Abstract group 31104.mi: $C_2\times C_3^3:S_4^2$

• Label: 31104.mi
• Order: \( 2^{7} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_2\times C_3^3:S_4^2$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_2\times C_3^3.A_4^2.C_2^4$, of order \(124416\)\(\medspace = 2^{9} \cdot 3^{5} \)
Composition factors:	$C_2$ x 7, $C_3$ x 5
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	919	890	5736	10502	1296	6144	3888	1728	31104
Conjugacy classes	1	17	11	14	83	4	24	12	4	170
Divisions	1	17	9	14	61	2	18	6	2	130
Autjugacy classes	1	12	9	7	43	2	9	5	1	89

Dimension	1	2	3	4	6	8	9	12	16	18	24	32	36	48
Irr. complex chars.	8	8	16	18	24	16	8	32	4	16	12	0	8	0	170
Irr. rational chars.	8	8	16	2	16	8	8	16	8	8	14	2	12	4	130

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{6}=d^{6}=e^{6}=f^{2}=g^{6}=[a,f]=[b,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(10,11,12)(13,14,15), (7,11,8,10)(9,12)(13,15), (7,14,11)(8,13,10)(9,15,12) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $S_4$ $\, \times\, $ $(C_3^3:S_4)$
Semidirect product:	$(C_6^3:S_4)$ $\,\rtimes\,$ $S_3$	$(C_6^3:S_3)$ $\,\rtimes\,$ $S_4$	$C_6^3$ $\,\rtimes\,$ $(S_3\times S_4)$	$(C_6^3:A_4)$ $\,\rtimes\,$ $D_6$	all 34
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 634620 subgroups in 7740 conjugacy classes, 53 normal (33 characteristic).

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

Special subgroups

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

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

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

Subgroup information Click on a subgroup in the diagram to see information about it.

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

All subgroups of index up to 128 (order at least 243) are shown, as well as all normal subgroups of any index.

Order 31104: $C_2\times C_3^3:S_4^2$

Order 15552: $C_3^3:S_4^2$ x 4, $C_6^3:(S_3\times A_4)$, $C_6^3:C_3:S_4$, $C_6^3:(C_3\times S_4)$

Order 10368: $S_3^2:D_6\times S_4$, $C_6^3:(C_2\times S_4)$

Order 7776: $C_3^3:A_4\times S_4$ x 2, $(A_4\times C_3^3):S_4$ x 2, $A_4\times C_3^3:S_4$ x 2, $C_2\times C_3^3:A_4^2$, $C_6^3:S_3^2$, $D_6\times C_3^3:S_4$

Order 5184: $D_6^2:S_3^2$ x 8, $D_4\times C_3^3:S_4$ x 8, $C_6^3:S_4$ x 2, $C_6^3:S_4$ x 2, $C_{12}:S_3^2:D_6$, $C_6^3:(C_2\times A_4)$, $C_6^3:(C_4\times S_3)$, $C_6:S_3^2:D_{12}$, $C_2\times C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_6^2$, $C_6^3:D_{12}$, $A_4\times S_3^2:D_6$, $C_2\times C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_6^2$, $C_6^3:S_4$

Order 3888: $C_3\wr S_3\times S_4$ x 4, $S_3\times C_3^3:S_4$ x 4, $A_4\times C_3^3:D_6$, $C_3^3:C_6\times S_4$, $\He_3:(C_6\times S_4)$, $C_3^3:A_4^2$, $C_2\times C_3^4:S_4$, $C_3^3:A_4\times D_6$, $C_2\times C_3\wr S_4$

Order 3456: $C_2\times D_6^2:D_6$, $C_2\times D_6^2:D_6$

Order 2592: $C_2^2\times C_3^3:S_4$ x 15, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_3\times S_4\times S_3^2$ x 6, $C_3^3:C_4\times S_4$ x 4, $(C_3\times C_6^2):D_{12}$ x 4, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 4, $S_4\times C_3:S_3^2$ x 4, $A_4\times S_3^2:S_3$ x 4, $(C_3^2\times C_{12}):S_4$ x 4, $C_4\times C_3^3:S_4$ x 4, $D_4\times C_3^3:A_4$ x 4, $C_6^3:A_4$ x 2, $(C_3\times C_6^2):S_4$ x 2, $C_6:S_3^2:C_{12}$, $C_6^3:C_3:C_4$, $C_6^3:C_{12}$, $C_6^3:D_6$, $A_4\times C_6:S_3^2$, $C_6^3:D_6$, $C_6^3:D_6$, $C_3^2:D_6\times S_4$, $C_6\times A_4\times S_3^2$, $C_6^3:D_6$, $(C_3\times C_6^2).S_4$, $C_6^3:D_6$, $C_6^3:D_6$, $S_3^3:D_6$, $C_6^3:A_4$

