Abstract group 93312.dm: $C_6^4:D_6:S_3$

• Label: 93312.dm
• Order: \( 2^{7} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $30$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^4:D_6:S_3$
Order:	\(93312\)\(\medspace = 2^{7} \cdot 3^{6} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^6:D_5$, of order \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \)
Composition factors:	$C_2$ x 7, $C_3$ x 6
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	895	6560	3456	39200	43200	93312
Conjugacy classes	1	7	37	6	213	50	314
Divisions	1	7	24	6	125	23	186
Autjugacy classes	1	7	20	5	99	17	149

Dimension	1	2	4	6	8	9	12	18	24	36	72
Irr. complex chars.	12	39	30	36	0	12	90	47	0	48	0	314
Irr. rational chars.	4	11	20	4	13	4	30	15	38	28	19	186

Minimal presentations

Permutation degree:	$30$
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:	${\langle a, b, c, d, e, f, g \mid a^{6}=b^{12}=c^{3}=d^{6}=e^{6}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $30$ $\langle(1,3,5)(4,7)(6,10,14,11,15,16)(8,12)(13,17)(19,21)(20,23)(22,25)(24,28) \!\cdots\! \rangle$
Transitive group:	36T19506				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_6^4:D_6)$ $\,\rtimes\,$ $S_3$	$C_6^4$ $\,\rtimes\,$ $(C_3:D_{12})$	$C_3^4$ $\,\rtimes\,$ $(A_4^2:D_4)$	$C_2^4$ $\,\rtimes\,$ $(\He_3^2:D_4)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_6^4)$ . $S_3^2$ (3)	$C_2^5$ . $(C_3^4:S_3^2)$	$C_3^3$ . $(C_3\times A_4^2.D_4)$	$(C_6^2:A_4)$ . $(C_3^3:D_4)$	all 32

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2} \times C_{6} \simeq C_{2}^{2} \times C_{3}$
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$2$

Subgroups

There are 1169058 subgroups in 13279 conjugacy classes, 66 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^4.C_3^4.(C_6\times S_3)$
Commutator:	$G' \simeq$ $C_2^4.C_3^4.C_6$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3\times C_6$	$G/\Phi \simeq$ $C_3\times A_4^2:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^4$	$G/\operatorname{Fit} \simeq$ $S_3^2$
Radical:	$R \simeq$ $C_6^4:D_6:S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_3\wr C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

All subgroups of index up to 128 (order at least 729) are shown, as well as all normal subgroups of any index.

Order 93312: $C_6^4:D_6:S_3$

Order 46656: $C_2^4.(C_6\times \He_3).C_3.C_6$, $C_2^4.C_3^4.C_6^2$, $C_6^4:(C_6\times S_3)$

Order 31104: $(C_2\times C_6^3).C_6.C_6.C_2$, $C_2^4.C_3^4.C_6.C_2^2$, $(C_2^2\times C_6^3).S_3^2$, $C_6^4:D_{12}$

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

Order 15552: $C_2^4.C_3^4.C_6.C_2$ x 2, $C_2^4.C_3^4:C_3.C_2^2$ x 2, $C_2^4.C_3^4:C_6.C_2$ x 2, $C_2^4.C_3^4:C_3.C_2^2$ x 2, $C_2^4.(C_6\times \He_3).C_6$, $C_2^4.(C_6\times \He_3).C_6$, $C_6^4.C_6.C_2$, $(C_3\times C_6\times A_4).C_6^2.C_2$, $C_2^4.(C_6\times \He_3).C_6$, $C_2^4.C_3^4.C_6.C_2$, $(C_2\times C_6^4).C_6$, $C_3^3:(A_4^2:C_4)$, $C_6^4:D_6$, $(C_3^2\times A_4^2):C_{12}$

Order 11664: $(C_2\times C_6^3).C_3^3$, $(C_3^3\times A_4).C_6^2$

Order 10368: $C_6^4.C_2^3$, $(C_2\times C_6^3):D_{12}$, $(C_3\times C_6^2).\GL(2,\mathbb{Z}/4)$, $(C_3\times A_4^2):D_{12}$, $(C_2\times C_6^3):D_{12}$, $(D_6\times C_6^2):S_4$

