Abstract group 1152.157508: $A_4\wr C_2\times C_4$

• Label: 1152.157508
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$A_4\wr C_2\times C_4$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^4:D_6^2$, of order \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	55	80	200	368	448	1152
Conjugacy classes	1	7	5	12	13	18	56
Divisions	1	7	3	7	7	5	30
Autjugacy classes	1	6	3	6	6	5	27

Dimension	1	2	4	6	8	9	12	18	24
Irr. complex chars.	24	12	0	12	0	8	0	0	0	56
Irr. rational chars.	4	8	5	2	1	4	3	2	1	30

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	$120$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{6}=b^{12}=c^{2}=d^{2}=e^{2}=f^{2}=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(1,2)(3,5)(4,6)(7,8), (9,10,11,12), (3,4,7)(5,6,8), (9,11)(10,12), (3,7,4)(5,6,8), (2,5)(6,8), (2,6)(5,8), (1,3)(4,7), (1,4)(3,7)\rangle$
Transitive group:	24T2691	36T1634			more information
Direct product:	$C_4$ $\, \times\, $ $(A_4\wr C_2)$
Semidirect product:	$(C_2^3.S_4)$ $\,\rtimes\,$ $C_6$	$(C_4\times A_4^2)$ $\,\rtimes\,$ $C_2$	$(A_4^2:C_4)$ $\,\rtimes\,$ $C_2$	$A_4^2$ $\,\rtimes\,$ $(C_2\times C_4)$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^3:S_4)$ . $C_6$	$C_2^5$ . $(C_6\times S_3)$	$(C_2^3:A_4)$ . $D_6$	$(A_4^2:C_2^2)$ . $C_2$	all 7

Elements of the group are displayed as permutations of degree 12.

Homology

Abelianization:	$C_{2} \times C_{12} \simeq C_{2} \times C_{4} \times C_{3}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 2954 subgroups in 257 conjugacy classes, 25 normal (21 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_4$	$G/Z \simeq$ $A_4\wr C_2$
Commutator:	$G' \simeq$ $C_2^2:A_4$	$G/G' \simeq$ $C_2\times C_{12}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_4^2:C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_4$	$G/\operatorname{Fit} \simeq$ $C_3\times S_3$
Radical:	$R \simeq$ $A_4\wr C_2\times C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_6\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3.C_2^4$
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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1152: $A_4\wr C_2\times C_4$

Order 576: $A_4^2:C_2^2$, $C_4\times A_4^2$, $A_4^2:C_4$

Order 384: $C_4\times C_2^3:A_4$, $C_2^5.D_6$

Order 288: $A_4\wr C_2$ x 2, $C_2\times A_4^2$

Order 192: $C_2^4:C_{12}$ x 2, $C_2^4:C_{12}$, $C_2^3:S_4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_2^4:A_4$

Order 144: $A_4^2$, $C_{12}\times A_4$

Order 128: $C_2^3.C_2^4$

Order 96: $C_2^3:C_{12}$ x 3, $C_2^3:A_4$ x 2, $C_2^3:A_4$ x 2, $C_2^2:S_4$ x 2, $C_2^3\times A_4$, $C_4\times S_4$

Order 72: $C_6\times A_4$, $S_3\times C_{12}$

Order 64: $C_2^3.D_4$, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2:C_4^2$, $C_2^4\times C_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 48: $C_4\times A_4$ x 7, $C_2^2\times A_4$ x 4, $C_2^2:A_4$ x 2, $C_2\times S_4$, $C_2^2\times C_{12}$, $A_4:C_4$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$, $C_3\times A_4$

Order 32: $C_2^3\times C_4$ x 5, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 4, $C_2^3:C_4$ x 4, $C_2.C_4^2$ x 2, $C_2^5$, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$

Order 24: $C_2\times A_4$ x 9, $C_2\times C_{12}$ x 3, $S_4$ x 2, $C_4\times S_3$, $C_2^2\times C_6$

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

Order 16: $C_2^2\times C_4$ x 18, $C_2^2:C_4$ x 10, $C_2\times D_4$ x 8, $C_2^4$ x 6, $C_4:C_4$ x 2, $C_4^2$ x 2

Order 12: $A_4$ x 5, $C_{12}$ x 5, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 20, $C_2^3$ x 19, $D_4$ x 8

Order 6: $C_6$ x 7, $S_3$ x 2

Order 4: $C_2^2$ x 19, $C_4$ x 7

Order 3: $C_3$ x 3

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $A_4\wr C_2\times C_4$

Order 576: $A_4^2:C_2^2$, $C_4\times A_4^2$, $A_4^2:C_4$

Order 384: $C_4\times C_2^3:A_4$, $C_2^5.D_6$

Order 288: $C_2\times A_4^2$, $A_4\wr C_2$

Order 192: $C_2^4:C_{12}$ x 2, $C_2^4:C_{12}$, $C_2^3:S_4$, $C_2^4:C_{12}$, $C_2^3.S_4$, $C_2^4:A_4$

