Abstract group 7776.ij: $D_6\times C_3^3:S_4$

• Label: 7776.ij
• Order: \( 2^{5} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $14$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$D_6\times C_3^3:S_4$
Order:	\(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2\times C_3^3:C_2^2.D_6^2$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ x 5
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	367	296	1296	3440	432	648	1296	7776
Conjugacy classes	1	11	11	4	61	4	2	8	102
Divisions	1	11	9	4	45	2	2	4	78
Autjugacy classes	1	7	9	2	30	2	1	3	55

Dimension	1	2	3	4	6	8	12	16	24	32
Irr. complex chars.	8	8	8	18	20	16	16	4	4	0	102
Irr. rational chars.	8	8	8	2	12	8	16	8	6	2	78

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$3669120$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid c^{6}=d^{6}=e^{3}=f^{6}=[a,f]=[c,e]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(1,2,4,8,9,5,6,3,7)(11,12)(13,14), (1,3)(2,6)(4,5,7)(8,9)(11,12), (2,5,9,4)(3,7)(6,8)(10,11,12)\rangle$
Transitive group:	36T7284				more information
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_3^3:S_4)$
Semidirect product:	$(C_6:S_3^3)$ $\,\rtimes\,$ $S_3$	$(C_3:S_3^3)$ $\,\rtimes\,$ $D_6$ (2)	$(D_6\times C_3^3)$ $\,\rtimes\,$ $S_4$	$C_3^3$ $\,\rtimes\,$ $(D_6\times S_4)$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 63372 subgroups in 1640 conjugacy classes, 42 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $S_3\times C_3^3:S_4$
Commutator:	$G' \simeq$ $C_3\wr A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_6\times C_3^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_6$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $D_6\times C_3^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 7776: $D_6\times C_3^3:S_4$

Order 3888: $S_3\times C_3^3:S_4$ x 4, $C_2\times C_3^4:S_4$, $C_3^3:A_4\times D_6$, $C_2\times C_3\wr S_4$

Order 2592: $S_3^3:D_6$, $C_2^2\times C_3^3:S_4$

Order 1944: $S_3\times C_3^3:A_4$ x 2, $C_3^4:S_4$ x 2, $C_3\wr S_4$ x 2, $C_2\times C_3\wr A_4$, $C_3\wr S_3\times D_6$

Order 1296: $S_3^3:S_3$ x 8, $C_2\times C_3^3:S_4$ x 7, $C_6:S_3^3$, $C_6\times S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^4:D_4$, $C_2\times C_3^4:D_4$, $C_2\times C_3\wr D_4$, $C_3^3:C_4\times D_6$, $C_2^2\times C_3^3:A_4$

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

Order 864: $D_6^2:S_3$, $S_3^3:C_2^2$

Order 648: $C_3\times S_3^3$ x 6, $C_3^3:S_4$ x 6, $C_3:S_3^3$ x 4, $C_3^3:D_{12}$ x 4, $C_3^4:D_4$ x 4, $C_3^4:D_4$ x 4, $C_3\wr D_4$ x 4, $C_3^3:C_4\times S_3$ x 4, $C_2\times C_3^3:A_4$ x 4, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_2\times C_3\wr C_4$, $S_3\times C_3^2:D_6$, $C_2^2\times C_3\wr S_3$

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

Order 432: $S_3^2:D_6$ x 15, $S_3^3:C_2$ x 8, $C_3\times D_6^2$ x 2, $C_2\times S_3^3$ x 2, $C_3:D_6^2$, $C_2^2\times C_3^3:C_4$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$

Order 324: $C_3^3:D_6$ x 8, $C_3^2:S_3^2$ x 8, $C_3^2\times S_3^2$ x 7, $C_3\wr C_2^2$ x 5, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:C_6^2$ x 2, $D_6\times C_3^3$ x 2, $C_3^4:C_4$ x 2, $C_3\wr C_4$ x 2, $C_3^3:A_4$ x 2, $C_3^2.C_6^2$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$

Order 288: $D_6^2:C_2$, $C_6^2:D_4$, $D_6\times S_4$

Order 243: $C_3^4:C_3$

Order 216: $C_6\times S_3^2$ x 29, $S_3^2:S_3$ x 28, $S_3^3$ x 12, $C_6:S_3^2$ x 9, $C_2\times C_3^3:C_4$ x 8, $C_3^2:D_{12}$ x 4, $S_3^2:C_6$ x 4, $C_3^2:C_4\times S_3$ x 4, $S_3\times C_6^2$ x 3, $C_6:S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_2\times C_3^2:C_{12}$, $C_6^2:S_3$

