Abstract group 11664.bf: $C_3^5:(C_2\times S_4)$

• Label: 11664.bf
• Order: \( 2^{4} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $18$
• Trans deg.: $18$
• Rank: $2$

Group information

Description:	$C_3^5:(C_2\times S_4)$
Order:	\(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_3^2\times S_3^3):D_6$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 6
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	603	890	324	4662	1296	2592	1296	11664
Conjugacy classes	1	5	15	2	18	4	10	2	57
Divisions	1	5	15	2	18	4	6	2	53
Autjugacy classes	1	5	15	2	18	2	6	1	50

Dimension	1	2	3	4	6	8	12	16	24	48
Irr. complex chars.	4	4	4	1	14	6	13	3	6	2	57
Irr. rational chars.	4	4	4	1	10	6	11	3	8	2	53

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$18$
Rank:	$2$
Inequivalent generating pairs:	$702$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{6}=d^{6}=e^{3}=f^{3}=g^{3}=[a,c]=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,14,12,5,2,13,10,6,3,15,11,4)(7,8,9)(16,17,18), (2,3)(4,9,5,8,6,7)(10,12)(13,16,15,18,14,17)\rangle$
Transitive group:	18T579	27T807	36T9273	36T9274	all 7
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:S_4)$ $\,\rtimes\,$ $S_3$	$(C_3^5:S_4)$ $\,\rtimes\,$ $C_2$	$(C_3^5:S_4)$ $\,\rtimes\,$ $C_2$	$(C_3^4:C_6)$ $\,\rtimes\,$ $S_4$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $(S_3\times S_4)$	$C_3$ . $(S_3\times C_3^3:S_4)$	$(C_3\wr C_2^2)$ . $S_3^2$		more information
Aut. group:	$\Aut(C_{12}:D_6.D_6)$

Elements of the group are displayed as permutations of degree 18.

Homology

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

Subgroups

There are 96092 subgroups in 1226 conjugacy classes, 19 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^5:(C_2\times S_4)$
Commutator:	$G' \simeq$ $C_3^5:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3\times C_3^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^5$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_3^5:(C_2\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $S_3\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$

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 11664: $C_3^5:(C_2\times S_4)$

Order 5832: $C_3^5:S_4$, $C_3\wr A_4:S_3$, $C_3^5:S_4$

Order 3888: $C_3^3:(S_3\times D_{12})$, $S_3\times C_3^3:S_4$

Order 2916: $C_3^5:D_6$, $C_3^5:A_4$

Order 1944: $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:S_4$ x 2, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^3:(S_3\times C_{12})$, $S_3\times C_3^3:A_4$, $C_3^4:S_4$

Order 1458: $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1296: $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$, $C_2\times C_3^3:S_4$

Order 972: $C_3^3:S_3^2$ x 3, $C_3^4:C_{12}$ x 2, $C_3\wr A_4$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 729: $C_3^5:C_3$

Order 648: $C_3^3:D_{12}$ x 8, $C_3:S_3^3$ x 5, $C_3^3:S_4$ x 3, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\wr D_4$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^3:A_4$

Order 486: $C_3^4:S_3$ x 5, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4.S_3$ x 2, $C_3^4.S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 2, $C_2\times C_3^2:D_{12}$, $D_6:S_3^2$, $C_6^2:D_6$

Order 324: $C_3^2:S_3^2$ x 16, $C_3\wr C_2^2$ x 11, $C_3^2:S_3^2$ x 9, $C_3\wr C_4$ x 6, $C_3^2:D_{18}$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^3:A_4$ x 2, $C_3^2.S_3^2$ x 2, $C_3^3:D_6$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^4:C_3$ x 5, $C_3^4.C_3$ x 2, $C_3^5$, $C_3^4:C_3$

Order 216: $S_3^3$ x 8, $C_3^2:D_{12}$ x 8, $S_3^2:S_3$ x 4, $C_6:S_3^2$ x 3, $S_3^2:C_6$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_6:S_3^2$, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$

Order 162: $C_3^2\wr C_2$ x 24, $C_3^3:C_6$ x 18, $C_3^3:S_3$ x 7, $S_3\times C_3^3$ x 4, $C_3^2:D_9$ x 4, $D_9:C_3^2$ x 4, $C_3^3:C_6$ x 3, $C_3^3:C_6$ x 3, $C_3^3.S_3$ x 2, $C_3^2:C_{18}$ x 2, $C_3^3.C_6$ x 2, $C_3^2:D_9$ x 2, $C_3^3:S_3$, $C_3^3:S_3$, $S_3\times \He_3$, $C_3^3:C_6$

