Abstract group 31104.jx: $S_3^4:S_4$

• Label: 31104.jx
• Order: \( 2^{7} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $16$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$S_3^4:S_4$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$(C_3\times C_6^2).A_4.C_6.C_2^3$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
Composition factors:	$C_2$ x 7, $C_3$ x 5
Derived length:	$4$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	1599	944	4032	13440	1728	7632	1728	31104
Conjugacy classes	1	23	9	12	75	2	24	1	147
Divisions	1	23	9	12	75	2	24	1	147
Autjugacy classes	1	19	9	8	61	2	19	1	120

Dimension	1	2	3	4	6	8	12	16	24	32	48
Irr. complex chars.	8	8	24	2	32	4	38	4	22	1	4	147
Irr. rational chars.	8	8	24	2	32	4	38	4	22	1	4	147

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

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:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{6}=d^{6}=e^{6}=f^{2}=g^{6}=[c,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,4)(3,6,7,9,5,8)(10,11,12,13)(14,15), (1,3,2,5,4,7)(11,12), (1,4,2)(3,6,7,9)(5,8)(10,12)(15,16)\rangle$
Transitive group:	36T13604	36T13607	36T13608	36T13611	more information
Direct product:	$S_3$ $\, \times\, $ $(S_3^3:S_4)$
Semidirect product:	$S_3^4$ $\,\rtimes\,$ $S_4$	$S_3^3$ $\,\rtimes\,$ $(S_3\times S_4)$	$(S_3^3:A_4)$ $\,\rtimes\,$ $D_6$	$(S_3^4:A_4)$ $\,\rtimes\,$ $C_2$	all 47
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1564084 subgroups in 19953 conjugacy classes, 62 normal (48 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $S_3^4:S_4$
Commutator:	$G' \simeq$ $(C_3^2\times C_6^2):A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3^4:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^2\times C_6^2$	$G/\operatorname{Fit} \simeq$ $C_2^2\times S_4$
Radical:	$R \simeq$ $S_3^4:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_2^2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
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 36 (order at least 864) are shown, as well as all normal subgroups of any index.

Order 31104: $S_3^4:S_4$

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

Order 10368: $C_3^4.C_2^5.C_2^2$, $D_6\wr S_3$

Order 7776: $S_3^4:S_3$ x 2, $(C_3^2\times C_6^2):S_4$, $(C_3^2\times C_6^2):S_4$, $(C_3^2\times C_6^2):S_4$, $C_3:S_3^3:A_4$, $C_3:C_6^3:A_4$, $(C_3^2\times C_6^2):S_4$, $C_6^2.S_3^3$, $C_3\times S_3^3:A_4$

Order 5184: $S_3^3:S_4$ x 5, $C_3^4.C_2^4.C_2^2$ x 4, $C_3^4.C_2^4.C_2^2$ x 4, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^5.C_2$ x 2, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$ x 2, $S_3^2:D_6^2$ x 2, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^6$, $C_3^4.C_2^5.C_2$, $S_3\times C_3^2.C_2^3:D_6$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^3.(C_2\times C_6\times D_4).C_2$, $C_3\times (C_2\times S_3^3).C_2^2$, $D_6^2:S_3^2$, $C_6^3:S_4$, $D_6\wr C_3$, $C_6^3:S_4$

