Abstract group 20736.je: $C_2^2\times S_3^3:S_4$

• Label: 20736.je
• Order: \( 2^{8} \cdot 3^{4} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$C_2^2\times S_3^3:S_4$
Order:	\(20736\)\(\medspace = 2^{8} \cdot 3^{4} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_5^4:(C_2\times C_4)$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \)
Composition factors:	$C_2$ x 8, $C_3$ x 4
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	314	4032	8742	576	3744	1728	20736
Conjugacy classes	1	47	4	24	92	1	24	3	196
Divisions	1	47	4	24	92	1	24	3	196
Autjugacy classes	1	13	4	4	27	1	5	1	56

Dimension	1	2	3	6	8	12	16	24
Irr. complex chars.	16	8	48	40	8	56	4	16	196
Irr. rational chars.	16	8	48	40	8	56	4	16	196

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

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:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{2}=d^{6}=e^{6}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(2,6,5,4,9,3)(7,8)(11,12,14,13)(16,17), (1,2,4,7,5,6,8,9,3)(10,11,13,15,14,12) \!\cdots\! \rangle$
Transitive group:	36T12049	36T12081			more information
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(S_3^3:S_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(S_3\times D_6^2)$ . $S_4$	$(D_6\wr S_3)$ . $C_2$	$C_2$ . $(D_6\wr S_3)$	$(D_6\times D_6^2)$ . $D_6$	all 43
Aut. group:	$\Aut(C_4.S_3^3)$	$\Aut(C_4.S_3^3)$	$\Aut(C_{24}:S_3^2)$	$\Aut(C_{24}.S_3^2)$	all 15

Elements of the group are displayed as permutations of degree 17.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 3337572 subgroups in 79044 conjugacy classes, 157 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $S_3^3:S_4$
Commutator:	$G' \simeq$ $(C_3\times C_6^2):A_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^2\times S_3^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_2^2\times S_3^3:S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^3$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 64 (order at least 324) are shown, as well as all normal subgroups of any index.

Order 20736: $C_2^2\times S_3^3:S_4$

Order 10368: $D_6\wr S_3$ x 12, $C_2\times C_6^3:S_4$, $C_2\times C_6^3:S_4$, $C_2^2\times S_3^3:A_4$

Order 6912: $C_2^2\times C_3^3.C_2^5.C_2$

Order 5184: $S_3^3:S_4$ x 16, $C_6^3:S_4$ x 6, $D_6\wr C_3$ x 6, $C_6^3:S_4$ x 6, $C_2^2\times S_3\wr S_3$ x 2, $C_2\times C_6^3:A_4$, $C_6^2.D_6^2$

Order 3456: $D_6\times D_6\wr C_2$ x 48, $C_2^2\times (C_2\times S_3^3).C_2$ x 2, $C_2^2\times (C_2\times S_3^3).C_2$ x 2, $C_2^2\times (C_2\times S_3^3).C_2$ x 2, $S_3^3:C_2^4$ x 2, $C_2^2\times C_3:S_3.(C_6\times D_4)$, $C_2^2\times (C_6\times C_3:S_3).C_2^3$, $C_2^2\times C_3^2.C_2^4:S_3$, $C_2^2\times C_3^2.C_2^3:D_6$, $C_2^2\times C_3.(C_2\times D_6:D_6)$, $C_2^3\times C_2\times S_3^3$, $C_2^2\times C_3^3.C_2^2\wr C_2$

Order 2592: $S_3^3:D_6$ x 24, $C_6^3:D_6$ x 12, $S_3^3:A_4$ x 4, $(C_3\times C_6^2):S_4$ x 4, $(C_3\times C_6^2):S_4$ x 4, $C_2^2\times S_3\wr C_3$ x 3, $C_6^3:A_4$ x 3, $C_2^2\times C_3^3:S_4$ x 2, $C_2^2\times C_3^3:S_4$ x 2, $C_6^3:D_6$, $C_2\times C_6^3:C_6$, $C_6^3:D_6$