Order 7776: $C_2^4.C_3^4:C_3.C_2$ x 7, $(C_3\times C_6\times A_4).C_6^2$ x 4, $C_2^4.C_3^4:C_3.C_2$ x 3, $C_2^4.C_3^4.C_6$ x 2, $C_2^4.C_3^4:C_3.C_2$ x 2, $C_2^4.C_3^4.C_6$ x 2, $C_2^4.C_3^4:C_3.C_2$ x 2, $C_3\times C_2^4:\He_3.C_6$ x 2, $C_2^4.C_3^4:C_3.C_2$ x 2, $(C_3^2\times A_4^2):C_6$ x 2, $C_3\times C_6^2.C_6^2.C_2$, $C_2^4.C_3^4:C_6$, $(C_3\times C_6\times A_4).C_6^2$, $C_6^4.C_6$, $C_6^2.(C_3\times C_{12}):C_6$, $(C_3^2\times A_4).C_6^2.C_2$, $C_2^4.C_3^4.C_6$, $C_6^2.(C_3\times D_6:S_3)$, $(C_3^2\times A_4^2):S_3$, $C_6^4:S_3$

Order 5832: $C_6^2.C_3^4.C_2$ x 2, $C_3^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2$

Order 5184: $C_2^4.C_3^3:S_3.C_2$ x 6, $(C_3^2\times A_4).C_6.C_2^3$ x 3, $(C_6\times A_4).C_6^2.C_2$ x 3, $(C_3\times A_4^2).C_6.C_2$ x 3, $C_2^4:\He_3.C_6.C_2$ x 2, $C_2^4.(C_6\times \He_3).C_2$ x 2, $C_6^2.(C_{12}\times A_4)$ x 2, $C_6^3.C_6.C_2^2$ x 2, $(C_3\times A_4):A_4.C_6.C_2$ x 2, $C_6^2:A_4:C_{12}$ x 2, $(C_3\times A_4^2):C_{12}$ x 2, $C_2^4.(C_6\times \He_3).C_2$, $C_2^4.C_3^4.C_2^2$, $(C_3^2\times C_6^2).C_2^4$, $C_6^4.C_2^2$, $C_2^4:\He_3.C_6.C_2$, $C_3\times (C_3\times C_6\times A_4).C_2^3$, $C_6^3.C_6.C_2^2$, $C_3\times C_6\times (C_2^3\times C_6):S_3$, $C_2\times C_6^2.C_6^2.C_2$, $(C_3\times A_4^2).D_6$, $(C_2^2\times C_6^3).C_6$, $C_2^4:\He_3.C_6.C_2$, $(C_2\times C_6^3).C_6.C_2$, $C_2^4:\He_3.D_6$, $C_6^2:S_4:C_6$, $C_6\times A_4^2:S_3$, $(C_2^2\times C_6^3):C_6$

Order 3888: $C_2^4.C_3^4:C_3$ x 7, $C_6^2.C_3^3.C_2^2$ x 4, $C_3\times C_6^2.C_6^2$ x 3, $C_2^4.C_3^4:C_3$ x 3, $(C_3^2\times A_4).C_6^2$ x 3, $C_3\times C_2^4:\He_3.C_3$ x 2, $(C_3^3\times A_4).C_6.C_2$ x 2, $C_2^4.C_3^4:C_3$ x 2, $C_3\times (C_3\times C_6\times A_4).C_6$ x 2, $C_6^2.C_3^3.C_2^2$, $(C_3\times C_6^2).C_6.C_6$, $(C_3^3\times A_4).C_6.C_2$, $C_6^2.\He_3.C_2^2$, $C_2^4.C_3^4.C_3$, $C_2\times C_3^3.C_6^2.C_2$, $C_2\times (C_3\times A_4).C_3^3.C_2$, $C_2\times C_3^4:S_4$, $(C_3^2\times C_6^2):C_{12}$