Order 162: $C_3\wr S_3$ x 8, $C_3^3:C_6$ x 6, $C_3^2\wr C_2$ x 4, $S_3\times C_3^3$ x 4, $C_3^3:C_6$ x 2, $C_3^3:S_3$ x 2, $C_3^3:S_3$ x 2, $C_3^3.C_6$ x 2, $S_3\times \He_3$ x 2, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_6\times \He_3$

Order 144: $S_3^2:C_2^2$ x 12, $D_6:D_6$ x 8, $D_6^2$ x 4, $S_3\times S_4$ x 4, $C_6^2:C_2^2$ x 2, $C_6^2:C_4$, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $D_6:D_6$, $D_6.D_6$

Order 108: $C_3\times S_3^2$ x 55, $C_3^2\times D_6$ x 29, $C_3^2:D_6$ x 20, $C_3:S_3^2$ x 14, $C_3:S_3^2$ x 12, $C_3^3:C_4$ x 8, $C_3^2:D_6$ x 8, $C_3\times C_6^2$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $C_{18}:C_6$, $C_2^2\times \He_3$, $S_3\times C_{18}$

Order 96: $C_2^2\times S_4$, $C_2^3:D_6$, $D_4\times D_6$

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

Order 72: $S_3\times D_6$ x 41, $C_6\times D_6$ x 34, $\SOPlus(4,2)$ x 16, $C_2\times C_3^2:C_4$ x 6, $C_6^2:C_2$ x 4, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $D_6:S_3$ x 4, $C_6.D_6$ x 4, $C_6:D_6$ x 3, $C_2\times C_6^2$ x 2, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_3\times S_4$ x 2, $C_6\times A_4$, $C_6.D_6$, $C_6:C_{12}$

Order 54: $S_3\times C_3^2$ x 34, $C_3^2:C_6$ x 24, $C_3^2\times C_6$ x 12, $C_3^2:S_3$ x 8, $C_9:C_6$ x 5, $C_2\times \He_3$ x 5, $S_3\times C_9$ x 2, $C_3^2:S_3$ x 2, $C_3\times C_{18}$

Order 48: $C_6:D_4$ x 15, $S_3\times D_4$ x 8, $C_2\times S_4$ x 7, $C_2^2\times D_6$ x 4, $C_2^3\times C_6$, $C_2^2\times A_4$, $C_6\times D_4$, $C_6.C_2^3$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $C_6\times S_3$ x 114, $S_3^2$ x 56, $C_6^2$ x 20, $C_6:S_3$ x 19, $C_3^2:C_4$ x 4, $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2, $C_2\times C_{18}$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

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

Order 24: $C_2\times D_6$ x 42, $C_3:D_4$ x 28, $C_2^2\times C_6$ x 16, $C_6:C_4$ x 8, $S_4$ x 6, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_2\times A_4$ x 4, $C_3\times D_4$ x 4, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 72, $C_3\times C_6$ x 43, $C_3:S_3$ x 16, $C_{18}$ x 4

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 82, $C_2\times C_6$ x 52, $C_3:C_4$ x 8, $A_4$ x 2, $C_{12}$ x 2

Order 9: $C_3^2$ x 17, $C_9$ x 2

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 45, $S_3$ x 30

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$ x 9

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 7776: $D_6\times C_3^3:S_4$

Order 3888: $C_2\times C_3^4:S_4$, $C_3^3:A_4\times D_6$, $C_2\times C_3\wr S_4$, $S_3\times C_3^3:S_4$

Order 2592: $S_3^3:D_6$, $C_2^2\times C_3^3:S_4$

Order 1944: $C_2\times C_3\wr A_4$, $S_3\times C_3^3:A_4$, $C_3^4:S_4$, $C_3\wr S_4$, $C_3\wr S_3\times D_6$

Order 1296: $C_2\times C_3^3:S_4$ x 4, $S_3^3:S_3$ x 2, $C_6:S_3^3$, $C_6\times S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^4:D_4$, $C_2\times C_3^4:D_4$, $C_2\times C_3\wr D_4$, $C_3^3:C_4\times D_6$, $C_2^2\times C_3^3:A_4$

Order 972: $C_3\wr A_4$, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^4:D_6$, $C_3^3:S_3^2$

Order 864: $D_6^2:S_3$, $S_3^3:C_2^2$

Order 648: $C_2\times C_3^3:A_4$ x 3, $C_3^3:S_4$ x 3, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3^4:C_2^3$ x 2, $C_3:S_3^3$ x 2, $C_3\times S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3\wr D_4$ x 2, $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_2\times C_3\wr C_4$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:C_4\times S_3$, $S_3\times C_3^2:D_6$, $C_2^2\times C_3\wr S_3$