Order 144: $A_4^2$, $C_{12}\times A_4$

Order 128: $C_2^3.C_2^4$

Order 96: $C_2^3:C_{12}$ x 3, $C_2^3:A_4$ x 2, $C_2^3:A_4$, $C_2^3\times A_4$, $C_2^2:S_4$, $C_4\times S_4$

Order 72: $C_6\times A_4$, $S_3\times C_{12}$

Order 64: $C_2^3.D_4$, $C_2^3.D_4$, $C_2^4:C_4$, $C_2^2:C_4^2$, $C_2^4\times C_4$, $C_2^3:D_4$, $C_4^2:C_2^2$

Order 48: $C_4\times A_4$ x 7, $C_2^2\times A_4$ x 4, $C_2^2:A_4$ x 2, $C_2\times S_4$, $C_2^2\times C_{12}$, $A_4:C_4$

Order 36: $C_3\times C_{12}$, $C_3:C_{12}$, $C_6\times S_3$, $C_3\times A_4$

Order 32: $C_2^3\times C_4$ x 5, $C_4\times D_4$ x 4, $C_2^3:C_4$ x 4, $C_2^2\wr C_2$ x 2, $C_2^5$, $C_2^2\times D_4$, $C_2^2.D_4$, $C_2\times C_4^2$, $C_2.C_4^2$

Order 24: $C_2\times A_4$ x 8, $C_2\times C_{12}$ x 3, $C_4\times S_3$, $C_2^2\times C_6$, $S_4$

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

Order 16: $C_2^2\times C_4$ x 17, $C_2^2:C_4$ x 8, $C_2^4$ x 6, $C_2\times D_4$ x 6, $C_4:C_4$ x 2, $C_4^2$ x 2

Order 12: $A_4$ x 5, $C_{12}$ x 5, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 17, $C_2\times C_4$ x 17, $D_4$ x 4

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

Order 4: $C_2^2$ x 17, $C_4$ x 6

Order 3: $C_3$ x 3

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $A_4\wr C_2\times C_4$ ($C_1$)

Order 576: $A_4^2:C_2^2$ ($C_2$), $C_4\times A_4^2$ ($C_2$), $A_4^2:C_4$ ($C_2$)

Order 384: $C_2^5.D_6$ ($C_3$)

Order 288: $A_4\wr C_2$ ($C_4$) x 2, $C_2\times A_4^2$ ($C_2^2$)

Order 192: $C_2^3:S_4$ ($C_6$), $C_2^4:C_{12}$ ($C_6$), $C_2^4:C_{12}$ ($S_3$), $C_2^3.S_4$ ($C_6$)

Order 144: $A_4^2$ ($C_2\times C_4$)

Order 96: $C_2^2:S_4$ ($C_{12}$) x 2, $C_2^3:A_4$ ($C_2\times C_6$), $C_2^3:A_4$ ($D_6$)

Order 64: $C_2^4\times C_4$ ($C_3\times S_3$)

Order 48: $C_2^2:A_4$ ($C_2\times C_{12}$), $C_2^2:A_4$ ($C_4\times S_3$)

Order 32: $C_2^5$ ($C_6\times S_3$)

Order 16: $C_2^4$ ($S_3\times C_{12}$)

Order 4: $C_4$ ($A_4\wr C_2$)

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

Order 1: $C_1$ ($A_4\wr C_2\times C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $A_4\wr C_2\times C_4$ ($C_1$)

Order 576: $A_4^2:C_2^2$ ($C_2$), $C_4\times A_4^2$ ($C_2$), $A_4^2:C_4$ ($C_2$)

Order 384: $C_2^5.D_6$ ($C_3$)

Order 288: $C_2\times A_4^2$ ($C_2^2$), $A_4\wr C_2$ ($C_4$)

Order 192: $C_2^3:S_4$ ($C_6$), $C_2^4:C_{12}$ ($C_6$), $C_2^4:C_{12}$ ($S_3$), $C_2^3.S_4$ ($C_6$)

Order 144: $A_4^2$ ($C_2\times C_4$)

Order 96: $C_2^3:A_4$ ($C_2\times C_6$), $C_2^3:A_4$ ($D_6$), $C_2^2:S_4$ ($C_{12}$)

Order 64: $C_2^4\times C_4$ ($C_3\times S_3$)

Order 48: $C_2^2:A_4$ ($C_2\times C_{12}$), $C_2^2:A_4$ ($C_4\times S_3$)

Order 32: $C_2^5$ ($C_6\times S_3$)

Order 16: $C_2^4$ ($S_3\times C_{12}$)

Order 4: $C_4$ ($A_4\wr C_2$)

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

Order 1: $C_1$ ($A_4\wr C_2\times C_4$)

Series

Derived series

$A_4\wr C_2\times C_4$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$A_4\wr C_2\times C_4$

$\rhd$

$A_4^2:C_2^2$

$\rhd$

$C_2\times A_4^2$

$\rhd$

$A_4^2$

$\rhd$

$C_2^2:A_4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$A_4\wr C_2\times C_4$

$\rhd$

$C_2^2:A_4$

Upper central series

$C_1$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 22 larger groups in the database.

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

Character theory

Complex character table

See the $56 \times 56$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $30 \times 30$ rational character table.