Abstract group 1152.153827: $C_3\times C_2^5:A_4$

• Label: 1152.153827
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3^{2} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_3\times C_2^5:A_4$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.C_2^6.C_3^2.D_6$, of order \(442368\)\(\medspace = 2^{14} \cdot 3^{3} \)
Composition factors:	$C_2$ x 7, $C_3$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	79	386	48	542	96	1152
Conjugacy classes	1	15	8	4	36	8	72
Divisions	1	15	4	4	18	4	46
Autjugacy classes	1	5	2	1	6	1	16

Dimension	1	2	3	6	12
Irr. complex chars.	18	0	30	24	0	72
Irr. rational chars.	2	8	10	18	8	46

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$1820$

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:	${\langle a, b, c, d, e, f, g \mid a^{6}=b^{6}=c^{2}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(3,8)(4,6)(5,7), (13,14,15), (3,4,5)(6,7,8)(10,11,12)(13,14,15), (1,3) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_2^5:A_4)$
Semidirect product:	$(C_2^4:A_4)$ $\,\rtimes\,$ $C_6$	$(C_2^4\times C_6)$ $\,\rtimes\,$ $A_4$	$(C_2^4:C_6)$ $\,\rtimes\,$ $A_4$	$C_2^5$ $\,\rtimes\,$ $(C_3\times A_4)$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(C_6\times A_4)$	$C_2^2$ . $(C_2^5:C_3^2)$	$(C_2^3\times C_6)$ . $(C_2\times A_4)$	$(C_2\times C_6)$ . $(C_2^3:A_4)$	more information

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{3} \times C_{6} \simeq C_{2} \times C_{3}^{2}$
Schur multiplier:	$C_{2}^{4} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 10752 subgroups in 1728 conjugacy classes, 46 normal (16 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_2^5:A_4$
Commutator:	$G' \simeq$ $C_2^6$	$G/G' \simeq$ $C_3\times C_6$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^5:C_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_6$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_3\times C_2^5:A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1152: $C_3\times C_2^5:A_4$

Order 576: $C_2^6:C_3^2$

Order 384: $C_2^5:A_4$ x 3, $C_2^6:C_6$

Order 288: $C_3\times C_2^3:A_4$ x 4, $C_2^5:C_3^2$

Order 192: $C_2^5:C_6$ x 8, $C_2^4:A_4$ x 3, $C_2^5\times C_6$, $C_{12}:C_2^4$, $C_2^4:C_{12}$

Order 144: $C_2^4:C_3^2$ x 13

Order 128: $C_2^4:D_4$

Order 96: $C_2^4:C_6$ x 24, $C_{12}:C_2^3$ x 20, $C_2^4\times C_6$ x 14, $C_2^3:A_4$ x 12, $C_2^3:C_{12}$ x 12, $C_2^3:A_4$ x 3, $C_2^3\times C_{12}$

Order 72: $C_6\times A_4$ x 5

Order 64: $C_2^3:D_4$ x 8, $C_2^6$, $D_4\times C_2^3$, $C_2^4:C_4$

Order 48: $C_2^3\times C_6$ x 133, $C_6\times D_4$ x 64, $C_2^2:A_4$ x 39, $C_2^2:C_{12}$ x 16, $C_2^2\times C_{12}$ x 10

Order 36: $C_3\times A_4$ x 13

Order 32: $C_2^2\wr C_2$ x 24, $C_2^2\times D_4$ x 20, $C_2^5$ x 14, $C_2^3:C_4$ x 12, $C_2^3\times C_4$

Order 24: $C_2^2\times C_6$ x 271, $C_3\times D_4$ x 32, $C_2\times C_{12}$ x 16, $C_2\times A_4$ x 15

Order 18: $C_3\times C_6$

Order 16: $C_2^4$ x 133, $C_2\times D_4$ x 64, $C_2^2:C_4$ x 16, $C_2^2\times C_4$ x 10

Order 12: $C_2\times C_6$ x 139, $A_4$ x 39, $C_{12}$ x 4

Order 9: $C_3^2$

Order 8: $C_2^3$ x 271, $D_4$ x 32, $C_2\times C_4$ x 16

Order 6: $C_6$ x 18

Order 4: $C_2^2$ x 139, $C_4$ x 4

Order 3: $C_3$ x 4

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $C_3\times C_2^5:A_4$

Order 576: $C_2^6:C_3^2$

Order 384: $C_2^6:C_6$, $C_2^5:A_4$

Order 288: $C_3\times C_2^3:A_4$, $C_2^5:C_3^2$

Order 192: $C_2^5:C_6$ x 2, $C_2^5\times C_6$, $C_2^4:A_4$, $C_{12}:C_2^4$, $C_2^4:C_{12}$