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

Order 3456: $D_6\times D_6\wr C_2$ x 2

Order 2592: $S_3^4:C_2$ x 16, $C_2\times S_3^4$ x 13, $S_3^3:D_6$ x 8, $C_3^4.C_2^3.C_2^2$ x 4, $C_3^3.C_2^3:D_6$ x 4, $C_3^4.C_2^3.C_2^2$ x 4, $C_3^3.C_2^3:D_6$ x 4, $C_3^4.C_2^2\wr C_2$ x 4, $C_3^4.C_2^2\wr C_2$ x 4, $C_3^3.C_2^3:D_6$ x 4, $C_3^4.C_2^2\wr C_2$ x 4, $C_2\times S_3^3:C_6$ x 4, $C_3^2:D_{12}\times D_6$ x 4, $(C_3\times S_3^2):D_{12}$ x 4, $C_2\times S_3^2:S_3^2$ x 4, $S_3^3:A_4$ x 3, $(C_3\times C_6^2):S_4$ x 3, $(C_3\times C_6^2):S_4$ x 3, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^3.(C_6\times D_4).C_2$ x 2, $C_3\times C_3^2.C_2^3:D_6$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^3.C_2^2$ x 2, $C_3^3.(C_2^3\times C_6).C_2$ x 2, $C_3\times (C_6\times C_3:S_3).C_2^3$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^2:(D_6:C_4)\times S_3$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^3.C_2^2$ x 2, $C_3^4.C_2^5$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^3.(C_6\times D_4).C_2$ x 2, $C_3\times C_3^3.C_2^2\wr C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^3:(D_4\times D_6)$ x 2, $(C_6\times S_3^2):D_6$ x 2, $C_3\wr C_4:C_2^3$ x 2, $(C_6\times S_3^2).D_6$ x 2, $C_6^2:\SOPlus(4,2)$ x 2, $(C_6\times S_3)\wr C_2$ x 2, $S_3^3:D_6$ x 2, $C_3^3.(C_6\times D_4).C_2$, $C_3\times C_3^3.C_2^2\wr C_2$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^4:S_3$, $(C_3^3\times C_6).C_2^4$, $C_3^4.C_2^5$, $C_2^2\times C_3^3:C_2^2\times S_3$, $C_3^4.C_2^5$, $C_3\times S_3^3.C_2^2$, $C_3^4.C_2^4.C_2$, $C_3^3.(C_6\times C_2^2:C_4)$, $C_3\times (C_6\times C_3:S_3).C_2^3$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^5$, $C_3\times C_3:S_3.(C_6\times D_4)$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^4:S_3$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $C_3^3.(C_6\times D_4).C_2$, $C_3\times C_3^2.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^2:(C_6.D_4)\times S_3$, $S_3\times C_3\times D_6:D_6$, $C_3\times (C_2\times S_3^3).C_2$, $C_3^4.C_2^2\wr C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^4.C_2$, $S_3\times C_6^2:D_6$, $C_6:D_6.S_3^2$, $(C_2\times C_6):S_3^3$, $C_6^3:D_6$, $C_6^3:A_4$

Order 1944: $S_3\times C_3^3:A_4$ x 3, $C_3^3:(S_3\times A_4)$ x 3, $S_3^3:C_3^2$ x 3, $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3^4:S_4$ x 2, $C_3\wr S_4$ x 2, $C_3^3:(S_3\times A_4)$, $S_3\times C_3^3:A_4$, $C_3^3:(C_6\times A_4)$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^2.S_3^3$

Order 1728: $D_6^2:D_6$ x 33, $D_6^2:D_6$ x 4, $D_6^2.D_6$ x 4, $D_6^2:D_6$ x 4, $D_6^2.D_6$ x 4, $D_6\times D_6^2$ x 2, $D_6:D_6^2$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^2:C_4\times D_6$ x 2, $D_4\times S_3^3$