Order 144: $S_3\times D_{12}$ x 2, $S_3^2:C_2^2$, $S_3\times S_4$, $D_6:D_6$

Order 108: $C_3:S_3^2$ x 27, $C_3:S_3^2$ x 27, $C_3\times S_3^2$ x 25, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 5, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_{18}:C_6$ x 2, $S_3\times D_9$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:D_6$, $C_3^2:A_4$, $C_3^2:D_6$

Order 81: $C_3^4$ x 13, $C_3\wr C_3$ x 10, $C_3^2:C_9$ x 4, $C_9:C_3^2$ x 4, $C_3\times \He_3$ x 4

Order 72: $S_3\times D_6$ x 11, $C_3:D_{12}$ x 8, $\SOPlus(4,2)$ x 4, $C_3\times S_4$ x 2, $C_3:D_{12}$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_6:D_6$, $C_2\times C_3^2:C_4$, $S_3\times A_4$, $C_3:S_4$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 88, $S_3\times C_3^2$ x 39, $C_3^2:S_3$ x 15, $C_9:C_6$ x 12, $C_3^2:C_6$ x 9, $C_3\times D_9$ x 4, $C_3^2:S_3$ x 2, $C_3:D_9$ x 2, $S_3\times C_9$ x 2, $C_3^2\times C_6$ x 2, $C_9:C_6$ x 2, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 2, $C_2\times S_4$, $C_2\times D_{12}$

Order 36: $S_3^2$ x 50, $C_6\times S_3$ x 21, $C_6:S_3$ x 15, $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $D_{18}$ x 2, $C_3\times A_4$ x 2, $C_6^2$

Order 27: $C_3^3$ x 64, $C_9:C_3$ x 10, $\He_3$ x 9, $C_3\times C_9$ x 4

Order 24: $D_{12}$ x 8, $C_2\times D_6$ x 7, $C_3:D_4$ x 4, $S_4$ x 3, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $C_2\times A_4$

Order 18: $C_3\times S_3$ x 85, $C_3:S_3$ x 51, $C_3\times C_6$ x 13, $D_9$ x 6, $C_{18}$ x 2

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 35, $C_2\times C_6$ x 6, $C_{12}$ x 6, $A_4$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 68, $C_9$ x 4

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

Order 6: $S_3$ x 33, $C_6$ x 18

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

Order 3: $C_3$ x 15

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11664: $C_3^5:(C_2\times S_4)$

Order 5832: $C_3^5:S_4$, $C_3\wr A_4:S_3$, $C_3^5:S_4$

Order 3888: $C_3^3:(S_3\times D_{12})$, $S_3\times C_3^3:S_4$

Order 2916: $C_3^5:D_6$, $C_3^5:A_4$

Order 1944: $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:S_4$ x 2, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^3:(S_3\times C_{12})$, $S_3\times C_3^3:A_4$, $C_3^4:S_4$

Order 1458: $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1296: $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$, $C_2\times C_3^3:S_4$

Order 972: $C_3^3:S_3^2$ x 3, $C_3^4:C_{12}$ x 2, $C_3\wr A_4$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^4:D_6$

Order 729: $C_3^5:C_3$

Order 648: $C_3^3:D_{12}$ x 8, $C_3:S_3^3$ x 5, $C_3^3:S_4$ x 3, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\wr D_4$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^3:A_4$

Order 486: $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 2, $C_2\times C_3^2:D_{12}$, $D_6:S_3^2$, $C_6^2:D_6$

Order 324: $C_3^2:S_3^2$ x 16, $C_3\wr C_2^2$ x 11, $C_3^2:S_3^2$ x 9, $C_3\wr C_4$ x 6, $C_3^2:S_3^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^3:A_4$ x 2, $C_3^3:D_6$, $C_3^2:D_{18}$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^2:S_3^2$, $C_3^2.S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^4:C_3$ x 4, $C_3^5$, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $S_3^3$ x 8, $C_3^2:D_{12}$ x 8, $S_3^2:S_3$ x 4, $C_6:S_3^2$ x 3, $S_3^2:C_6$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2:S_4$, $C_6:S_3^2$, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$

Order 162: $C_3^2\wr C_2$ x 24, $C_3^3:C_6$ x 18, $C_3^3:S_3$ x 6, $S_3\times C_3^3$ x 4, $C_3^3:C_6$ x 3, $C_3^3:C_6$ x 3, $C_3^2:D_9$ x 2, $D_9:C_3^2$ x 2, $C_3^3:S_3$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^2:D_9$

Order 144: $S_3\times D_{12}$ x 2, $S_3^2:C_2^2$, $S_3\times S_4$, $D_6:D_6$

Order 108: $C_3:S_3^2$ x 27, $C_3:S_3^2$ x 27, $C_3\times S_3^2$ x 25, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 5, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_{18}:C_6$, $C_3^2:D_6$, $C_3^2:A_4$, $C_3^2:D_6$, $S_3\times D_9$