Order 2304: $C_2^2\times D_6^2.C_2^2$

Order 1728: $D_6^2:D_6$ x 256, $D_6^2:D_6$ x 80, $D_6^2.D_6$ x 48, $D_6^2:D_6$ x 48, $D_6^2.D_6$ x 48, $D_6\times D_6^2$ x 37, $D_6:D_6^2$ x 36, $C_6^3:D_4$ x 24, $C_6^3:D_4$ x 24, $C_6^3:D_4$ x 24, $C_6^3:D_4$ x 24, $C_6^2:C_4\times D_6$ x 24, $C_6^3:D_4$ x 4, $C_6^3:D_4$ x 2, $C_6^3:D_4$ x 2, $C_6^2:C_4\times D_6$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $D_6^2:C_{12}$ x 2, $(D_6\times C_6^2):C_4$ x 2, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_2\times C_6^3:C_4$, $C_2\times C_6^3:C_4$, $D_6:D_6^2$, $D_6:D_6^2$, $D_6:D_6^2$, $D_6:D_6^2$, $D_6.D_6^2$, $C_2^2\times C_6^2:D_6$

Order 1296: $S_3\wr S_3$ x 32, $S_3^3:C_6$ x 18, $C_6^2.S_3^2$ x 16, $C_2\times C_3^3:S_4$ x 12, $C_2\times C_3^3:S_4$ x 12, $C_6^3:C_6$ x 6, $C_6^3:S_3$ x 6, $C_6\wr S_3$ x 6, $C_2^2\times C_3^3:A_4$ x 3, $(C_3\times C_6^2):A_4$, $C_6^3:C_6$, $C_6^2.S_3^2$

Order 1152: $D_6^2:D_4$ x 56, $D_6^2:C_2^3$ x 2, $C_2^3\times S_3^2:C_4$ x 2, $C_2^3\times D_6^2$, $C_2^3\times D_6:D_6$, $D_4\times D_6^2$, $C_2^3\times C_3:S_3.D_4$

Order 864: $S_3^3:C_2^2$ x 512, $S_3\times D_6^2$ x 482, $C_2^2:S_3^3$ x 160, $S_3^2:D_{12}$ x 128, $D_6:\SOPlus(4,2)$ x 128, $S_3^3:C_4$ x 128, $D_6^2:S_3$ x 80, $C_6^2:D_{12}$ x 64, $D_6^2:S_3$ x 64, $(S_3\times D_6):D_6$ x 64, $D_6^2:C_6$ x 64, $C_6^2:C_4\times S_3$ x 64, $C_6:D_6^2$ x 45, $C_6^2:D_{12}$ x 40, $D_6^2:C_6$ x 40, $C_6^2:C_4\times S_3$ x 40, $C_6\times D_6^2$ x 39, $C_6^3:C_2^2$ x 24, $D_6^2:C_6$ x 24, $C_6.D_6^2$ x 24, $D_6^2:S_3$ x 24, $C_6^2.D_{12}$ x 24, $D_6^2.S_3$ x 24, $(C_3\times C_6^2).D_4$ x 24, $D_6^2.C_6$ x 24, $(S_3\times C_6^2):C_4$ x 24, $C_6:D_6^2$ x 19, $C_6^3:C_2^2$ x 18, $D_6^2:C_6$ x 18, $C_6.D_6^2$ x 18, $D_6^2:S_3$ x 18, $C_6.D_6^2$ x 18, $C_6^3:C_4$ x 12, $C_6^3:C_4$ x 12, $C_2\times C_6^2:D_6$ x 12, $C_6^3:C_4$ x 2, $C_6^3:C_4$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6.D_6^2$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_2^2\times C_6^2:C_6$, $C_6^2:S_4$, $C_6^2:S_4$, $C_6^2.S_4$

Order 768: $C_2^5:S_4$, $C_2^6:D_6$

Order 648: $S_3\wr C_3$ x 12, $C_2\times C_3^3:D_6$ x 12, $C_2\times C_3^3:A_4$ x 9, $C_3^3:S_4$ x 8, $C_3^3:S_4$ x 8, $(C_3\times C_6^2):C_6$ x 4, $C_3^3:S_4$ x 4, $C_3^3:S_4$ x 4, $C_6\wr C_3$ x 3, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $C_2^2\times C_3\wr S_3$