Order 1296: $S_3^4$ x 60, $C_6:S_3^3$ x 33, $S_3^3:S_3$ x 32, $C_6:S_3^3$ x 23, $C_6\times S_3^3$ x 23, $S_3\times C_3^2:D_{12}$ x 16, $S_3^2:S_3^2$ x 16, $S_3^3:C_6$ x 16, $S_3\wr S_3$ x 10, $C_2\times C_3^3:D_{12}$ x 8, $C_2\times C_3\wr D_4$ x 8, $C_3^3:(S_3\times D_4)$ x 8, $(C_3\times S_3^2):D_6$ x 8, $C_3^2:C_4\times S_3^2$ x 8, $C_6:S_3^3$ x 7, $C_6^2.S_3^2$ x 6, $C_2\times C_3^4:D_4$ x 4, $C_2\times C_3^4:D_4$ x 4, $C_6\times C_3^2:D_{12}$ x 4, $C_3^3:C_4\times D_6$ x 4, $C_3^2:C_{12}\times D_6$ x 4, $C_6.S_3^3$ x 4, $C_6.S_3^3$ x 4, $(C_3^2\times C_6).D_{12}$ x 4, $C_3\wr C_2^2:C_4$ x 4, $C_3\wr C_2^2:C_4$ x 4, $C_3\wr C_2^2:C_4$ x 4, $(C_3\times S_3^2):C_{12}$ x 4, $C_3\wr C_2^2:C_4$ x 4, $(D_6\times C_3^3):C_4$ x 4, $(C_3^2\times D_6):C_{12}$ x 4, $C_6^2:S_3^2$ x 3, $S_3^3:C_6$ x 3, $C_6^2:S_3^2$ x 3, $C_6.S_3^3$ x 3, $C_6.S_3^3$ x 3, $C_6.S_3^3$ x 3, $C_6.S_3^3$ x 3, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2\times S_3^2$ x 2, $C_2\times C_3^3:D_{12}$ x 2, $S_3^2:C_6^2$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $(C_3\times C_6^2):A_4$ x 2, $C_2\times C_3^3:S_4$ x 2, $C_2\times C_3^3:S_4$ 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, $C_6^2:C_6^2$ x 2, $(C_3\times C_6^2):C_{12}$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $(C_3^2\times C_6).D_{12}$ x 2, $(C_3^2\times S_3^2):C_4$ x 2, $(C_3^3\times C_6).D_4$ x 2, $C_3\times S_3^2:C_{12}$ x 2, $C_3^2\wr C_2.D_4$ x 2, $C_3^2\wr C_2.D_4$ x 2, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:C_6^2$, $(C_3^2\times C_6^2):C_4$, $C_3\times C_6^2:C_{12}$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6^2:C_6\times S_3$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6\wr S_3$

Order 1152: $C_2^5:S_3^2$, $D_6^2:D_4$, $D_6^2:D_4$

Order 972: $C_3\wr A_4$ x 3, $C_3^4:A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 864: $S_3^3:C_2^2$ x 66, $S_3\times D_6^2$ x 55, $S_3^2:D_{12}$ x 34, $D_6:\SOPlus(4,2)$ x 34, $S_3^3:C_4$ x 34, $C_2^2:S_3^3$ x 33, $C_6^2:D_{12}$ x 17, $D_6^2:S_3$ x 17, $(S_3\times D_6):D_6$ x 17, $D_6^2:C_6$ x 17, $C_6^2:C_4\times S_3$ x 17, $D_6^2:S_3$ x 8, $D_6^2:S_3$ x 6, $C_6^2:D_{12}$ x 4, $D_6^2:C_6$ x 4, $C_6^2:C_4\times S_3$ x 4, $C_6^2.D_{12}$ x 4, $D_6^2.S_3$ x 4, $(C_3\times C_6^2).D_4$ x 4, $D_6^2.C_6$ x 4, $(S_3\times C_6^2):C_4$ x 4, $C_6:D_6^2$ x 3, $C_6\times D_6^2$ x 3, $C_6.D_6^2$ x 3, $D_6^2:C_6$ x 3, $C_4:S_3^3$ x 3, $D_{12}\times S_3^2$ x 3, $C_6:D_6^2$ x 2, $C_6^3:C_2^2$ 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, $C_6^3:C_4$ x 2, $C_2^2:S_3^3$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $D_6^2:S_3$ x 2, $D_6^2:S_3$ x 2, $C_6.D_6^2$ x 2, $S_4\times S_3^2$, $C_6.D_6^2$, $C_4:S_3^3$, $C_4\times S_3^3$, $C_2\times C_6^2:D_6$

Order 648: $C_3:S_3^3$ x 2, $C_3:S_3^3$ x 2, $C_3\times S_3^3$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_3:C_6^3$

Order 576: $C_2^4:S_3^2$ x 4

Order 432: $C_3:D_6^2$

Order 384: $C_2^2\wr S_3$

Order 324: $C_3\wr C_2^2$ x 2, $C_3^2\times C_6^2$, $C_3^2:S_3^2$

Order 288: $D_6\times S_4$ x 3

Order 243: $C_3^4:C_3$

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

Order 192: $C_2^3:S_4$ x 7

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $S_3\times C_3^3$

Order 144: $S_3\times S_4$ x 13

Order 128: $C_2^4:D_4$

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

Order 96: $C_2^2:S_4$ x 6, $C_2^2\times S_4$ x 3

Order 81: $C_3^4$

Order 72: $S_3\times D_6$

Order 54: $C_3^2:S_3$

Order 48: $C_2\times S_4$ x 21

Order 36: $S_3^2$ x 5

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 22, $S_4$ x 18

Order 12: $D_6$ x 136, $C_2\times C_6$

Order 8: $C_2^3$ x 16

Order 6: $S_3$ x 40

Order 4: $C_2^2$ x 76

Order 3: $C_3$

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 864) are shown, as well as all normal subgroups of any index.