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, $A_4\times C_3\wr S_3$ x 2, $C_3\wr C_3:S_4$ x 2, $C_3\wr C_3\times S_4$ x 2, $C_2\times C_3\wr A_4$, $C_6^3:C_3^2$, $C_3\wr S_3\times D_6$

Order 1728: $D_6^2:D_6$ x 16, $D_6^2:D_6$ x 8, $D_6^2:D_6$ x 3, $C_6^3:D_4$ x 2, $A_4:D_6^2$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6\times D_6\times S_4$, $C_2\times D_6^2:C_6$, $C_2\times C_6^2:D_{12}$, $C_6^2:(C_4\times D_6)$, $C_{12}:D_6^2$, $C_{12}:D_6^2$, $C_6^3:(C_2\times C_4)$, $D_6^2.D_6$, $D_6^2.D_6$, $D_6^2.D_6$, $D_6^2.D_6$

Order 1296: $C_2\times C_3^3:S_4$ x 23, $S_3^3:S_3$ x 8, $C_3\wr S_3\times D_4$ x 8, $C_2^2\times C_3^3:A_4$ x 7, $C_6^2:S_3^2$ x 4, $C_6^2:S_3^2$ x 4, $C_6^2:S_3^2$ x 4, $C_6^2:S_3^2$ x 4, $C_6^2:C_6^2$ x 3, $C_6^2:S_3^2$ x 3, $C_6^2:S_3^2$ x 3, $A_4\times C_3:S_3^2$ x 2, $A_4\times C_3^3:C_4$ x 2, $(A_4\times C_3^3):C_4$ x 2, $C_4\times C_3^3:A_4$ x 2, $(C_3^2\times C_6).S_4$ x 2, $C_6^3:S_3$ x 2, $C_6^3:S_3$ x 2, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^3:S_3$, $C_6^3:S_3$, $C_6:S_3^3$, $C_6\times S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^4:D_4$, $C_2\times C_3^4:D_4$, $C_2\times C_3\wr D_4$, $C_3^3:C_4\times D_6$, $(C_3\times C_6^2):A_4$, $C_3^2:D_6\times A_4$, $C_2\times \He_3:S_4$, $C_9:C_6\times S_4$, $C_2\times S_4\times \He_3$, $(C_3^2\times A_4):D_6$, $C_6\wr S_3$, $C_6^3:C_6$, $C_2\times C_3^3:D_{12}$, $C_4\times C_3^3:D_6$

Order 1152: $C_2\times S_4^2$, $C_2^5:S_3^2$, $D_6^2:C_2^3$

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

Order 864: $D_6^2:S_3$ x 33, $C_6\times S_3\times S_4$ x 17, $(C_3\times C_6^2):D_4$ x 16, $D_6^2:S_3$ x 16, $(S_3\times D_6):D_6$ x 16, $S_4\times S_3^2$ x 12, $S_3^2:S_4$ x 12, $D_6^2:C_6$ x 12, $C_6.D_6^2$ x 8, $C_3^3:C_4\times D_4$ x 8, $C_{12}:\SOPlus(4,2)$ x 8, $S_3^2:D_{12}$ x 8, $C_{12}:\SOPlus(4,2)$ x 8, $C_4\times S_3^2:S_3$ x 8, $D_6^2:C_6$ x 4, $C_6^2:D_{12}$ x 4, $C_3^2:C_4\times S_4$ x 4, $D_6^2:C_6$ x 3, $C_2\times A_4:S_3^2$ x 3, $D_6^2.S_3$ x 3, $(C_3\times C_6^2).D_4$ x 3, $C_6:D_6^2$ x 2, $C_6\times D_6^2$ x 2, $C_6^3:C_4$ x 2, $S_3\times C_6:S_4$ x 2, $C_6^2:C_6:C_4$ x 2, $C_6^3:C_2^2$ x 2, $D_6^2:C_6$ x 2, $D_6^2:C_6$ x 2, $C_6^3:C_4$ x 2, $C_6\times A_4\times D_6$, $C_6^2:S_4$, $S_4\times C_6^2$, $C_6:S_3\times S_4$, $C_2\times C_6^2:C_{12}$, $S_3^3:C_2^2$, $C_6^3:C_2^2$, $D_{12}:C_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $D_6^2:C_6$, $D_6^2:C_6$, $C_6.D_6^2$, $C_6:S_3.D_{12}$, $C_2\times C_3^3:C_4^2$, $(C_3\times C_6^2).D_4$, $C_2\times C_6^2:D_6$