Order 576: $D_6^2:C_2^2$ x 448, $D_6^2:C_2^2$ x 88, $C_2^2\times D_6^2$ x 80, $D_6^2:C_4$ x 56, $C_4:D_6^2$ x 48, $C_2^2:D_6^2$ x 45, $C_2^2\times C_6^2:C_4$ x 28, $C_2^2:D_6^2$ x 4, $C_2^2\times C_6^2:C_4$ x 2, $C_6^2:C_2^4$ x 2, $C_4:D_6^2$ x 2, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^2:D_6^2$, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_4:D_6^2$, $C_4\times D_6^2$

Order 432: $C_2\times S_3^3$ x 1600, $S_3^3:C_2$ x 512, $C_3:D_6^2$ x 364, $C_3\times D_6^2$ x 339, $S_3^2:D_6$ x 256, $C_3:D_6^2$ x 134, $C_2\times C_3^2:D_{12}$ x 128, $C_6\times \SOPlus(4,2)$ x 128, $C_3^2:C_4\times D_6$ x 128, $C_6^2:D_6$ x 64, $C_6^2:D_6$ x 64, $D_6:S_3^2$ x 64, $D_6:S_3^2$ x 64, $C_6^2:D_6$ x 40, $C_6^2:D_6$ x 40, $D_6:S_3^2$ x 40, $D_6:S_3^2$ x 40, $C_2.S_3^3$ x 40, $C_6.\SOPlus(4,2)$ x 32, $(C_3\times S_3^2):C_4$ x 32, $C_6.\SOPlus(4,2)$ x 32, $S_3^2:C_{12}$ x 32, $(C_3^2\times D_6):C_4$ x 32, $C_6^2.D_6$ x 24, $C_6^3:C_2$ x 23, $D_6\times C_6^2$ x 23, $C_2^2\times C_3^3:C_4$ x 20, $C_6^2:C_{12}$ x 20, $(C_3\times C_6^2):C_4$ x 16, $C_6^2:C_{12}$ x 16, $C_6^2:D_6$ x 16, $C_6^3:C_2$ x 14, $C_6^3:C_2$ x 12, $D_6:C_6^2$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 12, $C_6^2.D_6$ x 9, $C_6^2.D_6$ x 9, $C_2\times C_6^2:C_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 6, $C_2\times C_6^3$, $C_6^2:C_{12}$, $C_6^2:C_{12}$, $C_6^2:A_4$, $C_6^2.A_4$, $C_6^2:D_6$

Order 384: $C_2^5:D_6$ x 48, $C_2^2\wr S_3$ x 14, $C_{12}:C_2^5$ x 4, $C_2^5:D_6$ x 3, $C_2^4:D_{12}$ x 3, $C_2^5.D_6$ x 3, $D_6\times C_2^5$, $C_2^5:A_4$, $C_2^6:C_6$, $C_2^5:D_6$

Order 324: $C_3^3:D_6$ x 16, $C_3^3:D_6$ x 6, $\He_3:D_6$ x 6, $C_3^3:D_6$ x 6, $C_3^3:A_4$ x 3, $C_3^2.C_6^2$, $C_3^3:A_4$

Order 256: $C_2^5:D_4$

Order 216: $S_3^3$ x 8, $C_6:S_3^2$ x 6, $C_6^2:S_3$ x 5, $C_6^3$ x 3

Order 108: $C_3^2:D_6$ x 6, $C_3\times C_6^2$ x 2, $C_3:S_3^2$ x 2

Order 81: $C_3\wr C_3$

Order 54: $C_3^2:S_3$ x 4, $C_3^2\times C_6$ x 3

Order 27: $C_3^3$

Order 16: $C_2^4$

Order 8: $C_2^3$ x 3

Order 4: $C_2^2$ x 2

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 64 (order at least 324) are shown, as well as all normal subgroups of any index.