Order 31104: $S_3^4:S_4$

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

Order 10368: $C_3^4.C_2^5.C_2^2$, $D_6\wr S_3$

Order 7776: $(C_3^2\times C_6^2):S_4$, $(C_3^2\times C_6^2):S_4$, $(C_3^2\times C_6^2):S_4$, $C_3:S_3^3:A_4$, $C_3:C_6^3:A_4$, $(C_3^2\times C_6^2):S_4$, $C_6^2.S_3^3$, $C_3\times S_3^3:A_4$, $S_3^4:S_3$

Order 5184: $S_3^3:S_4$ x 5, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$ x 2, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^5.C_2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^6$, $C_3^4.C_2^5.C_2$, $S_3\times C_3^2.C_2^3:D_6$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^4.C_2^4.C_2^2$, $C_3^3.(C_2\times C_6\times D_4).C_2$, $C_3\times (C_2\times S_3^3).C_2^2$, $S_3^2:D_6^2$, $D_6^2:S_3^2$, $C_6^3:S_4$, $D_6\wr C_3$, $C_6^3:S_4$

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

Order 3456: $D_6\times D_6\wr C_2$ x 2

Order 2592: $C_2\times S_3^4$ x 9, $S_3^4:C_2$ x 8, $S_3^3:D_6$ x 4, $S_3^3:A_4$ x 3, $(C_3\times C_6^2):S_4$ x 3, $(C_3\times C_6^2):S_4$ x 3, $C_3^4.C_2^3.C_2^2$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^4.C_2^3.C_2^2$ x 2, $C_3^4.C_2^5$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_3^4.C_2^4.C_2$ x 2, $C_3^3.C_2^3:D_6$ x 2, $C_3^4.C_2^2\wr C_2$ x 2, $C_2\times S_3^3:C_6$ x 2, $C_3^2:D_{12}\times D_6$ x 2, $(C_3\times S_3^2):D_{12}$ x 2, $C_2\times S_3^2:S_3^2$ x 2, $C_6^2:\SOPlus(4,2)$ x 2, $(C_6\times S_3)\wr C_2$ x 2, $C_3^3.C_2^3:D_6$, $C_3^3.(C_6\times D_4).C_2$, $C_3\times C_3^3.C_2^2\wr C_2$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^4:S_3$, $C_3^3.(C_6\times D_4).C_2$, $C_3\times C_3^2.C_2^3:D_6$, $(C_3^3\times C_6).C_2^4$, $C_3^4.C_2^5$, $C_2^2\times C_3^3:C_2^2\times S_3$, $C_3^4.C_2^3.C_2^2$, $C_3^3.(C_2^3\times C_6).C_2$, $C_3\times (C_6\times C_3:S_3).C_2^3$, $C_3^4.C_2^4.C_2$, $C_3^2:(D_6:C_4)\times S_3$, $C_3^4.C_2^5$, $C_3\times S_3^3.C_2^2$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^3:D_6$, $C_3^3.(C_6\times C_2^2:C_4)$, $C_3\times (C_6\times C_3:S_3).C_2^3$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^5$, $C_3\times C_3:S_3.(C_6\times D_4)$, $C_3^4.C_2^2\wr C_2$, $C_3^4.C_2^4.C_2$, $C_3^3.C_2^4:S_3$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^2\wr C_2$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $C_3^3.(C_6\times D_4).C_2$, $C_3\times C_3^2.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^2\wr C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^2:(C_6.D_4)\times S_3$, $C_3^3.(C_6\times D_4).C_2$, $C_3\times C_3^3.C_2^2\wr C_2$, $S_3\times C_3\times D_6:D_6$, $C_3\times (C_2\times S_3^3).C_2$, $C_3^4.C_2^2\wr C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_2^4.C_2$, $C_3^4.C_2^4.C_2$, $C_3^3:(D_4\times D_6)$, $(C_6\times S_3^2):D_6$, $C_3\wr C_4:C_2^3$, $S_3\times C_6^2:D_6$, $(C_6\times S_3^2).D_6$, $C_6:D_6.S_3^2$, $(C_2\times C_6):S_3^3$, $S_3^3:D_6$, $C_6^3:D_6$, $C_6^3:A_4$

