Abstract group 93312.fs: $C_6^4.\SOPlus(4,2)$

• Label: 93312.fs
• Order: \( 2^{7} \cdot 3^{6} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 3 \)
• Perm deg.: $26$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^4.\SOPlus(4,2)$
Order:	\(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_3^3.A_4^2.C_6^2.C_2$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
Composition factors:	$C_2$ x 7, $C_3$ x 6
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	9	12	18	24
Elements	1	1851	1376	6804	19776	3888	5184	31104	15552	7776	93312
Conjugacy classes	1	6	12	6	54	1	3	19	3	2	107
Divisions	1	6	12	6	50	1	2	16	2	1	97
Autjugacy classes	1	6	8	6	42	1	3	17	3	2	89

Dimension	1	2	4	6	8	9	12	18	24	36	72	144
Irr. complex chars.	4	5	8	8	1	4	24	9	19	14	11	0	107
Irr. rational chars.	4	3	5	6	3	4	25	5	19	12	9	2	97

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	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:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=d^{9}=e^{6}=f^{6}=g^{2}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(1,3)(2,5)(4,8)(6,10)(9,13)(12,16)(20,22)(21,23)(24,26), (1,2,4,7,3,6,8,11,15,16,17,18)(5,9,12,13)(10,14)(19,20,21,22,23,24,25,26)\rangle$
Transitive group:	36T19592	36T19593	36T19594	36T19595	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^4.S_3^2)$ $\,\rtimes\,$ $C_2$	$(C_3^4.A_4^2:C_4)$ $\,\rtimes\,$ $C_2$	$((C_2\times C_6^3).S_3^2)$ $\,\rtimes\,$ $S_3$	$C_2^4$ $\,\rtimes\,$ $(C_3^4.\SOPlus(4,2))$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $(S_4\wr C_2)$	$C_6^4$ . $\SOPlus(4,2)$	$C_3^3$ . $(C_3.S_4\wr C_2)$	$(C_3^3.A_4^2.C_2)$ . $D_6$	all 10

Elements of the group are displayed as permutations of degree 26.

Homology

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

Subgroups

There are 1367914 subgroups in 6236 conjugacy classes, 19 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

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

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

All subgroups of index up to 243 (order at least 384) are shown, as well as all normal subgroups of any index.

Order 93312: $C_6^4.\SOPlus(4,2)$

Order 46656: $C_3^3.A_4^2.D_6$, $C_3^4.A_4^2:C_4$, $C_6^4.S_3^2$

Order 31104: $C_3^3.S_4\wr C_2$

Order 23328: $C_3^3.A_4^2.C_6$, $C_6^2.C_3^3.A_4.C_2$, $C_6^4.(C_3\times S_3)$

Order 15552: $C_6^2:C_3.D_6:D_6$, $C_3^3.A_4^2.C_2^2$, $C_3^3.A_4^2:C_4$, $(C_2\times C_6^3).S_3^2$, $C_6^4.D_6$

Order 11664: $C_6^4.C_3^2$, $C_3^3.C_6^2.D_6$

Order 10368: $C_6^4:D_4$

Order 7776: $C_6^2.C_6^2.C_6$ x 2, $C_3^3.A_4\wr C_2$ x 2, $C_6^2.C_3^3.D_4$, $C_2^4.C_3^2:C_9.C_6$, $C_6^2.C_3^3.D_4$, $C_3^3.A_4^2.C_2$, $C_6^2.C_3^3.C_2^3$, $C_6^3.C_6^2$, $C_6^2.C_3^3.C_2^3$, $C_6^2.C_3^3:C_2:C_4$, $(C_3\times C_6\times A_4).C_6^2$, $C_6^4.S_3$, $C_6^4.C_6$

Order 5832: $C_3.\He_3:A_4.C_6$, $C_3^3.C_6^2.C_6$, $C_6^2.C_3^3.C_6$, $C_3^4.\SOPlus(4,2)$

Order 5184: $C_6^2.D_6:D_6$ x 3, $C_6^2.C_6^2.C_2^2$ x 3, $C_6^4.C_2^2$, $C_6^2.C_6^2.C_2^2$, $(C_2^2\times C_6^2):D_{18}$, $C_6^2.(C_6\times S_4)$, $C_6^3.D_{12}$, $C_6^3.D_{12}$, $C_6^3:D_{12}$, $C_6^3.D_{12}$, $C_6\wr C_4$, $C_6\wr C_2^2$