Order 20736: $C_2^2\times S_3^3:S_4$

Order 10368: $C_2\times C_6^3:S_4$, $C_2\times C_6^3:S_4$, $C_2^2\times S_3^3:A_4$, $D_6\wr S_3$

Order 6912: $C_2^2\times C_3^3.C_2^5.C_2$

Order 5184: $C_2^2\times S_3\wr S_3$, $C_2\times C_6^3:A_4$, $C_6^2.D_6^2$, $C_6^3:S_4$, $D_6\wr C_3$, $C_6^3:S_4$, $S_3^3:S_4$

Order 3456: $D_6\times D_6\wr C_2$ x 3, $C_2^2\times (C_2\times S_3^3).C_2$, $C_2^2\times C_3:S_3.(C_6\times D_4)$, $C_2^2\times (C_6\times C_3:S_3).C_2^3$, $C_2^2\times C_3^2.C_2^4:S_3$, $C_2^2\times (C_2\times S_3^3).C_2$, $C_2^2\times C_3^2.C_2^3:D_6$, $C_2^2\times C_3.(C_2\times D_6:D_6)$, $C_2^3\times C_2\times S_3^3$, $C_2^2\times (C_2\times S_3^3).C_2$, $C_2^2\times C_3^3.C_2^2\wr C_2$, $S_3^3:C_2^4$

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

Order 2304: $C_2^2\times D_6^2.C_2^2$

Order 1728: $D_6\times D_6^2$ x 9, $D_6^2:D_6$ x 8, $D_6:D_6^2$ x 4, $D_6^2:D_6$ x 4, $C_6^3:D_4$ x 3, $C_6^3:D_4$ x 3, $C_6^3:D_4$ 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, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:D_4$, $C_6^3:D_4$, $C_6^2:C_4\times D_6$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_6^3:C_2^3$, $C_2\times C_6^3:C_4$, $C_2\times C_6^3:C_4$, $D_6:D_6^2$, $D_6:D_6^2$, $D_6:D_6^2$, $D_6:D_6^2$, $D_6.D_6^2$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $D_6^2:C_{12}$, $(D_6\times C_6^2):C_4$, $C_2^2\times C_6^2:D_6$

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

Order 1152: $D_6^2:D_4$ x 9, $C_2^3\times D_6^2$, $D_6^2:C_2^3$, $C_2^3\times D_6:D_6$, $D_4\times D_6^2$, $C_2^3\times S_3^2:C_4$, $C_2^3\times C_3:S_3.D_4$

Order 864: $S_3\times D_6^2$ x 51, $S_3^3:C_2^2$ x 13, $C_6:D_6^2$ x 12, $C_6\times D_6^2$ x 11, $D_6^2:S_3$ x 10, $C_6:D_6^2$ x 7, $C_6^2:D_{12}$ x 6, $C_2^2:S_3^3$ x 5, $D_6^2:C_6$ x 4, $C_6^2:C_4\times S_3$ x 4, $C_6^3:C_2^2$ x 4, $D_6^2:C_6$ x 4, $C_6^2:D_{12}$ x 4, $(S_3\times D_6):D_6$ x 4, $C_6^3:C_2^2$ x 2, $D_6^2:C_6$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3:C_4$ x 2, $C_6^3:C_4$ 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, $C_6^2:C_4\times S_3$ x 2, $S_3^2:D_{12}$ x 2, $D_6:\SOPlus(4,2)$ x 2, $S_3^3:C_4$ x 2, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_4$, $C_6^3:C_4$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6.D_6^2$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $D_6^2.S_3$, $D_6^2.C_6$, $(S_3\times C_6^2):C_4$, $C_2^2\times C_6^2:C_6$, $C_6^2:S_4$, $C_6^2:S_4$, $C_6^2.S_4$, $C_2\times C_6^2:D_6$

Order 768: $C_2^5:S_4$, $C_2^6:D_6$

Order 648: $C_2\times C_3^3:A_4$ x 2, $S_3\wr C_3$ x 2, $C_3^3:S_4$, $C_3^3:S_4$, $C_2\times C_3^3:D_6$, $C_2^2\times C_3^3:C_6$, $C_2^2\times C_3\wr S_3$, $C_6\wr C_3$, $C_2\times C_3^3:D_6$, $(C_3\times C_6^2):C_6$, $C_3^3:S_4$, $C_3^3:S_4$