Order 1944: $S_3\times C_3^3:A_4$ x 2, $C_3^3:(S_3\times A_4)$ x 2, $S_3^3:C_3^2$ x 2, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3\wr S_4$, $C_3^3:(S_3\times A_4)$, $S_3\times C_3^3:A_4$, $C_3^3:(C_6\times A_4)$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^2.S_3^3$

Order 1728: $D_6^2:D_6$ x 25, $D_6\times D_6^2$ x 2, $D_6^2:D_6$ x 2, $D_6:D_6^2$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^2:C_4\times D_6$ x 2, $D_6^2.D_6$ x 2, $D_6^2:D_6$ x 2, $D_6^2.D_6$ x 2, $D_4\times S_3^3$

Order 1296: $S_3^4$ x 34, $C_6:S_3^3$ x 23, $S_3^3:S_3$ x 16, $C_6:S_3^3$ x 15, $C_6\times S_3^3$ x 15, $S_3\times C_3^2:D_{12}$ x 8, $S_3^2:S_3^2$ x 8, $S_3^3:C_6$ x 8, $C_6^2.S_3^2$ x 6, $C_6:S_3^3$ x 5, $S_3\wr S_3$ x 5, $C_2\times C_3^3:D_{12}$ x 4, $C_2\times C_3\wr D_4$ x 4, $C_3^3:(S_3\times D_4)$ x 4, $(C_3\times S_3^2):D_6$ x 4, $C_3^2:C_4\times S_3^2$ x 4, $C_6.S_3^3$ x 4, $C_6.S_3^3$ x 4, $C_6^2:S_3^2$ x 3, $C_6^2:S_3^2$ x 3, $C_6.S_3^3$ x 3, $C_6.S_3^3$ x 3, $C_6.S_3^3$ x 3, $C_6.S_3^3$ x 3, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2\times S_3^2$ x 2, $C_2\times C_3^4:D_4$ x 2, $C_2\times C_3^4:D_4$ x 2, $C_6\times C_3^2:D_{12}$ x 2, $C_3^3:C_4\times D_6$ x 2, $C_3^2:C_{12}\times D_6$ x 2, $(C_3\times C_6^2):A_4$ x 2, $S_3^3:C_6$ 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, $C_6^2:C_6^2$ x 2, $(C_3\times C_6^2):C_{12}$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $(C_3^2\times C_6).D_{12}$ x 2, $C_3\wr C_2^2:C_4$ x 2, $C_3\wr C_2^2:C_4$ x 2, $C_3\wr C_2^2:C_4$ x 2, $(C_3\times S_3^2):C_{12}$ x 2, $C_3\wr C_2^2:C_4$ x 2, $(D_6\times C_3^3):C_4$ x 2, $(C_3^2\times D_6):C_{12}$ x 2, $C_6^2:S_3^2$, $C_2\times C_3^3:D_{12}$, $S_3^2:C_6^2$, $C_3^3:(C_4\times D_6)$, $C_3^3:(C_4\times D_6)$, $C_2\times C_3^3:S_4$, $C_2\times C_3^3:S_4$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:C_6^2$, $(C_3^2\times C_6^2):C_4$, $C_3\times C_6^2:C_{12}$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $C_6.S_3^3$, $(C_3^2\times C_6).D_{12}$, $(C_3^2\times S_3^2):C_4$, $(C_3^3\times C_6).D_4$, $C_3\times S_3^2:C_{12}$, $C_3^2\wr C_2.D_4$, $C_3^2\wr C_2.D_4$, $C_6^2:C_6\times S_3$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6\wr S_3$