Order 3456: $C_2^4:C_3.C_3^2:D_4$, $C_2^4.C_6^2:C_2:C_3$, $(C_2\times C_6^3).C_2^3$, $A_4^2.D_6.C_2$, $C_2^4:\He_3.D_4$, $C_2^4:\He_3.D_4$, $C_3\times C_6^2.C_2^4.C_2$, $(C_2\times C_6^3).C_2^3$, $C_2^4.(C_3\times C_{12}):C_6$, $C_2^4:C_3.C_3^2:D_4$, $C_6^3.C_2^3.C_2$, $A_4^2:D_{12}$

Order 2916: $(C_2\times C_3^4:C_3).C_6$, $C_2\times (C_3^4.C_3^2):C_2$, $C_2\times (C_3^3.C_3^3):C_2$, $C_6^2.C_3^4$

Order 2592: $C_3\times (C_3\times A_4):A_4.C_2$ x 21, $C_2^4:\He_3.C_6$ x 15, $C_3\times C_2^4:\He_3.C_2$ x 12, $(C_3\times C_6\times A_4).C_6.C_2$ x 10, $C_2^4.C_3^3:S_3$ x 8, $(C_3\times A_4^2).C_6$ x 6, $C_3\times (C_6\times A_4).(C_2\times C_6)$ x 6, $(C_6\times A_4).C_6^2$ x 6, $C_6^2.C_6^2.C_2$ x 6, $C_3\times C_6^2:C_3.C_2^3$ x 6, $(C_3\times A_4):A_4.C_6$ x 6, $C_3^2\times (C_2^3\times C_6):S_3$ x 4, $C_3^2\times C_2\times A_4^2$ x 4, $C_3\times A_4.C_6^2:C_2$ x 4, $(C_3\times C_6\times A_4).C_6.C_2$ x 3, $(C_3\times A_4).C_6^2.C_2$ x 3, $(C_6\times A_4).C_6^2$ x 3, $(C_3^2\times A_4).C_6.C_2^2$ x 3, $C_2^4:\He_3.C_6$ x 3, $(C_2\times C_6^3):C_6$ x 3, $C_6^3.C_6.C_2$ x 2, $(C_3^2\times C_6^2).C_2^3$ x 2, $C_2^4.C_3^4.C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $(C_3^2\times C_6^2).C_2^3$ x 2, $C_6^2\times C_6^2:C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_3\times C_6\times C_3:(C_6.D_4)$ x 2, $C_6^2.(C_6\times S_3).C_2$ x 2, $C_3^3\times C_2^3\times A_4$ x 2, $C_6^3.D_6$ x 2, $C_3\times C_3^2:S_4.C_2^2$, $C_2\times C_6^4$, $(C_3\times A_4).D_6:S_3$, $C_2^2\times C_3^4:D_4$, $C_3\times A_4:(C_3^2:C_4).C_2$, $C_2\times C_6^2.C_6^2$, $(C_2\times C_6^3).C_6$, $C_2^4:\He_3.S_3$, $C_3^3.(C_6\times D_4).C_2$, $(C_3^2\times C_6^2).C_2^3$, $(A_4\times C_6^2):S_3$, $C_6^3.D_6$, $C_6^3.D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $(C_3\times A_4)\wr C_2$

Order 1944: $C_6^3:C_3^2$ x 13, $C_6^3.C_3^2$ x 7, $C_3^4:S_4$ x 6, $C_6^3:C_3^2$ x 5, $C_6^2:(S_3\times C_3^2)$ x 4, $(A_4\times C_3^3):C_6$ x 4, $C_6^3:C_3^2$ x 2, $C_3^4:S_4$ x 2, $C_3^3:(C_3\times S_4)$ x 2, $C_3^4:S_4$ x 2, $C_3^2:C_6^3$, $C_2\times C_3^3:C_6^2$, $C_6\times C_3^3:C_{12}$, $(C_3^2\times C_6^2):C_6$, $C_3^4:S_4$, $(C_3\times \He_3):D_{12}$, $C_3^3:C_6\wr C_2$, $C_3^4:(C_3\times D_4)$, $C_3^4:D_{12}$