Order 3888: $C_3^3.A_4^2$ x 3, $C_6^2:C_3.S_3^2$ x 2, $C_6^2.C_3^3.C_2^2$ x 2, $\He_3.D_6:D_6$, $C_3^3.(C_{12}\times A_4)$, $(C_3\times C_6^2).C_6^2$, $C_6^2.C_3^3.C_2^2$, $C_3^3.(S_3\times S_4)$, $(C_3\times A_4).C_3^3.C_4$, $(C_3^2\times A_4).C_6^2$, $C_6^4.C_3$, $C_3^3.(C_6\times S_4)$

Order 3456: $(C_2\times C_6^3):D_4$, $(C_2\times C_6^3):D_4$

Order 2916: $C_3^3.C_6^2.C_3$, $C_3^4.S_3^2$, $C_3^3.C_3^3:C_4$, $C_3^4.S_3^2$

Order 2592: $C_6^2.C_6^2.C_2$ x 10, $C_6^3.C_6.C_2$ x 3, $C_6^2:C_3.D_6.C_2$ x 3, $(C_3\times A_4).C_6^2.C_2$ x 3, $C_6^2:C_3.D_6.C_2$ x 3, $C_6^3:D_6$ x 3, $C_6^3.D_6$ x 3, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 3, $C_6^3:D_6$ x 3, $C_3^3:\GL(2,\mathbb{Z}/4)$ x 3, $C_6^2:(C_6\times A_4)$ x 3, $C_6^2.C_6^2:C_2$ x 2, $C_6^4.C_2$ x 2, $C_2^4.C_3:D_9:C_3$ x 2, $C_6^2:C_3.D_6.C_2$ x 2, $(C_2^2\times C_6^2):C_{18}$ x 2, $C_6^3.D_6$ x 2, $C_6^4.C_2$, $C_3^4.C_2^2:D_4$, $(C_3^3\times A_4).C_2^3$, $C_3^4.C_4:D_4$, $(C_3^2\times A_4).D_{12}$, $(C_3^3\times A_4).D_4$, $C_6^3.C_6.C_2$, $(C_3^2\times A_4).D_6.C_2$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $(C_3^3\times A_4).D_4$, $C_3^4.C_2^2:D_4$, $C_3^4.C_2^4.C_2$, $(C_3\times A_4).C_6^2.C_2$, $(C_3\times C_2^4:C_9).C_6$, $A_4.C_3:(C_2.S_3^2)$, $C_3^4.(C_4\times D_4)$, $C_6^3:D_6$, $(C_2^2\times C_6^2):D_9$, $C_6^2.(C_6\times A_4)$, $C_3\times C_6^2.S_4$, $C_6^2.\SOPlus(4,2)$, $C_6^2.\SOPlus(4,2)$, $C_6^2.\SOPlus(4,2)$, $C_6^2:\SOPlus(4,2)$, $C_6^3.C_{12}$, $C_6^3:C_{12}$, $C_6^3:D_6$

Order 1944: $C_3^3.(S_3\times A_4)$ x 2, $(C_3\times \He_3):S_4$ x 2, $C_3^4:S_4$ x 2, $C_6^2:(S_3\times C_3^2)$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4.S_4$ x 2, $(C_3\times \He_3):S_4$, $C_3^3.\SOPlus(4,2)$, $C_2\times C_3^4:D_6$, $C_6^3:C_3^2$, $C_2\times C_3^3:S_3^2$, $C_3^3:(C_6\times A_4)$, $C_3^4:S_4$, $C_3^4:(C_4\times S_3)$, $C_3^3.(C_6\times A_4)$, $C_3^4.S_4$

Order 1728: $C_6^2:(C_2\times S_4)$ x 9, $C_6^2:(C_2\times S_4)$ x 3, $C_3^3.D_4^2$ x 3, $C_6^3.C_2^3$ x 2, $C_6^2:(C_2\times S_4)$, $C_6^2:(C_2\times S_4)$, $C_3^3.D_4^2$, $C_6^3.D_4$, $C_6^3:D_4$, $C_6^3.D_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $(C_2\times C_6^3):C_4$, $(C_2\times C_6^3):C_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^2.(C_2\times S_4)$, $C_2^4:(S_3\times D_9)$

Order 1458: $(C_3^2\times \He_3):S_3$, $(C_3^2\times \He_3):C_6$, $C_3^4.(C_3\times S_3)$