Order 144: $C_2^4:C_3^2$ x 3

Order 128: $C_2^4:D_4$

Order 96: $C_2^4\times C_6$ x 4, $C_2^4:C_6$ x 4, $C_{12}:C_2^3$ x 3, $C_2^3:A_4$, $C_2^3:A_4$, $C_2^3\times C_{12}$, $C_2^3:C_{12}$

Order 72: $C_6\times A_4$ x 2

Order 64: $C_2^3:D_4$ x 2, $C_2^6$, $D_4\times C_2^3$, $C_2^4:C_4$

Order 48: $C_2^3\times C_6$ x 13, $C_6\times D_4$ x 4, $C_2^2:A_4$ x 3, $C_2^2\times C_{12}$ x 2, $C_2^2:C_{12}$ x 2

Order 36: $C_3\times A_4$ x 3

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

Order 24: $C_2^2\times C_6$ x 17, $C_2\times C_{12}$ x 2, $C_2\times A_4$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times C_6$

Order 16: $C_2^4$ x 13, $C_2\times D_4$ x 4, $C_2^2:C_4$ x 2, $C_2^2\times C_4$ x 2

Order 12: $C_2\times C_6$ x 14, $A_4$ x 3, $C_{12}$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 17, $D_4$ x 2, $C_2\times C_4$ x 2

Order 6: $C_6$ x 6

Order 4: $C_2^2$ x 14, $C_4$

Order 3: $C_3$ x 2

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $C_3\times C_2^5:A_4$ ($C_1$)

Order 576: $C_2^6:C_3^2$ ($C_2$)

Order 384: $C_2^5:A_4$ ($C_3$) x 3, $C_2^6:C_6$ ($C_3$)

Order 192: $C_2^4:A_4$ ($C_6$) x 3, $C_2^5\times C_6$ ($C_6$)

Order 128: $C_2^4:D_4$ ($C_3^2$)

Order 96: $C_2^4:C_6$ ($A_4$) x 4, $C_2^4\times C_6$ ($A_4$)

Order 64: $C_2^6$ ($C_3\times C_6$)

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

Order 32: $C_2^2\wr C_2$ ($C_3\times A_4$) x 4, $C_2^5$ ($C_3\times A_4$)

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

Order 16: $C_2^4$ ($C_6\times A_4$) x 5

Order 12: $C_2\times C_6$ ($C_2^3:A_4$) x 4, $C_2\times C_6$ ($C_2^3:A_4$)

Order 8: $C_2^3$ ($C_2^4:C_3^2$)

Order 4: $C_2^2$ ($C_3\times C_2^3:A_4$) x 4, $C_2^2$ ($C_2^5:C_3^2$)

Order 3: $C_3$ ($C_2^5:A_4$)

Order 1: $C_1$ ($C_3\times C_2^5:A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $C_3\times C_2^5:A_4$ ($C_1$)

Order 576: $C_2^6:C_3^2$ ($C_2$)

Order 384: $C_2^6:C_6$ ($C_3$), $C_2^5:A_4$ ($C_3$)

Order 192: $C_2^5\times C_6$ ($C_6$), $C_2^4:A_4$ ($C_6$)

Order 128: $C_2^4:D_4$ ($C_3^2$)

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

Order 64: $C_2^6$ ($C_3\times C_6$)

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

Order 32: $C_2^5$ ($C_3\times A_4$), $C_2^2\wr C_2$ ($C_3\times A_4$)

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

Order 16: $C_2^4$ ($C_6\times A_4$) x 2

Order 12: $C_2\times C_6$ ($C_2^3:A_4$), $C_2\times C_6$ ($C_2^3:A_4$)

Order 8: $C_2^3$ ($C_2^4:C_3^2$)

Order 4: $C_2^2$ ($C_3\times C_2^3:A_4$), $C_2^2$ ($C_2^5:C_3^2$)

Order 3: $C_3$ ($C_2^5:A_4$)

Order 1: $C_1$ ($C_3\times C_2^5:A_4$)

Series

Derived series

$C_3\times C_2^5:A_4$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$C_3\times C_2^5:A_4$

$\rhd$

$C_2^6:C_3^2$

$\rhd$

$C_2^4:A_4$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_3\times C_2^5:A_4$

$\rhd$

$C_2^6$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.