Order 648: $C_2^2\times C_3\wr S_3$ x 15, $C_3^3:S_4$ x 10, $(C_3\times C_6^2):S_3$ x 8, $C_2\times C_3^3:A_4$ x 7, $C_3\times S_3^3$ 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, $C_3\wr C_3\times D_4$ x 4, $C_3^3:D_{12}$ x 4, $C_4\times C_3\wr S_3$ 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, $C_3\times C_6^2:C_6$ x 2, $S_3\times C_3^2\times A_4$ x 2, $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $S_4\times C_3^3$ x 2, $A_4\times C_3^2:S_3$ x 2, $\He_3:S_4$ x 2, $C_9:C_3\times S_4$ x 2, $S_4\times \He_3$ x 2, $C_6^3:C_3$ x 2, $(C_3^2\times A_4):S_3$ x 2, $C_3^3:S_4$ x 2, $C_6^3:C_3$, $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_2\times C_3\wr C_4$, $C_9:C_6\times A_4$, $C_2\times A_4\times \He_3$, $S_3\times C_3^2:D_6$, $C_6\wr C_3$, $(C_3^2\times A_4):C_6$, $C_2\times \He_3:A_4$, $C_2\times C_3^3:C_{12}$, $C_2\times \He_3:C_{12}$

Order 576: $C_3^2:D_4^2$ x 16, $C_2^4:S_3^2$ x 8, $S_4^2$ x 4, $D_6^2:C_2^2$ x 4, $D_6^2:C_2^2$ x 2, $C_2^2:D_6^2$ x 2, $A_4^2:C_2^2$ x 2, $C_4:D_6^2$ x 2, $D_6^2.C_2^2$ x 2, $D_6^2.C_2^2$ x 2, $C_2^2\times C_6\times S_4$, $A_4^2:C_2^2$, $C_6^2:C_2^4$, $C_6^2:(C_2^2\times C_4)$, $C_6:D_4\times A_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_2^4.S_3^2$

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 102, $C_6^2:D_6$ x 36, $C_3\times D_6^2$ x 24, $C_2^2\times C_3^3:C_4$ x 16, $C_6^2:D_6$ x 16, $C_3:D_6^2$ x 15, $C_6\times S_3\times A_4$ x 13, $C_6^2:D_6$ x 12, $C_6^2:D_6$ x 12, $C_6^2:D_6$ x 12, $(C_3\times S_3^2):C_4$ x 12, $C_6.\SOPlus(4,2)$ x 12, $A_4:C_6^2$ x 10, $A_4\times S_3^2$ x 9, $A_4:S_3^2$ x 9, $A_4:S_3^2$ x 8, $S_3^3:C_2$ x 8, $C_6^2:D_6$ x 8, $C_{12}:C_6^2$ x 8, $C_{12}:S_3^2$ x 8, $(C_3\times C_6^2):C_4$ x 8, $C_6^2:D_6$ x 8, $C_{12}:S_3^2$ x 6, $C_{12}:S_3^2$ x 6, $C_{12}\times S_3^2$ x 6, $C_6^2:D_6$ x 4, $(C_3^2\times A_4):C_4$ x 4, $C_{12}:S_3^2$ x 4, $(C_3^2\times C_{12}):C_4$ x 4, $C_3^3:C_4^2$ x 4, $C_6.\SOPlus(4,2)$ x 4, $C_6^3:C_2$ x 2, $D_6\times C_6^2$ x 2, $C_6:S_3\times A_4$ x 2, $C_2\times S_3^3$ x 2, $C_6^2:C_{12}$ x 2, $C_6^3:C_2$ x 2, $D_6:C_6^2$ x 2, $C_6^2.D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $A_4\times C_6^2$, $C_6^2:D_6$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_{12}:C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{18}\times S_4$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_6:D_4^2$

Order 324: $C_3^3:D_6$ x 23, $C_3^2:S_3^2$ x 8, $C_3^2.C_6^2$ x 7, $C_3^2\times S_3^2$ 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, $C_3^3:A_4$ x 3, $C_3^3:C_{12}$ x 2, $C_3^2:C_6^2$ x 2, $D_6\times C_3^3$ x 2, $C_3^4:C_4$ x 2, $C_3\wr C_4$ x 2, $\He_3:C_{12}$ x 2, $C_3^3:A_4$, $C_6^2.C_3^2$, $\He_3:A_4$, $C_3^3:D_6$, $A_4\times C_3^3$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $C_6^2.C_3^2$, $A_4\times \He_3$