Order 1296: $C_6^3:S_3$ x 42, $C_6^3:C_6$ x 10, $C_6^2.S_3^2$ x 6, $C_6^2:S_3^2$ x 6, $C_6^2:S_3^2$ x 6, $C_6^3:S_3$ x 5, $C_6^2:S_3^2$ x 5, $C_6^3:C_6$ x 4, $C_6^3:C_6$ x 3, $C_6^3:C_6$ x 3, $C_6^3:C_6$ x 3, $C_6^3:S_3$ x 3, $C_6^3:S_3$ x 3, $C_6^3.C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^3.S_3$ x 3, $C_6^3.S_3$ x 3, $(C_2^2\times C_6^2):C_9$ 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, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^4$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^2:S_3^2$, $C_6^3:C_6$, $C_2\times C_3^3:D_{12}$, $C_6^3.C_6$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $(C_3\times C_6^2):C_{12}$, $C_3^4:\OD_{16}$, $C_6^3.C_6$, $C_6^3.S_3$, $C_3^4:\SD_{16}$, $C_3^3:D_{24}$, $(C_3^2\times C_6).D_{12}$, $C_3\times C_6^2.A_4$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2:D_{18}$

Order 1152: $D_6^2:D_4$, $(C_2^3\times C_6):D_{12}$

Order 972: $C_6^2.\He_3$ x 4, $C_3^4:D_6$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4.A_4$ x 2, $C_3^3.C_3^2:C_4$, $\He_3\times C_6^2$, $C_3^4:A_4$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$

Order 864: $C_6^2:S_4$ x 18, $C_2\times C_6^2:D_6$ x 9, $C_6^2:(C_2\times A_4)$ x 9, $C_6^2:S_4$ x 9, $C_6^2:D_{12}$ x 9, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 9, $C_3^2:(C_4\times S_4)$ x 9, $C_6^2:S_4$ x 8, $C_6^3:C_2^2$ x 6, $C_6^2:S_4$ x 5, $C_6^3.C_2^2$ x 5, $C_6^3:C_2^2$ x 4, $C_6^3.C_2^2$ x 4, $C_6^2:D_{12}$ x 4, $C_6^3.C_2^2$ x 4, $C_6^3.C_2^2$ x 4, $C_6^2:D_{12}$ x 4, $C_2\times A_4:S_3^2$ x 3, $A_4\times C_6^2:C_2$ x 3, $C_6\wr C_2\times A_4$ x 3, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_6^2:D_{12}$ x 3, $C_3^2:(C_4\times S_4)$ x 3, $C_6^2.S_4$ x 3, $C_6^3:C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^2.D_{12}$ x 3, $C_2^4:(S_3\times C_9)$ x 2, $S_3\times C_6:S_4$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:S_4$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^2:S_4$, $C_6^2:D_{12}$, $C_6^2.D_{12}$, $(S_3\times D_6):D_6$, $(C_3\times C_6^2).D_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:(C_4\times S_4)$, $C_3^2:(C_4\times S_4)$, $C_6^2.S_4$, $C_6^2.S_4$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_4$, $C_6^3:C_4$, $C_6^3.C_4$, $C_6^3:C_4$, $C_6^2.D_{12}$, $C_6^2.D_{12}$, $(C_3\times C_6^2).D_4$, $(C_6\times S_3^2):C_4$, $(C_2\times C_6^2).A_4$

Order 729: $C_3^3.C_3^3$

Order 648: $C_3^3:S_4$ x 31, $C_6^3:C_3$ x 21, $C_2\times C_3^3:D_6$ x 10, $C_3^4:D_4$ x 6, $C_3^4:D_4$ x 6, $C_2\times C_3^2:S_3^2$ x 6, $(C_3\times C_6^2):C_6$ x 6, $C_3\times C_6^2:C_6$ x 6, $C_3^3:D_{12}$ x 6, $C_3^3:D_{12}$ x 6, $C_3^3:(C_4\times S_3)$ x 6, $C_3^4:C_2^3$ x 5, $C_6^2:D_9$ x 5, $C_3^4:C_2^3$ x 4, $C_3^3:S_4$ x 4, $C_3^3:D_{12}$ x 4, $C_6\times C_3^2:C_{12}$ x 3, $C_3^3:D_{12}$ x 3, $C_3^3:(C_4\times S_3)$ x 3, $(C_3\times C_6^2):C_6$ x 3, $C_3\times C_6^2:C_6$ x 3, $C_3^3:S_4$ x 3, $C_3^3:S_4$ x 3, $C_3\times C_6^3$ x 2, $C_3^4:C_2^3$ x 2, $C_6^3:C_3$ x 2, $C_3^3:D_{12}$ x 2, $C_2\times C_3^3:C_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\times C_6.S_3^2$ x 2, $C_3^3.S_4$ x 2, $C_3^3.S_4$ x 2, $C_6^2:C_{18}$ x 2, $C_3^4:C_2^3$, $C_2\times C_3\wr C_4$, $C_3\times C_6^2:C_6$, $C_3^4:Q_8$, $C_3^3:C_{24}$, $S_3\times C_3^2.A_4$, $C_6^2:D_9$

