:C_4$ ($C_2$) x 2, $C_4\times A_4$ ($C_2$)

Order 24: $C_2\times A_4$ ($C_2^2$)

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

Order 12: $A_4$ ($Q_8$)

Order 8: $C_2^3$ ($D_6$)

Order 4: $C_2^2$ ($C_3:Q_8$), $C_4$ ($S_4$)

Order 2: $C_2$ ($C_2\times S_4$)

Order 1: $C_1$ ($A_4:Q_8$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 96: $A_4:Q_8$ ($C_1$)

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

Order 24: $C_2\times A_4$ ($C_2^2$)

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

Order 12: $A_4$ ($Q_8$)

Order 8: $C_2^3$ ($D_6$)

Order 4: $C_2^2$ ($C_3:Q_8$), $C_4$ ($S_4$)

Order 2: $C_2$ ($C_2\times S_4$)

Order 1: $C_1$ ($A_4:Q_8$)

Series

Derived series	$A_4:Q_8$	$\rhd$	$C_2\times A_4$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Chief series	$A_4:Q_8$	$\rhd$	$A_4:C_4$	$\rhd$	$C_2\times A_4$	$\rhd$	$A_4$	$\rhd$	$C_2^2$	$\rhd$	$C_1$
Lower central series	$A_4:Q_8$	$\rhd$	$C_2\times A_4$	$\rhd$	$A_4$
Upper central series	$C_1$	$\lhd$	$C_2$	$\lhd$	$C_4$

Supergroups

This group is a maximal subgroup of 46 larger groups in the database.

This group is a maximal quotient of 36 larger groups in the database.

Character theory

Complex character table

		1A	2A	2B	2C	3A	4A	4B	4C	4D	4E	4F	6A	12A1	12A5
	Size	1	1	3	3	8	2	6	12	12	12	12	8	8	8
	2 P	1A	1A	1A	1A	3A	2A	2A	2C	2C	2A	2A	3A	6A	6A
	3 P	1A	2A	2B	2C	1A	4A	4B	4C	4D	4E	4F	2A	4A	4A
	Type
96.185.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.185.1b	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
96.185.1c	R	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$
96.185.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$
96.185.2a	R	$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$
96.185.2b	R	$2$	$2$	$2$	$2$	$-1$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-1$	$1$	$1$
96.185.2c	S	$2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$	$0$
96.185.2d1	S	$2$	$-2$	$2$	$-2$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$	${\zeta }_{12}^{-1}+{\zeta }_{12}$
96.185.2d2	S	$2$	$-2$	$2$	$-2$	$-1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	${\zeta }_{12}^{-1}+{\zeta }_{12}$	$-{\zeta }_{12}^{-1}-{\zeta }_{12}$
96.185.3a	R	$3$	$3$	$-1$	$-1$	$0$	$3$	$-1$	$-1$	$-1$	$1$	$1$	$0$	$0$	$0$
96.185.3b	R	$3$	$3$	$-1$	$-1$	$0$	$3$	$-1$	$1$	$1$	$-1$	$-1$	$0$	$0$	$0$
96.185.3c	R	$3$	$3$	$-1$	$-1$	$0$	$-3$	$1$	$-1$	$1$	$1$	$-1$	$0$	$0$	$0$
96.185.3d	R	$3$	$3$	$-1$	$-1$	$0$	$-3$	$1$	$1$	$-1$	$-1$	$1$	$0$	$0$	$0$
96.185.6a	S	$6$	$-6$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Rational character table

		1A	2A	2B	2C	3A	4A	4B	4C	4D	4E	4F	6A	12A
	Size	1	1	3	3	8	2	6	12	12	12	12	8	16
	2 P	1A	1A	1A	1A	3A	2A	2A	2C	2C	2A	2A	3A	6A
	3 P	1A	2A	2B	2C	1A	4A	4B	4C	4D	4E	4F	2A	4A
	Schur
96.185.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.185.1b		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$
96.185.1c		$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
96.185.1d		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$
96.185.2a		$2$	$2$	$2$	$2$	$-1$	$2$	$2$	$0$	$0$	$0$	$0$	$-1$	$-1$
96.185.2b		$2$	$2$	$2$	$2$	$-1$	$-2$	$-2$	$0$	$0$	$0$	$0$	$-1$	$1$
96.185.2c	2	$2$	$-2$	$2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$-2$	$0$
96.185.2d	2	$4$	$-4$	$4$	$-4$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$0$
96.185.3a		$3$	$3$	$-1$	$-1$	$0$	$3$	$-1$	$-1$	$-1$	$1$	$1$	$0$	$0$
96.185.3b		$3$	$3$	$-1$	$-1$	$0$	$3$	$-1$	$1$	$1$	$-1$	$-1$	$0$	$0$
96.185.3c		$3$	$3$	$-1$	$-1$	$0$	$-3$	$1$	$-1$	$1$	$1$	$-1$	$0$	$0$
96.185.3d		$3$	$3$	$-1$	$-1$	$0$	$-3$	$1$	$1$	$-1$	$-1$	$1$	$0$	$0$
96.185.6a	2	$6$	$-6$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$