Order 288: $C_6^2:D_4$ x 33, $D_6\wr C_2$ x 32, $C_6^2:C_2^3$ x 28, $D_4\times S_3^2$ x 24, $D_6\times S_4$ x 24, $C_6^2:D_4$ x 16, $S_3^2:D_4$ x 16, $C_2\times C_6\times S_4$ x 13, $C_3^2:C_4\times D_4$ x 8, $C_4:\SOPlus(4,2)$ x 8, $C_4\times \SOPlus(4,2)$ x 8, $C_6^2.D_4$ x 6, $D_6^2:C_2$ x 5, $D_6^2:C_2$ x 4, $A_4\times C_3:D_4$ x 4, $C_3:\GL(2,\mathbb{Z}/4)$ x 4, $D_6:S_4$ x 4, $C_2^3.S_3^2$ x 4, $C_2^3.S_3^2$ x 4, $C_2\times D_6^2$ x 4, $A_4\times S_4$ x 4, $C_2\times A_4\times D_6$ x 3, $D_{12}:D_6$ x 2, $D_6\times D_{12}$ x 2, $C_2.D_6^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_6^2:C_2^3$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_2\times C_6:S_4$ x 2, $\PSOPlus(4,3)$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $C_6^2.C_2^3$, $(C_6\times C_{12}):C_4$, $C_2\times C_3^2:C_4^2$, $C_6:C_4\times A_4$, $C_2\times C_6.S_4$, $C_6^2.D_4$, $C_2^3:C_6^2$, $C_2\times A_4^2$, $D_4\times C_6^2$, $C_6^2:C_2^3$

Order 243: $C_3^4:C_3$

Order 216: $C_6^3$, $C_6:S_3^2$

Order 144: $S_3\times S_4$ x 8

Order 128: $C_2\times D_4^2$

Order 108: $C_3\times C_6^2$, $C_3:S_3^2$

Order 96: $C_2^2\times S_4$ x 9

Order 72: $S_3\times D_6$

Order 54: $C_3^2\times C_6$

Order 48: $C_2\times S_4$ x 41

Order 36: $S_3^2$ x 4

Order 27: $C_3^3$

Order 24: $S_4$ x 24, $C_2\times D_6$ x 9, $C_2\times A_4$

Order 12: $D_6$ x 39, $A_4$

Order 8: $C_2^3$ x 17

Order 6: $S_3$ x 20

Order 4: $C_2^2$ x 41

Order 2: $C_2$ x 14

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 243) are shown, as well as all normal subgroups of any index.

Order 31104: $C_2\times C_3^3:S_4^2$

Order 15552: $C_6^3:(S_3\times A_4)$, $C_6^3:C_3:S_4$, $C_6^3:(C_3\times S_4)$, $C_3^3:S_4^2$

Order 10368: $S_3^2:D_6\times S_4$, $C_6^3:(C_2\times S_4)$

Order 7776: $C_2\times C_3^3:A_4^2$, $C_6^3:S_3^2$, $C_3^3:A_4\times S_4$, $(A_4\times C_3^3):S_4$, $A_4\times C_3^3:S_4$, $D_6\times C_3^3:S_4$

Order 5184: $C_6^3:S_4$ x 2, $D_6^2:S_3^2$ x 2, $C_6^3:S_4$ x 2, $D_4\times C_3^3:S_4$ x 2, $C_{12}:S_3^2:D_6$, $C_6^3:(C_2\times A_4)$, $C_6^3:(C_4\times S_3)$, $C_6:S_3^2:D_{12}$, $C_2\times C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_6^2$, $C_6^3:D_{12}$, $A_4\times S_3^2:D_6$, $C_2\times C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_6^2$, $C_6^3:S_4$

Order 3888: $A_4\times C_3^3:D_6$, $C_3^3:C_6\times S_4$, $\He_3:(C_6\times S_4)$, $C_3\wr S_3\times S_4$, $C_3^3:A_4^2$, $C_2\times C_3^4:S_4$, $C_3^3:A_4\times D_6$, $C_2\times C_3\wr S_4$, $S_3\times C_3^3:S_4$