Order 576: $C_3^2:D_4^2$ x 4, $\GL(2,\mathbb{Z}/4):S_3$ x 3, $C_2^4:S_3^2$ x 3, $C_3\times C_2^4:D_6$ x 2, $C_2^4:S_3^2$ x 2, $D_4\times C_3:S_4$, $C_3^2:D_4^2$, $C_2^4:S_3^2$, $C_2^3.\SOPlus(4,2)$, $D_6^2:C_2^2$, $(C_2\times C_{12}):D_{12}$, $(C_3\times D_{12}):D_4$, $(C_2^2\times C_6^2):C_4$, $C_2^3.\SOPlus(4,2)$, $D_6^2:C_4$, $C_2^4:D_{18}$, $(C_2^3\times C_6):C_{12}$, $(C_6\times D_4).D_6$, $(C_2\times C_{12}).D_{12}$

Order 486: $C_3^3.(C_3\times S_3)$ x 2, $(C_3\times \He_3):C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $(C_3\times \He_3):S_3$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.S_3$, $C_3^4.C_6$

Order 432: $C_6^2:D_6$ x 90, $C_6^3:C_2$ x 46, $C_6^2:D_6$ x 45, $C_6^2:D_6$ x 21, $C_6^2:A_4$ x 18, $C_6^2:D_6$ x 18, $A_4\times C_6^2$ x 12, $C_6^2.D_6$ x 11, $C_6^2.D_6$ x 11, $C_6^2:C_{12}$ x 10, $C_2\times C_6^2:C_6$ x 9, $C_6^2:D_6$ x 9, $C_6^2:D_6$ x 9, $C_6^2:C_{12}$ x 9, $(C_3\times A_4):C_{12}$ x 9, $D_6:C_6^2$ x 8, $C_6^2:C_{12}$ x 8, $C_2\times C_6^3$ x 6, $C_6^3:C_2$ x 6, $A_4:S_3^2$ x 6, $C_6^2:C_{12}$ x 6, $C_6^2:D_6$ x 5, $C_6^2:D_6$ x 5, $C_6^2:D_6$ x 5, $C_3\times C_6.S_4$ x 5, $A_4:S_3^2$ x 4, $C_6^3:C_2$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_3:D_6^2$ x 3, $C_6:S_3\times A_4$ x 3, $C_6\times S_3\times A_4$ x 3, $C_6^2.D_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.A_4$ x 2, $C_6^2.A_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^3:C_2$, $C_3:D_6^2$, $C_2\times C_3^2:D_{12}$, $S_3^2:D_6$, $C_6^3.C_2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $(C_3\times C_6^2):C_4$, $C_3^3:\OD_{16}$, $C_6^2:C_{12}$, $C_6^2.C_{12}$, $(C_3^2\times A_4):C_4$, $C_3^3:\SD_{16}$, $C_3^2:D_{24}$, $C_6.\SOPlus(4,2)$, $C_3^3:\SD_{16}$, $C_3^3:D_8$, $C_6.\SOPlus(4,2)$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$

Order 384: $(C_2\times D_{12}):D_4$, $C_2^4:D_{12}$

Order 144: $C_2^2\times C_6^2$

Order 128: $C_2\wr D_4$

Order 81: $C_3^4$

Order 27: $C_3^3$ x 2

Order 24: $C_3:D_4$ x 2

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $D_4$ x 2

Order 6: $S_3$ x 22

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 243 (order at least 384) are shown, as well as all normal subgroups of any index.

Order 93312: $C_6^4.\SOPlus(4,2)$

Order 46656: $C_3^3.A_4^2.D_6$, $C_3^4.A_4^2:C_4$, $C_6^4.S_3^2$

Order 31104: $C_3^3.S_4\wr C_2$

Order 23328: $C_3^3.A_4^2.C_6$, $C_6^2.C_3^3.A_4.C_2$, $C_6^4.(C_3\times S_3)$

Order 15552: $C_6^2:C_3.D_6:D_6$, $C_3^3.A_4^2.C_2^2$, $C_3^3.A_4^2:C_4$, $(C_2\times C_6^3).S_3^2$, $C_6^4.D_6$

