Abstract group 864.4609: $C_3^2\times Q_8:A_4$

• Label: 864.4609
• Order: \( 2^{5} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3^{3} \)
• $\card{Z(G)}$: \( 2 \cdot 3^{2} \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{4} \)
• Perm deg.: $14$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_3^2\times Q_8:A_4$
Order:	\(864\)\(\medspace = 2^{5} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3:S_3.C_2^6.S_3^3$, of order \(248832\)\(\medspace = 2^{10} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ x 3
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	12
Elements	1	19	296	12	440	96	864
Conjugacy classes	1	4	26	2	50	16	99
Divisions	1	4	13	2	25	8	53
Autjugacy classes	1	2	2	1	3	1	10

Dimension	1	2	3	4	6	8
Irr. complex chars.	27	0	45	27	0	0	99
Irr. rational chars.	1	13	5	1	20	13	53

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$1040$

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:	${\langle a, b, c, d, e \mid a^{3}=b^{2}=c^{6}=d^{4}=e^{6}=[b,c]=[b,d]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(2,3,5)(6,7,8)(9,10,11), (9,10,11)(12,13,14), (9,11,10), (1,2)(3,5)(4,6)(7,8), (1,3)(2,5)(4,7)(6,8), (3,7)(5,8), (2,6)(5,8), (1,4)(2,6)(3,7)(5,8)\rangle$
Direct product:	$C_3$ ${}^2$ $\, \times\, $ $(Q_8:A_4)$
Semidirect product:	$(Q_8\times C_3^2)$ $\,\rtimes\,$ $A_4$	$Q_8$ $\,\rtimes\,$ $(C_3^2\times A_4)$	$(C_2\times C_6^2)$ $\,\rtimes\,$ $A_4$	$(D_4:C_6^2)$ $\,\rtimes\,$ $C_3$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2$ . $(C_6^2:A_4)$	$C_6$ . $(C_2^4:C_3^2)$	$(C_3\times C_6)$ . $(C_2^2:A_4)$		more information

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{3}^{3} $
Schur multiplier:	$C_{6}^{3}$
Commutator length:	$1$

Subgroups

There are 2090 subgroups in 452 conjugacy classes, 70 normal (7 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3\times C_6$	$G/Z \simeq$ $C_2^2:A_4$
Commutator:	$G' \simeq$ $D_4:C_2^2$	$G/G' \simeq$ $C_3^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_6^2$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_3^2\times Q_8:A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 864: $C_3^2\times Q_8:A_4$

Order 288: $C_3\times Q_8:A_4$ x 12, $D_4:C_6^2$

Order 216: $C_6^2:C_6$ x 3, $C_3^2\times \SL(2,3)$ x 2

Order 144: $C_4:C_6^2$ x 3, $C_4.C_6^2$ x 2

Order 108: $C_3^2\times A_4$ x 3

Order 96: $Q_8:A_4$ x 9, $C_6.C_2^4$ x 4

Order 72: $C_6\times A_4$ x 36, $C_3\times \SL(2,3)$ x 24, $D_4\times C_3^2$ x 6, $C_2\times C_6^2$ x 4, $C_6\times C_{12}$ x 3, $Q_8\times C_3^2$ x 2

Order 54: $C_3^2\times C_6$

Order 48: $C_6\times D_4$ x 12, $D_4:C_6$ x 8

Order 36: $C_3\times A_4$ x 36, $C_6^2$ x 7, $C_3\times C_{12}$ x 2

Order 32: $D_4:C_2^2$

Order 27: $C_3^3$

Order 24: $C_2\times A_4$ x 27, $C_3\times D_4$ x 24, $\SL(2,3)$ x 18, $C_2^2\times C_6$ x 16, $C_2\times C_{12}$ x 12, $C_3\times Q_8$ x 8

Order 18: $C_3\times C_6$ x 16

Order 16: $C_2\times D_4$ x 3, $D_4:C_2$ x 2

Order 12: $C_2\times C_6$ x 28, $A_4$ x 27, $C_{12}$ x 8

Order 9: $C_3^2$ x 13

Order 8: $D_4$ x 6, $C_2^3$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2