Order 3456: $C_2\times D_6^2:D_6$, $C_2\times D_6^2:D_6$

Order 2592: $C_2^2\times C_3^3:S_4$ x 9, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 3, $C_6^3:A_4$ x 2, $C_3\times S_4\times S_3^2$ x 2, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 2, $S_4\times C_3:S_3^2$ x 2, $A_4\times S_3^2:S_3$ x 2, $D_4\times C_3^3:A_4$ x 2, $C_6:S_3^2:C_{12}$, $C_6^3:C_3:C_4$, $C_6^3:C_{12}$, $C_6^3:D_6$, $A_4\times C_6:S_3^2$, $C_3^3:C_4\times S_4$, $C_6^3:D_6$, $C_6^3:D_6$, $C_3^2:D_6\times S_4$, $(C_3\times C_6^2):D_{12}$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6\times A_4\times S_3^2$, $C_6^3:D_6$, $(C_3\times C_6^2).S_4$, $C_6^3:D_6$, $C_6^3:D_6$, $S_3^3:D_6$, $(C_3^2\times C_{12}):S_4$, $C_4\times C_3^3:S_4$, $(C_3\times C_6^2):S_4$, $C_6^3:A_4$

Order 1944: $C_2\times C_3\wr A_4$, $S_3\times C_3^3:A_4$, $C_3^4:S_4$, $C_3\wr S_4$, $C_6^3:C_3^2$, $C_3\wr S_3\times D_6$, $A_4\times C_3\wr S_3$, $C_3\wr C_3:S_4$, $C_3\wr C_3\times S_4$

Order 1728: $D_6^2:D_6$ x 4, $D_6^2:D_6$ x 3, $C_6^3:D_4$ x 2, $A_4:D_6^2$ x 2, $D_6^2:D_6$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6\times D_6\times S_4$, $C_2\times D_6^2:C_6$, $C_2\times C_6^2:D_{12}$, $C_6^2:(C_4\times D_6)$, $C_{12}:D_6^2$, $C_{12}:D_6^2$, $C_6^3:(C_2\times C_4)$, $D_6^2.D_6$, $D_6^2.D_6$, $D_6^2.D_6$, $D_6^2.D_6$

Order 1296: $C_2\times C_3^3:S_4$ x 13, $C_2^2\times C_3^3:A_4$ x 6, $A_4\times C_3:S_3^2$ x 2, $C_6^2:C_6^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $S_3^3:S_3$ x 2, $C_6^3:S_3$ x 2, $C_6^3:S_3$ x 2, $C_3\wr S_3\times D_4$ x 2, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^3:S_3$, $C_6^3:S_3$, $C_6:S_3^3$, $C_6\times S_3^3$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^4:D_4$, $C_2\times C_3^4:D_4$, $C_2\times C_3\wr D_4$, $C_3^3:C_4\times D_6$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $(C_3\times C_6^2):A_4$, $A_4\times C_3^3:C_4$, $(A_4\times C_3^3):C_4$, $C_4\times C_3^3:A_4$, $(C_3^2\times C_6).S_4$, $C_3^2:D_6\times A_4$, $C_2\times \He_3:S_4$, $C_9:C_6\times S_4$, $C_2\times S_4\times \He_3$, $C_6^2:S_3^2$, $(C_3^2\times A_4):D_6$, $C_6\wr S_3$, $C_6^3:C_6$, $C_2\times C_3^3:D_{12}$, $C_4\times C_3^3:D_6$

Order 1152: $C_2\times S_4^2$, $C_2^5:S_3^2$, $D_6^2:C_2^3$

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

Order 864: $D_6^2:S_3$ x 21, $C_6\times S_3\times S_4$ x 11, $(S_3\times D_6):D_6$ x 6, $S_4\times S_3^2$ x 5, $S_3^2:S_4$ x 4, $C_6.D_6^2$ x 4, $D_6^2:C_6$ x 4, $(C_3\times C_6^2):D_4$ x 4, $D_6^2:S_3$ x 4, $D_6^2:C_6$ x 3, $C_2\times A_4:S_3^2$ x 3, $D_6^2.S_3$ x 3, $(C_3\times C_6^2).D_4$ x 3, $C_6:D_6^2$ x 2, $C_6\times D_6^2$ x 2, $C_6^3:C_4$ x 2, $S_3\times C_6:S_4$ x 2, $C_6^2:C_6:C_4$ x 2, $D_6^2:C_6$ x 2, $C_6^3:C_2^2$ x 2, $D_6^2:C_6$ x 2, $D_6^2:C_6$ x 2, $C_6^3:C_4$ x 2, $C_3^3:C_4\times D_4$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $S_3^2:D_{12}$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $C_4\times S_3^2:S_3$ x 2, $C_6\times A_4\times D_6$, $C_6^2:S_4$, $S_4\times C_6^2$, $C_6:S_3\times S_4$, $C_2\times C_6^2:C_{12}$, $S_3^3:C_2^2$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$, $C_6^3:C_2^2$, $D_{12}:C_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $D_6^2:C_6$, $D_6^2:C_6$, $C_6.D_6^2$, $C_6:S_3.D_{12}$, $C_2\times C_3^3:C_4^2$, $(C_3\times C_6^2).D_4$, $C_2\times C_6^2:D_6$

