Abstract group 576.8673: $C_2^2:A_4\times D_6$

• Label: 576.8673
• Order: \( 2^{6} \cdot 3^{2} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \cdot 5 \)
• Perm deg.: $13$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_2^2:A_4\times D_6$
Order:	\(576\)\(\medspace = 2^{6} \cdot 3^{2} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$S_3\times C_2^4.(C_6\times A_5).C_2$, of order \(69120\)\(\medspace = 2^{9} \cdot 3^{3} \cdot 5 \)
Composition factors:	$C_2$ x 6, $C_3$ x 2
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), metabelian, and an A-group.

Group statistics

Order	1	2	3	6
Elements	1	127	98	350	576
Conjugacy classes	1	23	5	19	48
Divisions	1	23	3	15	42
Autjugacy classes	1	5	3	6	15

Dimension	1	2	3	4	6
Irr. complex chars.	12	6	20	0	10	48
Irr. rational chars.	4	6	20	2	10	42

Minimal presentations

Permutation degree:	$13$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$1092$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{6}=c^{2}=d^{2}=e^{2}=f^{6}=[a,b]= \!\cdots\! \rangle}$
Permutation group:	Degree $13$ $\langle(10,11)(12,13), (12,13), (2,4,3)(6,8,7), (1,2)(3,4)(5,6)(7,8), (1,3)(2,4)(5,7)(6,8), (1,3)(2,4), (1,4)(2,3), (9,10,11)\rangle$
Transitive group:	36T823	36T831			more information
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_2^2:A_4)$
Semidirect product:	$C_2^3$ $\,\rtimes\,$ $(S_3\times A_4)$	$C_2^5$ $\,\rtimes\,$ $(C_3\times S_3)$	$C_2^4$ $\,\rtimes\,$ $(C_6\times S_3)$	$(C_2^2\times D_6)$ $\,\rtimes\,$ $A_4$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 13.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 11292 subgroups in 2014 conjugacy classes, 56 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times C_2^2:A_4$
Commutator:	$G' \simeq$ $C_2^3\times C_6$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2:A_4\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_6$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_2^2:A_4\times D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4\times C_6$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 576: $C_2^2:A_4\times D_6$

Order 288: $S_3\times C_2^2:A_4$ x 2, $C_2^5:C_3^2$

Order 192: $D_6\times C_2^4$, $C_2^2\wr C_3$

Order 144: $A_4\times D_6$ x 5, $C_2^4:C_3^2$

Order 96: $C_2^3\times D_6$ x 22, $C_2^3:A_4$ x 4, $C_2^4\times C_6$

Order 72: $S_3\times A_4$ x 10, $C_6\times A_4$ x 5

Order 64: $C_2^6$

Order 48: $C_2^2\times D_6$ x 210, $C_2^3\times C_6$ x 11, $C_2^2\times A_4$ x 5, $C_2^2:A_4$ x 2

Order 36: $C_3\times A_4$ x 5, $C_6\times S_3$

Order 32: $C_2^5$ x 23

Order 24: $C_2\times D_6$ x 420, $C_2^2\times C_6$ x 55, $C_2\times A_4$ x 20

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

Order 16: $C_2^4$ x 221

Order 12: $D_6$ x 166, $C_2\times C_6$ x 56, $A_4$ x 10

Order 9: $C_3^2$

Order 8: $C_2^3$ x 475

Order 6: $C_6$ x 15, $S_3$ x 12

Order 4: $C_2^2$ x 221

Order 3: $C_3$ x 3

Order 2: $C_2$ x 23

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 576: $C_2^2:A_4\times D_6$

Order 288: $C_2^5:C_3^2$, $S_3\times C_2^2:A_4$

Order 192: $D_6\times C_2^4$, $C_2^2\wr C_3$

Order 144: $C_2^4:C_3^2$, $A_4\times D_6$

Order 96: $C_2^3\times D_6$ x 4, $C_2^3:A_4$ x 3, $C_2^4\times C_6$

Order 72: $C_6\times A_4$, $S_3\times A_4$

Order 64: $C_2^6$

Order 48: $C_2^2\times D_6$ x 10, $C_2^3\times C_6$ x 3, $C_2^2:A_4$ x 2, $C_2^2\times A_4$