Order 11664: $C_6^4.C_3^2$, $C_3^3.C_6^2.D_6$

Order 10368: $C_6^4:D_4$

Order 7776: $C_6^2.C_6^2.C_6$ x 2, $C_3^3.A_4\wr C_2$ x 2, $C_6^2.C_3^3.D_4$, $C_2^4.C_3^2:C_9.C_6$, $C_6^2.C_3^3.D_4$, $C_3^3.A_4^2.C_2$, $C_6^2.C_3^3.C_2^3$, $C_6^3.C_6^2$, $C_6^2.C_3^3.C_2^3$, $C_6^2.C_3^3:C_2:C_4$, $(C_3\times C_6\times A_4).C_6^2$, $C_6^4.S_3$, $C_6^4.C_6$

Order 5832: $C_3.\He_3:A_4.C_6$, $C_3^3.C_6^2.C_6$, $C_6^2.C_3^3.C_6$, $C_3^4.\SOPlus(4,2)$

Order 5184: $C_6^2.C_6^2.C_2^2$ x 3, $C_6^2.D_6:D_6$, $C_6^4.C_2^2$, $C_6^2.C_6^2.C_2^2$, $(C_2^2\times C_6^2):D_{18}$, $C_6^2.(C_6\times S_4)$, $C_6^3.D_{12}$, $C_6^3.D_{12}$, $C_6^3:D_{12}$, $C_6^3.D_{12}$, $C_6\wr C_4$, $C_6\wr C_2^2$

Order 3888: $C_3^3.A_4^2$ x 3, $C_6^2:C_3.S_3^2$ x 2, $C_6^2.C_3^3.C_2^2$ x 2, $\He_3.D_6:D_6$, $C_3^3.(C_{12}\times A_4)$, $(C_3\times C_6^2).C_6^2$, $C_6^2.C_3^3.C_2^2$, $C_3^3.(S_3\times S_4)$, $(C_3\times A_4).C_3^3.C_4$, $(C_3^2\times A_4).C_6^2$, $C_6^4.C_3$, $C_3^3.(C_6\times S_4)$

Order 3456: $(C_2\times C_6^3):D_4$, $(C_2\times C_6^3):D_4$

Order 2916: $C_3^3.C_6^2.C_3$, $C_3^4.S_3^2$, $C_3^3.C_3^3:C_4$, $C_3^4.S_3^2$

Order 2592: $C_6^2.C_6^2.C_2$ x 6, $(C_3\times A_4).C_6^2.C_2$ x 3, $C_6^3:D_6$ x 3, $C_6^3.D_6$ x 3, $C_6^2:(C_6\times A_4)$ x 3, $C_6^2.C_6^2:C_2$ x 2, $C_6^4.C_2$ x 2, $C_2^4.C_3:D_9:C_3$ x 2, $C_6^2:C_3.D_6.C_2$ x 2, $(C_2^2\times C_6^2):C_{18}$ x 2, $C_6^3.D_6$ x 2, $C_6^4.C_2$, $C_3^4.C_2^2:D_4$, $(C_3^3\times A_4).C_2^3$, $C_3^4.C_4:D_4$, $(C_3^2\times A_4).D_{12}$, $(C_3^3\times A_4).D_4$, $C_6^3.C_6.C_2$, $C_6^3.C_6.C_2$, $(C_3^2\times A_4).D_6.C_2$, $C_6^2:C_3.D_6.C_2$, $C_3^3.C_2^3:D_6$, $C_3^4.C_2^4.C_2$, $(C_3^3\times A_4).D_4$, $C_3^4.C_2^2:D_4$, $C_6^2:C_3.D_6.C_2$, $C_3^4.C_2^4.C_2$, $(C_3\times A_4).C_6^2.C_2$, $(C_3\times C_2^4:C_9).C_6$, $A_4.C_3:(C_2.S_3^2)$, $C_3^4.(C_4\times D_4)$, $C_6^3:D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^3:D_6$, $(C_2^2\times C_6^2):D_9$, $C_6^2.(C_6\times A_4)$, $C_3\times C_6^2.S_4$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_6^2.\SOPlus(4,2)$, $C_6^2.\SOPlus(4,2)$, $C_6^2.\SOPlus(4,2)$, $C_6^2:\SOPlus(4,2)$, $C_6^3.C_{12}$, $C_6^3:C_{12}$, $C_6^3:D_6$

Order 1944: $C_3^3.(S_3\times A_4)$ x 2, $(C_3\times \He_3):S_4$ x 2, $C_3^4:S_4$ x 2, $C_6^2:(S_3\times C_3^2)$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4.S_4$ x 2, $(C_3\times \He_3):S_4$, $C_3^3.\SOPlus(4,2)$, $C_2\times C_3^4:D_6$, $C_6^3:C_3^2$, $C_2\times C_3^3:S_3^2$, $C_3^3:(C_6\times A_4)$, $C_3^4:S_4$, $C_3^4:(C_4\times S_3)$, $C_3^3.(C_6\times A_4)$, $C_3^4.S_4$