Order 81: $C_3^4$ x 13, $C_3\wr C_3$ x 8, $C_3\times \He_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$ x 2

Order 72: $S_3\times D_6$ x 11, $C_3:D_{12}$ x 8, $\SOPlus(4,2)$ x 4, $C_3\times S_4$ x 2, $C_3:D_{12}$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_6:D_6$, $C_2\times C_3^2:C_4$, $S_3\times A_4$, $C_3:S_4$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 88, $S_3\times C_3^2$ x 39, $C_3^2:S_3$ x 15, $C_3^2:C_6$ x 8, $C_9:C_6$ x 6, $C_3^2:S_3$ x 2, $C_3\times D_9$ x 2, $C_3^2\times C_6$ x 2, $C_3:D_9$, $S_3\times C_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 2, $C_2\times S_4$, $C_2\times D_{12}$

Order 36: $S_3^2$ x 50, $C_6\times S_3$ x 21, $C_6:S_3$ x 15, $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_3\times A_4$ x 2, $D_{18}$, $C_6^2$

Order 27: $C_3^3$ x 64, $\He_3$ x 8, $C_9:C_3$ x 5, $C_3\times C_9$ x 2

Order 24: $D_{12}$ x 8, $C_2\times D_6$ x 7, $C_3:D_4$ x 4, $S_4$ x 3, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $C_2\times A_4$

Order 18: $C_3\times S_3$ x 85, $C_3:S_3$ x 51, $C_3\times C_6$ x 13, $D_9$ x 3, $C_{18}$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 35, $C_2\times C_6$ x 6, $C_{12}$ x 6, $A_4$ x 2, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 68, $C_9$ x 2

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

Order 6: $S_3$ x 33, $C_6$ x 18

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

Order 3: $C_3$ x 15

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11664: $C_3^5:(C_2\times S_4)$ ($C_1$)

Order 5832: $C_3^5:S_4$ ($C_2$), $C_3\wr A_4:S_3$ ($C_2$), $C_3^5:S_4$ ($C_2$)

Order 2916: $C_3^5:A_4$ ($C_2^2$)

Order 1944: $C_3^2:S_3^3$ ($S_3$), $C_3^4:S_4$ ($S_3$)

Order 972: $C_3^3:C_6^2$ ($D_6$), $C_3\wr A_4$ ($D_6$)

Order 486: $C_3^4:C_6$ ($S_4$)

Order 324: $C_3\wr C_2^2$ ($S_3^2$)

Order 243: $C_3^5$ ($C_2\times S_4$)

Order 108: $C_3:S_3^2$ ($C_3^2:D_6$)

Order 81: $C_3^4$ ($S_3\times S_4$)

Order 27: $C_3^3$ ($C_6^2:D_6$)

Order 18: $C_3:S_3$ ($C_3^3:S_4$)

Order 9: $C_3^2$ ($C_2\times C_3^3:S_4$)

Order 3: $C_3$ ($S_3\times C_3^3:S_4$)

Order 1: $C_1$ ($C_3^5:(C_2\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11664: $C_3^5:(C_2\times S_4)$ ($C_1$)

Order 5832: $C_3^5:S_4$ ($C_2$), $C_3\wr A_4:S_3$ ($C_2$), $C_3^5:S_4$ ($C_2$)

Order 2916: $C_3^5:A_4$ ($C_2^2$)

Order 1944: $C_3^2:S_3^3$ ($S_3$), $C_3^4:S_4$ ($S_3$)

Order 972: $C_3^3:C_6^2$ ($D_6$), $C_3\wr A_4$ ($D_6$)

Order 486: $C_3^4:C_6$ ($S_4$)

Order 324: $C_3\wr C_2^2$ ($S_3^2$)

Order 243: $C_3^5$ ($C_2\times S_4$)

Order 108: $C_3:S_3^2$ ($C_3^2:D_6$)

Order 81: $C_3^4$ ($S_3\times S_4$)

Order 27: $C_3^3$ ($C_6^2:D_6$)

Order 18: $C_3:S_3$ ($C_3^3:S_4$)

Order 9: $C_3^2$ ($C_2\times C_3^3:S_4$)

Order 3: $C_3$ ($S_3\times C_3^3:S_4$)

Order 1: $C_1$ ($C_3^5:(C_2\times S_4)$)

Series

Derived series

$C_3^5:(C_2\times S_4)$

$\rhd$

$C_3^5:(C_2\times S_4)$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^5:(C_2\times S_4)$

$\rhd$

$C_3^5:(C_2\times S_4)$

$\rhd$

$C_3\wr A_4:S_3$

$\rhd$

$C_3\wr A_4:S_3$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3\wr A_4$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^5:(C_2\times S_4)$

$\rhd$

$C_3^5:(C_2\times S_4)$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 13 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 $57 \times 57$ 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.