Order 576: $C_2^2\times D_6^2$ x 21, $D_6^2:C_2^2$ x 18, $C_2^2:D_6^2$ x 13, $D_6^2:C_2^2$ x 12, $D_6^2:C_4$ x 6, $C_2^2\times C_6^2:C_4$ x 5, $C_2^2:D_6^2$ x 4, $C_4:D_6^2$ x 4, $C_6^2:C_2^4$ x 2, $C_4:D_6^2$ x 2, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^2\times C_6^2:C_4$, $C_2^2:D_6^2$, $C_6^2:C_2^4$, $C_6^2:C_2^4$, $C_2^2.D_6^2$, $C_2^2.D_6^2$, $C_4:D_6^2$, $C_4\times D_6^2$

Order 432: $C_2\times S_3^3$ x 82, $C_3:D_6^2$ x 39, $C_3\times D_6^2$ x 38, $C_3:D_6^2$ x 32, $S_3^2:D_6$ x 13, $C_6^3:C_2$ x 9, $C_2\times C_3^2:D_{12}$ x 9, $S_3^3:C_2$ x 8, $D_6\times C_6^2$ x 7, $C_6^2:D_6$ x 5, $C_6^2:D_6$ x 5, $C_6^3:C_2$ x 4, $C_6\times \SOPlus(4,2)$ x 4, $C_3^2:C_4\times D_6$ x 4, $C_2^2\times C_3^3:C_4$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2.D_6$ x 3, $C_6^3:C_2$ 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, $C_6^2.D_6$ x 2, $(C_3\times C_6^2):C_4$ x 2, $C_6^2:C_{12}$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $C_2.S_3^3$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_6.\SOPlus(4,2)$ x 2, $C_2\times C_6^3$, $C_6^2:C_{12}$, $C_6^2:C_{12}$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $(C_3\times S_3^2):C_4$, $S_3^2:C_{12}$, $(C_3^2\times D_6):C_4$, $C_6^2:A_4$, $C_6^2.A_4$, $C_6^2:D_6$, $C_2\times C_6^2:C_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2:D_6$

Order 384: $C_2^2\wr S_3$ x 3, $C_{12}:C_2^5$ x 3, $C_2^5:D_6$ x 3, $C_2^5:D_6$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $D_6\times C_2^5$, $C_2^5:A_4$, $C_2^6:C_6$, $C_2^5:D_6$

Order 324: $C_3^3:A_4$ x 2, $C_3^2.C_6^2$, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:A_4$, $C_3^3:D_6$

Order 256: $C_2^5:D_4$

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

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

Order 81: $C_3\wr C_3$

Order 54: $C_3^2\times C_6$, $C_3^2:S_3$

Order 27: $C_3^3$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 20736: $C_2^2\times S_3^3:S_4$ ($C_1$)

Order 10368: $D_6\wr S_3$ ($C_2$) x 12, $C_2\times C_6^3:S_4$ ($C_2$), $C_2\times C_6^3:S_4$ ($C_2$), $C_2^2\times S_3^3:A_4$ ($C_2$)

Order 5184: $S_3^3:S_4$ ($C_2^2$) x 16, $C_6^3:S_4$ ($C_2^2$) x 6, $D_6\wr C_3$ ($C_2^2$) x 6, $C_6^3:S_4$ ($C_2^2$) x 6, $C_2\times C_6^3:A_4$ ($C_2^2$)

Order 3456: $C_2^3\times C_2\times S_3^3$ ($S_3$)

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

Order 1728: $D_6\times D_6^2$ ($D_6$) x 6, $C_6^3:C_2^3$ ($D_6$)

Order 1296: $(C_3\times C_6^2):A_4$ ($C_2^4$)

Order 864: $S_3\times D_6^2$ ($C_2\times D_6$) x 4, $C_6:D_6^2$ ($C_2\times D_6$) x 3, $S_3\times D_6^2$ ($S_4$) x 2, $C_6^3:C_2^2$ ($S_4$)

Order 432: $C_2\times S_3^3$ ($C_2\times S_4$) x 12, $C_6^3:C_2$ ($C_2\times S_4$) x 6, $C_3:D_6^2$ ($C_2\times S_4$) x 2, $C_2\times C_6^3$ ($C_2\times S_4$), $C_3:D_6^2$ ($C_2^2\times D_6$)