Order 1728: $C_3^3.D_4^2$ x 3, $C_6^2:(C_2\times S_4)$ x 3, $C_6^3.C_2^3$ x 2, $C_6^2:(C_2\times S_4)$, $C_6^2:(C_2\times S_4)$, $C_6^2:(C_2\times S_4)$, $C_3^3.D_4^2$, $C_6^3.D_4$, $C_6^3:D_4$, $C_6^3.D_4$, $C_6^3:D_4$, $C_6^3.C_2^3$, $(C_2\times C_6^3):C_4$, $(C_2\times C_6^3):C_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^3.D_4$, $C_6^2.(C_2\times S_4)$, $C_2^4:(S_3\times D_9)$

Order 1458: $(C_3^2\times \He_3):S_3$, $(C_3^2\times \He_3):C_6$, $C_3^4.(C_3\times S_3)$

Order 1296: $C_6^3:S_3$ x 22, $C_6^3:C_6$ x 6, $C_6^2.S_3^2$ x 6, $C_6^3:S_3$ x 5, $C_6^2:S_3^2$ x 5, $C_6^3:C_6$ x 4, $C_6^2:S_3^2$ x 4, $C_6^3:C_6$ x 3, $C_6^3:C_6$ x 3, $C_6^3:S_3$ x 3, $C_6^3.C_6$ x 3, $C_6^3.C_6$ x 3, $C_6^3.S_3$ x 3, $(C_2^2\times C_6^2):C_9$ 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, $C_6^2.S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^4$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^2:S_3^2$, $C_6^3:C_6$, $C_2\times C_3^3:D_{12}$, $C_6^3.C_6$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $(C_3\times C_6^2):C_{12}$, $C_3^4:\OD_{16}$, $C_6^3.C_6$, $C_6^3.S_3$, $C_3^4:\SD_{16}$, $C_3^3:D_{24}$, $(C_3^2\times C_6).D_{12}$, $C_3\times C_6^2.A_4$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^3.C_6$, $C_6^3.S_3$, $C_6^2:D_{18}$

Order 1152: $D_6^2:D_4$, $(C_2^3\times C_6):D_{12}$

Order 972: $C_6^2.\He_3$ x 4, $C_3^4:D_6$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4.A_4$ x 2, $C_3^3.C_3^2:C_4$, $\He_3\times C_6^2$, $C_3^4:A_4$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$

Order 864: $C_6^3:C_2^2$ x 6, $C_6^2:S_4$ x 6, $C_6^3.C_2^2$ x 5, $C_6^2:S_4$ x 4, $C_6^3:C_2^2$ x 4, $C_6^3.C_2^2$ x 4, $C_6^2:D_{12}$ x 4, $C_6^3.C_2^2$ x 4, $C_6^3.C_2^2$ x 4, $C_6^2:D_{12}$ x 4, $C_6\wr C_2\times A_4$ x 3, $C_6^2:S_4$ x 3, $C_6^2.S_4$ x 3, $C_2\times C_6^2:D_6$ x 3, $C_6^3:C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^2:(C_2\times A_4)$ x 3, $C_6^2:S_4$ x 3, $C_6^2:D_{12}$ x 3, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 3, $C_3^2:(C_4\times S_4)$ x 3, $C_6^2.D_{12}$ x 3, $C_2^4:(S_3\times C_9)$ x 2, $S_3\times C_6:S_4$ x 2, $C_6^3:C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:S_4$, $C_2\times A_4:S_3^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $C_6^3:C_2^2$, $A_4\times C_6^2:C_2$, $C_6^2:S_4$, $C_6^2:D_{12}$, $C_6^2.D_{12}$, $(S_3\times D_6):D_6$, $(C_3\times C_6^2).D_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_3^2:(C_4\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:(C_4\times S_4)$, $C_3^2:(C_4\times S_4)$, $C_6^2.S_4$, $C_6^2.S_4$, $C_6^3:C_2^2$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_2^2$, $C_6^3.C_4$, $C_6^3:C_4$, $C_6^3.C_4$, $C_6^3:C_4$, $C_6^2.D_{12}$, $C_6^2.D_{12}$, $(C_3\times C_6^2).D_4$, $(C_6\times S_3^2):C_4$, $(C_2\times C_6^2).A_4$