Order 1152: $C_2^5:S_3^2$, $D_6^2:D_4$, $D_6^2:D_4$

Order 972: $C_3\wr A_4$ x 2, $C_3^4:A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 864: $S_3\times D_6^2$ x 39, $S_3^3:C_2^2$ x 33, $C_2^2:S_3^3$ x 33, $S_3^2:D_{12}$ x 17, $D_6:\SOPlus(4,2)$ x 17, $S_3^3:C_4$ x 17, $C_6^2:D_{12}$ x 13, $D_6^2:S_3$ x 13, $(S_3\times D_6):D_6$ x 13, $D_6^2:C_6$ x 13, $C_6^2:C_4\times S_3$ x 13, $D_6^2:S_3$ x 6, $D_6^2:S_3$ x 4, $C_6:D_6^2$ x 3, $C_6\times D_6^2$ x 3, $C_6.D_6^2$ x 3, $D_6^2:C_6$ x 3, $C_4:S_3^3$ x 3, $D_{12}\times S_3^2$ x 3, $C_6:D_6^2$ x 2, $C_6^2:D_{12}$ x 2, $D_6^2:C_6$ x 2, $C_6^2:C_4\times S_3$ x 2, $C_6^3:C_2^2$ 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, $C_6^3:C_4$ x 2, $C_2^2:S_3^3$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $D_6^2:S_3$ x 2, $D_6^2:S_3$ x 2, $C_6.D_6^2$ x 2, $C_6^2.D_{12}$ x 2, $D_6^2.S_3$ x 2, $(C_3\times C_6^2).D_4$ x 2, $D_6^2.C_6$ x 2, $(S_3\times C_6^2):C_4$ x 2, $S_4\times S_3^2$, $C_6.D_6^2$, $C_4:S_3^3$, $C_4\times S_3^3$, $C_2\times C_6^2:D_6$

Order 648: $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_3:C_6^3$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3\times S_3^3$

Order 576: $C_2^4:S_3^2$ x 4

Order 432: $C_3:D_6^2$

Order 384: $C_2^2\wr S_3$

Order 324: $C_3^2\times C_6^2$, $C_3^2:S_3^2$, $C_3\wr C_2^2$

Order 288: $D_6\times S_4$ x 2

Order 243: $C_3^4:C_3$

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

Order 192: $C_2^3:S_4$ x 7

Order 162: $C_3^3:S_3$, $C_3^3:C_6$, $S_3\times C_3^3$

Order 144: $S_3\times S_4$ x 9

Order 128: $C_2^4:D_4$

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

Order 96: $C_2^2:S_4$ x 6, $C_2^2\times S_4$ x 2

Order 81: $C_3^4$

Order 72: $S_3\times D_6$

Order 54: $C_3^2:S_3$

Order 48: $C_2\times S_4$ x 14

Order 36: $S_3^2$ x 5

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 14, $S_4$ x 12

Order 12: $D_6$ x 87, $C_2\times C_6$

Order 8: $C_2^3$ x 10

Order 6: $S_3$ x 34

Order 4: $C_2^2$ x 50

Order 3: $C_3$

Order 2: $C_2$ x 16

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 31104: $S_3^4:S_4$ ($C_1$)

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

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

Order 5184: $C_3^4.C_2^6$ ($S_3$), $S_3^3:S_4$ ($S_3$)

Order 3888: $(C_3^2\times C_6^2):A_4$ ($C_2^3$)

Order 2592: $C_2^2\times C_3^3:C_2^2\times S_3$ ($D_6$), $C_3\times S_3^3.C_2^2$ ($D_6$), $C_3^4.C_2^5$ ($D_6$), $S_3^3:A_4$ ($D_6$), $(C_3\times C_6^2):S_4$ ($D_6$), $(C_3\times C_6^2):S_4$ ($D_6$)

Order 1296: $S_3^4$ ($S_4$) x 2, $C_6^2:S_3^2$ ($S_4$), $C_6^2:S_3^2$ ($C_2\times D_6$), $(C_3\times C_6^2):A_4$ ($C_2\times D_6$)

Order 864: $S_3\times D_6^2$ ($S_3^2$)