Order 6: $C_6$ x 25

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

Order 3: $C_3$ x 13

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 864: $C_3^2\times Q_8:A_4$

Order 288: $C_3\times Q_8:A_4$, $D_4:C_6^2$

Order 216: $C_6^2:C_6$, $C_3^2\times \SL(2,3)$

Order 144: $C_4.C_6^2$, $C_4:C_6^2$

Order 108: $C_3^2\times A_4$

Order 96: $C_6.C_2^4$, $Q_8:A_4$

Order 72: $C_2\times C_6^2$ x 2, $C_6\times A_4$, $Q_8\times C_3^2$, $D_4\times C_3^2$, $C_6\times C_{12}$, $C_3\times \SL(2,3)$

Order 54: $C_3^2\times C_6$

Order 48: $D_4:C_6$, $C_6\times D_4$

Order 36: $C_6^2$ x 3, $C_3\times C_{12}$, $C_3\times A_4$

Order 32: $D_4:C_2^2$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $\SL(2,3)$, $C_2\times A_4$, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times C_6$ x 3

Order 16: $D_4:C_2$, $C_2\times D_4$

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

Order 9: $C_3^2$ x 2

Order 8: $C_2^3$ x 2, $Q_8$, $D_4$, $C_2\times C_4$

Order 6: $C_6$ x 3

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

Order 3: $C_3$ x 2

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 864: $C_3^2\times Q_8:A_4$ ($C_1$)

Order 288: $C_3\times Q_8:A_4$ ($C_3$) x 12, $D_4:C_6^2$ ($C_3$)

Order 96: $Q_8:A_4$ ($C_3^2$) x 9, $C_6.C_2^4$ ($C_3^2$) x 4

Order 72: $C_2\times C_6^2$ ($A_4$) x 3, $Q_8\times C_3^2$ ($A_4$) x 2

Order 32: $D_4:C_2^2$ ($C_3^3$)

Order 24: $C_2^2\times C_6$ ($C_3\times A_4$) x 12, $C_3\times Q_8$ ($C_3\times A_4$) x 8

Order 18: $C_3\times C_6$ ($C_2^2:A_4$)

Order 9: $C_3^2$ ($Q_8:A_4$)

Order 8: $C_2^3$ ($C_3^2\times A_4$) x 3, $Q_8$ ($C_3^2\times A_4$) x 2

Order 6: $C_6$ ($C_2^4:C_3^2$) x 4

Order 3: $C_3$ ($C_3\times Q_8:A_4$) x 4

Order 2: $C_2$ ($C_6^2:A_4$)

Order 1: $C_1$ ($C_3^2\times Q_8:A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 864: $C_3^2\times Q_8:A_4$ ($C_1$)

Order 288: $C_3\times Q_8:A_4$ ($C_3$), $D_4:C_6^2$ ($C_3$)

Order 96: $C_6.C_2^4$ ($C_3^2$), $Q_8:A_4$ ($C_3^2$)

Order 72: $C_2\times C_6^2$ ($A_4$), $Q_8\times C_3^2$ ($A_4$)

Order 32: $D_4:C_2^2$ ($C_3^3$)

Order 24: $C_2^2\times C_6$ ($C_3\times A_4$), $C_3\times Q_8$ ($C_3\times A_4$)

Order 18: $C_3\times C_6$ ($C_2^2:A_4$)

Order 9: $C_3^2$ ($Q_8:A_4$)

Order 8: $C_2^3$ ($C_3^2\times A_4$), $Q_8$ ($C_3^2\times A_4$)

Order 6: $C_6$ ($C_2^4:C_3^2$)

Order 3: $C_3$ ($C_3\times Q_8:A_4$)

Order 2: $C_2$ ($C_6^2:A_4$)

Order 1: $C_1$ ($C_3^2\times Q_8:A_4$)

Series

Derived series

$C_3^2\times Q_8:A_4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3\times Q_8:A_4$

$\rhd$

$Q_8:A_4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_3^2\times Q_8:A_4$

$\rhd$

$D_4:C_2^2$

Upper central series

$C_1$

$\lhd$

$C_3\times C_6$

Supergroups

This group is a maximal subgroup of 25 larger groups in the database.

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

Character theory

Complex character table

See the $99 \times 99$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $53 \times 53$ rational character table.