Order 648: $C_2^2\times C_3\wr S_3$ x 9, $C_2\times C_3^3:A_4$ x 6, $C_3^3:S_4$ x 5, $(C_3\times C_6^2):S_3$ 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, $C_3\times C_6^2:C_6$ 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, $C_6^3:C_3$ x 2, $C_3\wr C_3\times D_4$ x 2, $C_6^3:C_3$, $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_2\times C_3\wr C_4$, $S_3\times C_3^2\times A_4$, $C_3^3:S_4$, $C_3^3:S_4$, $S_4\times C_3^3$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:C_4\times S_3$, $C_9:C_6\times A_4$, $C_2\times A_4\times \He_3$, $S_3\times C_3^2:D_6$, $A_4\times C_3^2:S_3$, $\He_3:S_4$, $C_9:C_3\times S_4$, $S_4\times \He_3$, $C_6\wr C_3$, $(C_3^2\times A_4):C_6$, $C_2\times \He_3:A_4$, $(C_3^2\times A_4):S_3$, $C_3^3:S_4$, $C_2\times C_3^3:C_{12}$, $C_2\times \He_3:C_{12}$, $C_3^3:D_{12}$, $C_4\times C_3\wr S_3$

Order 576: $D_6^2:C_2^2$ x 4, $C_3^2:D_4^2$ x 4, $D_6^2:C_2^2$ x 2, $C_2^2:D_6^2$ x 2, $A_4^2:C_2^2$ x 2, $C_4:D_6^2$ x 2, $D_6^2.C_2^2$ x 2, $D_6^2.C_2^2$ x 2, $C_2^4:S_3^2$ x 2, $C_2^2\times C_6\times S_4$, $A_4^2:C_2^2$, $S_4^2$, $C_6^2:C_2^4$, $C_6^2:(C_2^2\times C_4)$, $C_6:D_4\times A_4$, $C_6:\GL(2,\mathbb{Z}/4)$, $D_6^2.C_2^2$, $D_6^2.C_2^2$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_2^4.S_3^2$

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 51, $C_3\times D_6^2$ x 17, $C_3:D_6^2$ x 13, $C_6^2:D_6$ x 12, $C_6\times S_3\times A_4$ x 11, $C_2^2\times C_3^3:C_4$ x 10, $C_6^2:D_6$ x 9, $A_4:C_6^2$ x 7, $A_4\times S_3^2$ x 7, $C_6^2:D_6$ x 7, $A_4:S_3^2$ x 6, $C_6^2:D_6$ x 6, $C_6.\SOPlus(4,2)$ x 5, $C_6^2:D_6$ x 4, $C_6^2:D_6$ x 4, $A_4:S_3^2$ x 3, $C_{12}:S_3^2$ x 3, $(C_3\times C_6^2):C_4$ x 3, $(C_3\times S_3^2):C_4$ x 3, $C_6^3:C_2$ x 2, $D_6\times C_6^2$ x 2, $C_6:S_3\times A_4$ x 2, $C_2\times S_3^3$ x 2, $C_6^2:D_6$ x 2, $(C_3^2\times A_4):C_4$ x 2, $S_3^3:C_2$ x 2, $C_6^3:C_2$ x 2, $D_6:C_6^2$ x 2, $C_{12}:C_6^2$ x 2, $C_{12}:S_3^2$ x 2, $C_6^2.D_6$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}\times S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $A_4\times C_6^2$, $C_6^2:D_6$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $C_6^2:C_{12}$, $C_{12}:C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $(C_3^2\times C_{12}):C_4$, $C_3^3:C_4^2$, $C_6.\SOPlus(4,2)$, $C_{18}\times S_4$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$

Order 384: $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_6:D_4^2$

Order 324: $C_3^3:D_6$ x 13, $C_3^2.C_6^2$ x 6, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2\times S_3^2$ x 3, $C_3^3:A_4$ 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, $C_3^3:A_4$, $C_6^2.C_3^2$, $\He_3:A_4$, $C_3^3:C_{12}$, $C_3^3:D_6$, $A_4\times C_3^3$, $C_3^4:C_4$, $C_3\wr C_4$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $C_6^2.C_3^2$, $A_4\times \He_3$, $\He_3:C_{12}$, $C_3^2:S_3^2$