Order 216: $S_3^3$ ($C_2^2\times S_4$) x 8, $C_6:S_3^2$ ($C_2^2\times S_4$) x 6, $C_6^2:S_3$ ($C_2^2\times S_4$) x 4, $C_6^3$ ($C_2^2\times S_4$) x 3, $C_6^2:S_3$ ($C_2^2:S_4$)

Order 108: $C_3^2:D_6$ ($C_2^3:S_4$) x 6, $C_3:S_3^2$ ($C_2^3\times S_4$) x 2, $C_3\times C_6^2$ ($C_2^3:S_4$), $C_3\times C_6^2$ ($C_2^3\times S_4$)

Order 54: $C_3^2:S_3$ ($C_2^2\wr S_3$) x 4, $C_3^2\times C_6$ ($C_2^2\wr S_3$) x 3

Order 27: $C_3^3$ ($C_2^5:S_4$)

Order 16: $C_2^4$ ($S_3\wr S_3$)

Order 8: $C_2^3$ ($C_3^3.(C_2^2\times S_4)$) x 3

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

Order 2: $C_2$ ($D_6\wr S_3$) x 3

Order 1: $C_1$ ($C_2^2\times S_3^3:S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 20736: $C_2^2\times S_3^3:S_4$ ($C_1$)

Order 10368: $C_2\times C_6^3:S_4$ ($C_2$), $C_2\times C_6^3:S_4$ ($C_2$), $C_2^2\times S_3^3:A_4$ ($C_2$), $D_6\wr S_3$ ($C_2$)

Order 5184: $C_2\times C_6^3:A_4$ ($C_2^2$), $C_6^3:S_4$ ($C_2^2$), $D_6\wr C_3$ ($C_2^2$), $C_6^3:S_4$ ($C_2^2$), $S_3^3:S_4$ ($C_2^2$)

Order 3456: $C_2^3\times C_2\times S_3^3$ ($S_3$)

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

Order 1728: $C_6^3:C_2^3$ ($D_6$), $D_6\times D_6^2$ ($D_6$)

Order 1296: $(C_3\times C_6^2):A_4$ ($C_2^4$)

Order 864: $C_6^3:C_2^2$ ($S_4$), $C_6:D_6^2$ ($C_2\times D_6$), $S_3\times D_6^2$ ($C_2\times D_6$), $S_3\times D_6^2$ ($S_4$)

Order 432: $C_2\times C_6^3$ ($C_2\times S_4$), $C_6^3:C_2$ ($C_2\times S_4$), $C_3:D_6^2$ ($C_2^2\times D_6$), $C_3:D_6^2$ ($C_2\times S_4$), $C_2\times S_3^3$ ($C_2\times S_4$)

Order 216: $C_6^3$ ($C_2^2\times S_4$), $C_6^2:S_3$ ($C_2^2:S_4$), $C_6^2:S_3$ ($C_2^2\times S_4$), $C_6:S_3^2$ ($C_2^2\times S_4$), $S_3^3$ ($C_2^2\times S_4$)

Order 108: $C_3\times C_6^2$ ($C_2^3:S_4$), $C_3\times C_6^2$ ($C_2^3\times S_4$), $C_3^2:D_6$ ($C_2^3:S_4$), $C_3:S_3^2$ ($C_2^3\times S_4$)

Order 54: $C_3^2\times C_6$ ($C_2^2\wr S_3$), $C_3^2:S_3$ ($C_2^2\wr S_3$)

Order 27: $C_3^3$ ($C_2^5:S_4$)

Order 16: $C_2^4$ ($S_3\wr S_3$)

Order 8: $C_2^3$ ($C_3^3.(C_2^2\times S_4)$)

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

Order 2: $C_2$ ($D_6\wr S_3$)

Order 1: $C_1$ ($C_2^2\times S_3^3:S_4$)

Series

Derived series

$C_2^2\times S_3^3:S_4$

$\rhd$

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

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_2^2\times S_3^3:S_4$

$\rhd$

$C_2^2\times S_3^3:A_4$

$\rhd$

$C_2\times C_6^3:A_4$

$\rhd$

$C_6^3:A_4$

$\rhd$

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

$\rhd$

$C_3:D_6^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Lower central series

$C_2^2\times S_3^3:S_4$

$\rhd$

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

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 8 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 $196 \times 196$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.