Order 729: $C_3^3.C_3^3$

Order 648: $C_3^3:S_4$ x 17, $C_6^3:C_3$ x 11, $C_3^4:D_4$ x 6, $C_3^4:D_4$ x 6, $C_2\times C_3^3:D_6$ x 6, $C_3^4:C_2^3$ x 5, $C_6^2:D_9$ x 5, $C_3^4:C_2^3$ x 4, $C_3^3:S_4$ x 4, $C_3^3:D_{12}$ x 4, $C_2\times C_3^2:S_3^2$ x 4, $(C_3\times C_6^2):C_6$ x 4, $C_3\times C_6^2:C_6$ x 4, $C_3^3:D_{12}$ x 4, $C_3^3:D_{12}$ x 4, $C_3^3:(C_4\times S_3)$ x 4, $C_6\times C_3^2:C_{12}$ x 3, $C_3^3:D_{12}$ x 3, $C_3^3:(C_4\times S_3)$ x 3, $C_3\times C_6^2:C_6$ x 3, $C_3^3:S_4$ x 3, $C_3\times C_6^3$ x 2, $C_3^4:C_2^3$ x 2, $C_6^3:C_3$ x 2, $C_3^3:D_{12}$ x 2, $C_2\times C_3^3:C_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\times C_6.S_3^2$ x 2, $C_3^3.S_4$ x 2, $C_3^3.S_4$ x 2, $C_6^2:C_{18}$ x 2, $C_3^4:C_2^3$, $C_2\times C_3\wr C_4$, $C_3\times C_6^2:C_6$, $C_3^4:Q_8$, $C_3^3:C_{24}$, $(C_3\times C_6^2):C_6$, $S_3\times C_3^2.A_4$, $C_3^3:S_4$, $C_6^2:D_9$

Order 576: $C_3^2:D_4^2$ x 4, $C_3\times C_2^4:D_6$ x 2, $C_2^4:S_3^2$ x 2, $D_4\times C_3:S_4$, $\GL(2,\mathbb{Z}/4):S_3$, $C_2^4:S_3^2$, $C_3^2:D_4^2$, $C_2^4:S_3^2$, $C_2^3.\SOPlus(4,2)$, $D_6^2:C_2^2$, $(C_2\times C_{12}):D_{12}$, $(C_3\times D_{12}):D_4$, $(C_2^2\times C_6^2):C_4$, $C_2^3.\SOPlus(4,2)$, $D_6^2:C_4$, $C_2^4:D_{18}$, $(C_2^3\times C_6):C_{12}$, $(C_6\times D_4).D_6$, $(C_2\times C_{12}).D_{12}$

Order 486: $C_3^3.(C_3\times S_3)$ x 2, $(C_3\times \He_3):C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $(C_3\times \He_3):S_3$, $C_3^4:C_6$, $C_3^4.S_3$, $C_3^4.S_3$, $C_3^4.C_6$

Order 432: $C_6^3:C_2$ x 46, $C_6^2:D_6$ x 30, $C_6^2:D_6$ x 23, $C_6^2:D_6$ x 19, $C_6^2.D_6$ x 11, $C_6^2.D_6$ x 11, $C_6^2:C_{12}$ x 10, $A_4\times C_6^2$ x 8, $D_6:C_6^2$ x 8, $C_6^2:C_{12}$ x 8, $C_2\times C_6^3$ x 6, $C_6^3:C_2$ x 6, $C_6^2:C_{12}$ x 6, $C_6^2:A_4$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 5, $C_6^2:D_6$ x 5, $C_6^2:D_6$ x 5, $A_4:S_3^2$ x 4, $C_6^3:C_2$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_3:D_6^2$ x 3, $C_6\times S_3\times A_4$ x 3, $C_6^2.D_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2:C_{12}$ x 3, $C_3\times C_6.S_4$ x 3, $C_2\times C_6^2:C_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2:C_{12}$ x 3, $(C_3\times A_4):C_{12}$ x 3, $A_4:S_3^2$ x 2, $C_6^2.D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2.A_4$ x 2, $C_6^2.A_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^3:C_2$, $C_3:D_6^2$, $C_6:S_3\times A_4$, $C_2\times C_3^2:D_{12}$, $S_3^2:D_6$, $C_6^3.C_2$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $(C_3\times C_6^2):C_4$, $C_3^3:\OD_{16}$, $C_6^2:C_{12}$, $C_6^2.C_{12}$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$, $C_3^3:\SD_{16}$, $C_3^2:D_{24}$, $C_6.\SOPlus(4,2)$, $C_3^3:\SD_{16}$, $C_3^3:D_8$, $C_6.\SOPlus(4,2)$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$, $C_6^2.D_6$