Order 648: $C_3:S_3^3$ ($C_2\times S_4$) x 2, $C_3:S_3^3$ ($C_2\times S_4$) x 2, $C_3\times S_3^3$ ($C_2\times S_4$) x 2, $C_3^4:C_2^3$ ($C_2\times S_4$), $C_3^4:C_2^3$ ($C_2\times S_4$), $C_3:C_6^3$ ($C_2\times S_4$)

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

Order 324: $C_3\wr C_2^2$ ($C_2^2\times S_4$) x 2, $C_3^2\times C_6^2$ ($C_2^2\times S_4$), $C_3^2:S_3^2$ ($C_2^2:S_4$)

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

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

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

Order 81: $C_3^4$ ($C_2^2\wr S_3$)

Order 54: $C_3^2:S_3$ ($C_2^4:S_3^2$)

Order 27: $C_3^3$ ($C_2^5:S_3^2$)

Order 24: $C_2\times D_6$ ($S_3\wr S_3$)

Order 12: $C_2\times C_6$ ($S_3^3:D_6$)

Order 6: $S_3$ ($S_3^3:S_4$)

Order 4: $C_2^2$ ($S_3^4:S_3$)

Order 3: $C_3$ ($D_6\wr S_3$)

Order 1: $C_1$ ($S_3^4:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 31104: $S_3^4:S_4$ ($C_1$)

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

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

Order 5184: $C_3^4.C_2^6$ ($S_3$), $S_3^3:S_4$ ($S_3$)

Order 3888: $(C_3^2\times C_6^2):A_4$ ($C_2^3$)

Order 2592: $C_2^2\times C_3^3:C_2^2\times S_3$ ($D_6$), $C_3\times S_3^3.C_2^2$ ($D_6$), $C_3^4.C_2^5$ ($D_6$), $S_3^3:A_4$ ($D_6$), $(C_3\times C_6^2):S_4$ ($D_6$), $(C_3\times C_6^2):S_4$ ($D_6$)

Order 1296: $C_6^2:S_3^2$ ($S_4$), $C_6^2:S_3^2$ ($C_2\times D_6$), $(C_3\times C_6^2):A_4$ ($C_2\times D_6$), $S_3^4$ ($S_4$)

Order 864: $S_3\times D_6^2$ ($S_3^2$)

Order 648: $C_3^4:C_2^3$ ($C_2\times S_4$), $C_3^4:C_2^3$ ($C_2\times S_4$), $C_3:C_6^3$ ($C_2\times S_4$), $C_3:S_3^3$ ($C_2\times S_4$), $C_3:S_3^3$ ($C_2\times S_4$), $C_3\times S_3^3$ ($C_2\times S_4$)

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

Order 324: $C_3^2\times C_6^2$ ($C_2^2\times S_4$), $C_3^2:S_3^2$ ($C_2^2:S_4$), $C_3\wr C_2^2$ ($C_2^2\times S_4$)

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

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

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

Order 81: $C_3^4$ ($C_2^2\wr S_3$)

Order 54: $C_3^2:S_3$ ($C_2^4:S_3^2$)

Order 27: $C_3^3$ ($C_2^5:S_3^2$)

Order 24: $C_2\times D_6$ ($S_3\wr S_3$)

Order 12: $C_2\times C_6$ ($S_3^3:D_6$)

Order 6: $S_3$ ($S_3^3:S_4$)

Order 4: $C_2^2$ ($S_3^4:S_3$)

Order 3: $C_3$ ($D_6\wr S_3$)

Order 1: $C_1$ ($S_3^4:S_4$)

Series

Derived series

$S_3^4:S_4$

$\rhd$

$(C_3^2\times C_6^2):A_4$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$S_3^4:S_4$

$\rhd$

$S_3^4:A_4$

$\rhd$

$C_3:C_6^3:A_4$

$\rhd$

$(C_3^2\times C_6^2):A_4$

$\rhd$

$C_6^2:S_3^2$

$\rhd$

$(C_3\times C_6^2):A_4$

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$S_3^4:S_4$

$\rhd$

$(C_3^2\times C_6^2):A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 14 larger groups in the database.

This group is a maximal quotient of 12 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $147 \times 147$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.