Order 486: $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $S_3^2:D_6$ x 9, $C_3\times D_6^2$ x 2, $C_2\times S_3^3$ x 2, $S_3^3:C_2$ x 2, $C_3:D_6^2$, $C_2^2\times C_3^3:C_4$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$

Order 324: $C_3^3:D_6$ x 5, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2\times S_3^2$ x 3, $C_3^2:C_6^2$ x 2, $D_6\times C_3^3$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:A_4$ x 2, $C_3^2.C_6^2$, $C_3^3:D_6$, $C_3^4:C_4$, $C_3\wr C_4$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $C_3^2:S_3^2$

Order 288: $D_6^2:C_2$, $C_6^2:D_4$, $D_6\times S_4$

Order 243: $C_3^4:C_3$

Order 216: $C_6\times S_3^2$ x 19, $S_3^2:S_3$ x 10, $C_6:S_3^2$ x 7, $C_2\times C_3^3:C_4$ x 5, $S_3^3$ x 5, $S_3\times C_6^2$ x 3, $C_6:S_3^2$ x 3, $C_6^2:C_6$ x 2, $S_3^2:C_6$ x 2, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$, $C_3^2:C_4\times S_3$, $C_6^2:S_3$

Order 162: $C_3^3:C_6$ x 5, $C_3\wr S_3$ x 4, $C_3^2\wr C_2$ x 3, $S_3\times C_3^3$ x 2, $C_3^3\times C_6$, $C_3^3:C_6$, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^3:S_3$, $C_3^3:S_3$, $C_3^3.C_6$, $S_3\times \He_3$

Order 144: $S_3^2:C_2^2$ x 6, $D_6^2$ x 4, $C_6^2:C_2^2$ x 2, $D_6:D_6$ x 2, $C_6^2:C_4$, $A_4\times D_6$, $C_6:S_4$, $C_6\times S_4$, $S_3\times S_4$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $D_6:D_6$, $D_6.D_6$

Order 108: $C_3\times S_3^2$ x 24, $C_3^2\times D_6$ x 21, $C_3^2:D_6$ x 15, $C_3:S_3^2$ x 10, $C_3:S_3^2$ x 5, $C_3^2:D_6$ x 5, $C_3^3:C_4$ x 4, $C_3\times C_6^2$ x 2, $C_3^2:D_6$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $S_3\times C_{18}$

Order 96: $C_2^2\times S_4$, $C_2^3:D_6$, $D_4\times D_6$

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

Order 72: $S_3\times D_6$ x 25, $C_6\times D_6$ x 24, $\SOPlus(4,2)$ x 6, $C_6:D_6$ x 3, $C_2\times C_3^2:C_4$ x 3, $C_2\times C_6^2$ x 2, $C_6\wr C_2$ x 2, $D_6:S_3$ x 2, $C_6\times A_4$, $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$, $C_6^2:C_2$, $C_6.D_6$, $C_6:C_{12}$, $C_3:D_{12}$, $C_6.D_6$

Order 54: $S_3\times C_3^2$ x 19, $C_3^2:C_6$ x 15, $C_3^2\times C_6$ x 10, $C_3^2:S_3$ x 4, $C_9:C_6$ x 4, $C_2\times \He_3$ x 4, $C_3\times C_{18}$, $S_3\times C_9$, $C_3^2:S_3$

Order 48: $C_6:D_4$ x 9, $C_2^2\times D_6$ x 4, $C_2\times S_4$ x 4, $S_3\times D_4$ x 2, $C_2^3\times C_6$, $C_2^2\times A_4$, $C_6\times D_4$, $C_6.C_2^3$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $C_6\times S_3$ x 66, $S_3^2$ x 26, $C_6^2$ x 16, $C_6:S_3$ x 13, $C_3^2:C_4$ x 2, $C_3^2:C_4$, $C_3:C_{12}$, $C_2\times C_{18}$, $C_3\times A_4$

Order 32: $C_2^2\times D_4$

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

Order 24: $C_2\times D_6$ x 27, $C_2^2\times C_6$ x 12, $C_3:D_4$ x 10, $C_6:C_4$ x 5, $C_2\times A_4$ x 3, $S_4$ x 3, $C_3\times D_4$ x 2, $C_2\times C_{12}$, $D_{12}$, $C_4\times S_3$

Order 18: $C_3\times S_3$ x 42, $C_3\times C_6$ x 31, $C_3:S_3$ x 9, $C_{18}$ x 3

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

Order 12: $D_6$ x 45, $C_2\times C_6$ x 34, $C_3:C_4$ x 4, $A_4$ x 2, $C_{12}$

Order 9: $C_3^2$ x 17, $C_9$ x 2

Order 8: $C_2^3$ x 11, $D_4$ x 6, $C_2\times C_4$ x 3

Order 6: $C_6$ x 30, $S_3$ x 17

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7776: $D_6\times C_3^3:S_4$ ($C_1$)