Order 384: $(C_2\times D_{12}):D_4$, $C_2^4:D_{12}$

Order 144: $C_2^2\times C_6^2$

Order 128: $C_2\wr D_4$

Order 81: $C_3^4$

Order 27: $C_3^3$ x 2

Order 24: $C_3:D_4$ x 2

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $D_4$ x 2

Order 6: $S_3$ x 14

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 93312: $C_6^4.\SOPlus(4,2)$ ($C_1$)

Order 46656: $C_3^3.A_4^2.D_6$ ($C_2$), $C_3^4.A_4^2:C_4$ ($C_2$), $C_6^4.S_3^2$ ($C_2$)

Order 23328: $C_3^3.A_4^2.C_6$ ($C_2^2$)

Order 15552: $(C_2\times C_6^3).S_3^2$ ($S_3$)

Order 11664: $C_6^4.C_3^2$ ($D_4$)

Order 7776: $C_3^3.A_4^2.C_2$ ($D_6$)

Order 3888: $C_3^3.A_4^2$ ($C_3:D_4$)

Order 1296: $C_6^4$ ($\SOPlus(4,2)$)

Order 432: $C_2\times C_6^3$ ($\He_3:D_4$), $C_2\times C_6^3$ ($S_3^2:S_3$)

Order 144: $C_2^2\times C_6^2$ ($C_3^2.\SOPlus(4,2)$)

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

Order 27: $C_3^3$ ($C_3.S_4\wr C_2$), $C_3^3$ ($(C_3\times A_4^2):D_4$)

Order 16: $C_2^4$ ($C_3^4.\SOPlus(4,2)$)

Order 9: $C_3^2$ ($C_3^2.S_4\wr C_2$)

Order 1: $C_1$ ($C_6^4.\SOPlus(4,2)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 93312: $C_6^4.\SOPlus(4,2)$ ($C_1$)

Order 46656: $C_3^3.A_4^2.D_6$ ($C_2$), $C_3^4.A_4^2:C_4$ ($C_2$), $C_6^4.S_3^2$ ($C_2$)

Order 23328: $C_3^3.A_4^2.C_6$ ($C_2^2$)

Order 15552: $(C_2\times C_6^3).S_3^2$ ($S_3$)

Order 11664: $C_6^4.C_3^2$ ($D_4$)

Order 7776: $C_3^3.A_4^2.C_2$ ($D_6$)

Order 3888: $C_3^3.A_4^2$ ($C_3:D_4$)

Order 1296: $C_6^4$ ($\SOPlus(4,2)$)

Order 432: $C_2\times C_6^3$ ($\He_3:D_4$), $C_2\times C_6^3$ ($S_3^2:S_3$)

Order 144: $C_2^2\times C_6^2$ ($C_3^2.\SOPlus(4,2)$)

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

Order 27: $C_3^3$ ($C_3.S_4\wr C_2$), $C_3^3$ ($(C_3\times A_4^2):D_4$)

Order 16: $C_2^4$ ($C_3^4.\SOPlus(4,2)$)

Order 9: $C_3^2$ ($C_3^2.S_4\wr C_2$)

Order 1: $C_1$ ($C_6^4.\SOPlus(4,2)$)

Series

Derived series

$C_6^4.\SOPlus(4,2)$

$\rhd$

$C_6^4.\SOPlus(4,2)$

$\rhd$

$C_3^3.A_4^2.C_6$

$\rhd$

$C_3^3.A_4^2.C_6$

$\rhd$

$C_3^3.A_4^2$

$\rhd$

$C_3^3.A_4^2$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_6^4.\SOPlus(4,2)$

$\rhd$

$C_6^4.\SOPlus(4,2)$

$\rhd$

$C_3^3.A_4^2.D_6$

$\rhd$

$C_3^3.A_4^2.D_6$

$\rhd$

$C_3^3.A_4^2.C_6$

$\rhd$

$C_3^3.A_4^2.C_6$

$\rhd$

$C_6^4.C_3^2$

$\rhd$

$C_6^4.C_3^2$

$\rhd$

$C_3^3.A_4^2$

$\rhd$

$C_3^3.A_4^2$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_6^4.\SOPlus(4,2)$

$\rhd$

$C_6^4.\SOPlus(4,2)$

$\rhd$

$C_3^3.A_4^2.C_6$

$\rhd$

$C_3^3.A_4^2.C_6$

$\rhd$

$C_6^4.C_3^2$

$\rhd$

$C_6^4.C_3^2$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $107 \times 107$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $97 \times 97$ rational character table.