Order 288: $C_6^2:D_4$ x 21, $C_6^2:C_2^3$ x 16, $D_6\times S_4$ x 15, $D_4\times S_3^2$ x 10, $D_6\wr C_2$ x 10, $C_2\times C_6\times S_4$ x 9, $C_6^2.D_4$ x 6, $D_6^2:C_2$ x 5, $D_6^2:C_2$ x 4, $C_6^2:D_4$ x 4, $S_3^2:D_4$ x 4, $C_2\times D_6^2$ x 4, $C_2\times A_4\times D_6$ x 3, $D_{12}:D_6$ x 2, $D_6\times D_{12}$ x 2, $C_2.D_6^2$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_3^2:C_4\times D_4$ x 2, $A_4\times C_3:D_4$ x 2, $C_4:\SOPlus(4,2)$ x 2, $C_4\times \SOPlus(4,2)$ x 2, $D_6:S_4$ x 2, $C_6^2:C_2^3$ x 2, $C_2\times C_6^2:C_4$ x 2, $C_2\times C_6:S_4$ x 2, $A_4\times S_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$, $C_6^2.C_2^3$, $(C_6\times C_{12}):C_4$, $C_2\times C_3^2:C_4^2$, $C_6:C_4\times A_4$, $C_3:\GL(2,\mathbb{Z}/4)$, $C_2\times C_6.S_4$, $C_6^2.D_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2^3:C_6^2$, $C_2\times A_4^2$, $\PSOPlus(4,3)$, $D_4\times C_6^2$, $C_6^2:C_2^3$

Order 243: $C_3^4:C_3$

Order 216: $C_6^3$, $C_6:S_3^2$

Order 144: $S_3\times S_4$ x 2

Order 128: $C_2\times D_4^2$

Order 108: $C_3\times C_6^2$, $C_3:S_3^2$

Order 96: $C_2^2\times S_4$ x 7

Order 72: $S_3\times D_6$

Order 54: $C_3^2\times C_6$

Order 48: $C_2\times S_4$ x 24

Order 36: $S_3^2$

Order 27: $C_3^3$

Order 24: $S_4$ x 12, $C_2\times D_6$ x 7, $C_2\times A_4$

Order 12: $D_6$ x 22, $A_4$

Order 8: $C_2^3$ x 10

Order 6: $S_3$ x 10

Order 4: $C_2^2$ x 25

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 31104: $C_2\times C_3^3:S_4^2$ ($C_1$)

Order 15552: $C_3^3:S_4^2$ ($C_2$) x 4, $C_6^3:(S_3\times A_4)$ ($C_2$), $C_6^3:C_3:S_4$ ($C_2$), $C_6^3:(C_3\times S_4)$ ($C_2$)

Order 7776: $C_3^3:A_4\times S_4$ ($C_2^2$) x 2, $(A_4\times C_3^3):S_4$ ($C_2^2$) x 2, $A_4\times C_3^3:S_4$ ($C_2^2$) x 2, $C_2\times C_3^3:A_4^2$ ($C_2^2$)

Order 5184: $C_6^2:D_6^2$ ($S_3$), $C_6^3:S_4$ ($S_3$)

Order 3888: $C_3^3:A_4^2$ ($C_2^3$)

Order 2592: $S_4\times C_3:S_3^2$ ($D_6$) x 2, $C_2^2\times C_3^3:S_4$ ($D_6$) x 2, $C_6^3:A_4$ ($D_6$), $A_4\times C_6:S_3^2$ ($D_6$)

Order 1296: $C_6^3:S_3$ ($S_4$), $A_4\times C_3:S_3^2$ ($C_2\times D_6$), $C_2^2\times C_3^3:A_4$ ($C_2\times D_6$), $C_2\times C_3^3:S_4$ ($S_4$)

Order 864: $C_6:D_6^2$ ($S_3^2$)

Order 648: $S_4\times C_3^3$ ($C_2\times S_4$) x 2, $C_3^3:S_4$ ($C_2\times S_4$) x 2, $C_6^3:C_3$ ($C_2\times S_4$), $C_2\times C_3^3:A_4$ ($C_2\times S_4$)

Order 432: $C_3:D_6^2$ ($S_3\times D_6$)

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

Order 216: $C_6^3$ ($S_3\times S_4$), $C_6:S_3^2$ ($S_3\times S_4$)