Order 36: $C_6\times S_3$, $C_3\times A_4$

Order 32: $C_2^5$ x 5

Order 24: $C_2\times D_6$ x 12, $C_2^2\times C_6$ x 5, $C_2\times A_4$ x 3

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

Order 16: $C_2^4$ x 13

Order 12: $D_6$ x 8, $C_2\times C_6$ x 6, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $C_2^3$ x 17

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

Order 4: $C_2^2$ x 13

Order 3: $C_3$ x 3

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 576: $C_2^2:A_4\times D_6$ ($C_1$)

Order 288: $S_3\times C_2^2:A_4$ ($C_2$) x 2, $C_2^5:C_3^2$ ($C_2$)

Order 192: $D_6\times C_2^4$ ($C_3$)

Order 144: $C_2^4:C_3^2$ ($C_2^2$)

Order 96: $C_2^3\times D_6$ ($C_6$) x 2, $C_2^4\times C_6$ ($C_6$), $C_2^3:A_4$ ($S_3$)

Order 48: $C_2^2\times D_6$ ($A_4$) x 5, $C_2^3\times C_6$ ($C_2\times C_6$), $C_2^2:A_4$ ($D_6$)

Order 32: $C_2^5$ ($C_3\times S_3$)

Order 24: $C_2\times D_6$ ($C_2\times A_4$) x 10, $C_2^2\times C_6$ ($C_2\times A_4$) x 5

Order 16: $C_2^4$ ($C_6\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^2\times A_4$) x 5, $D_6$ ($C_2^2:A_4$)

Order 8: $C_2^3$ ($S_3\times A_4$) x 5

Order 6: $S_3$ ($C_2^3:A_4$) x 2, $C_6$ ($C_2^3:A_4$)

Order 4: $C_2^2$ ($A_4\times D_6$) x 5

Order 3: $C_3$ ($C_2^2\wr C_3$)

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

Order 1: $C_1$ ($C_2^2:A_4\times D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 576: $C_2^2:A_4\times D_6$ ($C_1$)

Order 288: $C_2^5:C_3^2$ ($C_2$), $S_3\times C_2^2:A_4$ ($C_2$)

Order 192: $D_6\times C_2^4$ ($C_3$)

Order 144: $C_2^4:C_3^2$ ($C_2^2$)

Order 96: $C_2^4\times C_6$ ($C_6$), $C_2^3\times D_6$ ($C_6$), $C_2^3:A_4$ ($S_3$)

Order 48: $C_2^3\times C_6$ ($C_2\times C_6$), $C_2^2\times D_6$ ($A_4$), $C_2^2:A_4$ ($D_6$)

Order 32: $C_2^5$ ($C_3\times S_3$)

Order 24: $C_2^2\times C_6$ ($C_2\times A_4$), $C_2\times D_6$ ($C_2\times A_4$)

Order 16: $C_2^4$ ($C_6\times S_3$)

Order 12: $C_2\times C_6$ ($C_2^2\times A_4$), $D_6$ ($C_2^2:A_4$)

Order 8: $C_2^3$ ($S_3\times A_4$)

Order 6: $C_6$ ($C_2^3:A_4$), $S_3$ ($C_2^3:A_4$)

Order 4: $C_2^2$ ($A_4\times D_6$)

Order 3: $C_3$ ($C_2^2\wr C_3$)

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

Order 1: $C_1$ ($C_2^2:A_4\times D_6$)

Series

Derived series

$C_2^2:A_4\times D_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_1$

Chief series

$C_2^2:A_4\times D_6$

$\rhd$

$C_2^5:C_3^2$

$\rhd$

$C_2^4:C_3^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^2:A_4\times D_6$

$\rhd$

$C_2^3\times C_6$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

This group is a maximal quotient of 18 larger groups in the database.

Character theory

Complex character table

See the $48 \times 48$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $42 \times 42$ rational character table.