Order 3888: $S_3\times C_3^3:S_4$ ($C_2$) x 4, $C_2\times C_3^4:S_4$ ($C_2$), $C_3^3:A_4\times D_6$ ($C_2$), $C_2\times C_3\wr S_4$ ($C_2$)

Order 1944: $S_3\times C_3^3:A_4$ ($C_2^2$) x 2, $C_3^4:S_4$ ($C_2^2$) x 2, $C_3\wr S_4$ ($C_2^2$) x 2, $C_2\times C_3\wr A_4$ ($C_2^2$)

Order 1296: $C_6:S_3^3$ ($S_3$), $C_2\times C_3^3:S_4$ ($S_3$)

Order 972: $C_3\wr A_4$ ($C_2^3$)

Order 648: $C_3:S_3^3$ ($D_6$) x 2, $C_3^3:S_4$ ($D_6$) x 2, $C_3^4:C_2^3$ ($D_6$), $C_2\times C_3^3:A_4$ ($D_6$)

Order 324: $D_6\times C_3^3$ ($S_4$), $C_3\wr C_2^2$ ($C_2\times D_6$), $C_3^3:A_4$ ($C_2\times D_6$)

Order 216: $C_6:S_3^2$ ($S_3^2$)

Order 162: $S_3\times C_3^3$ ($C_2\times S_4$) x 2, $C_3^3\times C_6$ ($C_2\times S_4$)

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

Order 81: $C_3^4$ ($C_2^2\times S_4$)

Order 54: $C_3^2\times C_6$ ($S_3\times S_4$)

Order 27: $C_3^3$ ($D_6\times S_4$)

Order 12: $D_6$ ($C_3^3:S_4$)

Order 6: $S_3$ ($C_2\times C_3^3:S_4$) x 2, $C_6$ ($C_2\times C_3^3:S_4$)

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

Order 2: $C_2$ ($S_3\times C_3^3:S_4$)

Order 1: $C_1$ ($D_6\times C_3^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7776: $D_6\times C_3^3:S_4$ ($C_1$)

Order 3888: $C_2\times C_3^4:S_4$ ($C_2$), $C_3^3:A_4\times D_6$ ($C_2$), $C_2\times C_3\wr S_4$ ($C_2$), $S_3\times C_3^3:S_4$ ($C_2$)

Order 1944: $C_2\times C_3\wr A_4$ ($C_2^2$), $S_3\times C_3^3:A_4$ ($C_2^2$), $C_3^4:S_4$ ($C_2^2$), $C_3\wr S_4$ ($C_2^2$)

Order 1296: $C_6:S_3^3$ ($S_3$), $C_2\times C_3^3:S_4$ ($S_3$)

Order 972: $C_3\wr A_4$ ($C_2^3$)

Order 648: $C_3^4:C_2^3$ ($D_6$), $C_3:S_3^3$ ($D_6$), $C_2\times C_3^3:A_4$ ($D_6$), $C_3^3:S_4$ ($D_6$)

Order 324: $D_6\times C_3^3$ ($S_4$), $C_3\wr C_2^2$ ($C_2\times D_6$), $C_3^3:A_4$ ($C_2\times D_6$)

Order 216: $C_6:S_3^2$ ($S_3^2$)

Order 162: $C_3^3\times C_6$ ($C_2\times S_4$), $S_3\times C_3^3$ ($C_2\times S_4$)

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

Order 81: $C_3^4$ ($C_2^2\times S_4$)

Order 54: $C_3^2\times C_6$ ($S_3\times S_4$)

Order 27: $C_3^3$ ($D_6\times S_4$)

Order 12: $D_6$ ($C_3^3:S_4$)

Order 6: $C_6$ ($C_2\times C_3^3:S_4$), $S_3$ ($C_2\times C_3^3:S_4$)

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

Order 2: $C_2$ ($S_3\times C_3^3:S_4$)

Order 1: $C_1$ ($D_6\times C_3^3:S_4$)

Series

Derived series

$D_6\times C_3^3:S_4$

$\rhd$

$C_3\wr A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$D_6\times C_3^3:S_4$

$\rhd$

$S_3\times C_3^3:S_4$

$\rhd$

$S_3\times C_3^3:A_4$

$\rhd$

$C_3\wr A_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$D_6\times C_3^3:S_4$

$\rhd$

$C_3\wr A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 6 larger groups in the database.

Character theory

Complex character table

See the $102 \times 102$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $78 \times 78$ rational character table.