Order 108: $C_3\times C_6^2$ ($D_6\times S_4$), $C_3:S_3^2$ ($D_6\times S_4$)

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

Order 48: $C_2\times S_4$ ($C_3^3:S_4$)

Order 27: $C_3^3$ ($C_2\times S_4^2$)

Order 24: $S_4$ ($C_2\times C_3^3:S_4$) x 2, $C_2\times A_4$ ($C_2\times C_3^3:S_4$)

Order 12: $A_4$ ($C_2^2\times C_3^3:S_4$)

Order 8: $C_2^3$ ($S_3\times C_3^3:S_4$)

Order 4: $C_2^2$ ($D_6\times C_3^3:S_4$)

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

Order 1: $C_1$ ($C_2\times C_3^3:S_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 31104: $C_2\times C_3^3:S_4^2$ ($C_1$)

Order 15552: $C_6^3:(S_3\times A_4)$ ($C_2$), $C_6^3:C_3:S_4$ ($C_2$), $C_6^3:(C_3\times S_4)$ ($C_2$), $C_3^3:S_4^2$ ($C_2$)

Order 7776: $C_2\times C_3^3:A_4^2$ ($C_2^2$), $C_3^3:A_4\times S_4$ ($C_2^2$), $(A_4\times C_3^3):S_4$ ($C_2^2$), $A_4\times C_3^3:S_4$ ($C_2^2$)

Order 5184: $C_6^2:D_6^2$ ($S_3$), $C_6^3:S_4$ ($S_3$)

Order 3888: $C_3^3:A_4^2$ ($C_2^3$)

Order 2592: $C_6^3:A_4$ ($D_6$), $A_4\times C_6:S_3^2$ ($D_6$), $S_4\times C_3:S_3^2$ ($D_6$), $C_2^2\times C_3^3:S_4$ ($D_6$)

Order 1296: $C_6^3:S_3$ ($S_4$), $A_4\times C_3:S_3^2$ ($C_2\times D_6$), $C_2^2\times C_3^3:A_4$ ($C_2\times D_6$), $C_2\times C_3^3:S_4$ ($S_4$)

Order 864: $C_6:D_6^2$ ($S_3^2$)

Order 648: $C_6^3:C_3$ ($C_2\times S_4$), $S_4\times C_3^3$ ($C_2\times S_4$), $C_2\times C_3^3:A_4$ ($C_2\times S_4$), $C_3^3:S_4$ ($C_2\times S_4$)

Order 432: $C_3:D_6^2$ ($S_3\times D_6$)

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

Order 216: $C_6^3$ ($S_3\times S_4$), $C_6:S_3^2$ ($S_3\times S_4$)

Order 108: $C_3\times C_6^2$ ($D_6\times S_4$), $C_3:S_3^2$ ($D_6\times S_4$)

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

Order 48: $C_2\times S_4$ ($C_3^3:S_4$)

Order 27: $C_3^3$ ($C_2\times S_4^2$)

Order 24: $C_2\times A_4$ ($C_2\times C_3^3:S_4$), $S_4$ ($C_2\times C_3^3:S_4$)

Order 12: $A_4$ ($C_2^2\times C_3^3:S_4$)

Order 8: $C_2^3$ ($S_3\times C_3^3:S_4$)

Order 4: $C_2^2$ ($D_6\times C_3^3:S_4$)

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

Order 1: $C_1$ ($C_2\times C_3^3:S_4^2$)

Series

Derived series

$C_2\times C_3^3:S_4^2$

$\rhd$

$C_2\times C_3^3:S_4^2$

$\rhd$

$C_3^3:A_4^2$

$\rhd$

$C_3^3:A_4^2$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_2\times C_3^3:S_4^2$

$\rhd$

$C_2\times C_3^3:S_4^2$

$\rhd$

$C_6^3:(C_3\times S_4)$

$\rhd$

$C_6^3:(C_3\times S_4)$

$\rhd$

$A_4\times C_3^3:S_4$

$\rhd$

$A_4\times C_3^3:S_4$

$\rhd$

$C_3^3:A_4^2$

$\rhd$

$C_3^3:A_4^2$

$\rhd$

$A_4\times C_3:S_3^2$

$\rhd$

$C_2^2\times C_3^3:A_4$

$\rhd$

$C_2^2\times C_3^3:A_4$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_2\times C_3^3:S_4^2$

$\rhd$

$C_2\times C_3^3:S_4^2$

$\rhd$

$C_3^3:A_4^2$

$\rhd$

$C_3^3:A_4^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $170 \times 170$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $130 \times 130$ rational character table (warning: may be slow to load).