Order 1728: $C_2\times C_6^2:S_4$ x 9, $C_2\times C_6^2:S_4$ x 8, $C_3\times C_6:\GL(2,\mathbb{Z}/4)$ x 6, $A_4^2:D_6$ x 5, $C_6^3:D_4$ x 5, $C_6^3:D_4$ x 5, $C_6^3.D_4$ x 4, $C_6.A_4\wr C_2$ x 3, $C_6.A_4\wr C_2$ x 3, $C_2\times C_6^2:S_4$ x 2, $C_2\times C_6^2:S_4$ x 2, $A_4^2:D_6$ x 2, $A_4\wr C_2\times C_6$ x 2, $(C_2\times C_6^2).S_4$ x 2, $(C_3\times A_4^2):C_4$ x 2, $A_4^2:C_{12}$ x 2, $C_6^3:D_4$ x 2, $(C_2^3\times C_6^2):C_6$ x 2, $(C_2\times C_6^2).S_4$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $(C_2^3\times C_6^2):C_6$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_6^3:D_4$, $C_6^3:D_4$, $(C_2\times C_6^2).S_4$, $(C_2^2\times C_6^2):A_4$, $C_6^3.D_4$, $C_6^3.D_4$

Order 1458: $C_2\times \He_3^2$, $\He_3^2:C_2$, $\He_3\wr C_2$

Order 1296: $C_6^3:C_6$ x 54, $C_6^3:S_3$ x 29, $C_3^2.A_4^2$ x 21, $C_6^3:C_6$ x 20, $C_3^2:A_4^2$ x 15, $C_6^3:S_3$ x 14, $C_6^3.C_6$ x 14, $C_3\times C_6^2:A_4$ x 12, $C_6^3.S_3$ x 8, $C_6^3:C_6$ x 6, $C_6^3.S_3$ x 6, $C_6^4$ x 5, $C_6^3.C_6$ x 5, $C_6^3:S_3$ x 4, $C_3^2\times A_4^2$ x 4, $C_6^2.S_3^2$ x 4, $C_2\times \He_3:S_4$ x 3, $C_6^3:C_6$ x 3, $C_6^3:C_6$ x 3, $C_6^3:C_6$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.S_3$ x 2, $C_6^3:C_6$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.C_6$ x 2, $C_6^3:S_3$, $C_6^3:C_6$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^3.C_6$, $C_6^3.S_3$

Order 1152: $A_4^2:D_4$ x 3, $C_6^2.C_2^4.C_2$ x 2, $(C_6^2\times D_4).C_2^2$ x 2, $C_6^2.C_2^4.C_2$, $C_3\times C_3:(C_2^4.C_2^3)$, $(C_2^3\times C_6):S_4$, $(C_2^2\times C_{12}):S_4$, $C_2^5.S_3^2$, $C_2^5.S_3^2$

Order 972: $C_6^2:\He_3$ x 13, $C_3^4:A_4$ x 7, $C_3^3:C_6^2$ x 4, $\He_3\times C_6^2$ x 3, $C_3^4:A_4$ x 3, $C_3^4:D_6$ x 3, $C_3^3.C_6^2$ x 3, $(C_3\times \He_3):C_{12}$ x 3, $C_3^4:C_{12}$ x 3, $C_6^2:C_3^3$ x 2, $C_3^4:C_{12}$ x 2, $C_2\times C_3^4:C_6$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:D_6$

Order 864: $C_6.A_4^2$ x 39, $C_6^2:S_4$ x 37, $C_6^2:S_4$ x 24, $C_6\times A_4^2$ x 23, $C_6^3.C_2^2$ x 22, $C_6^3.C_2^2$ x 22, $C_2^5:\He_3$ x 21, $C_2^2:C_6^3$ x 20, $C_2^5:\He_3$ x 18, $C_6^3:C_4$ x 16, $C_6^3:C_4$ x 16, $C_6^2:S_4$ x 11, $C_6^3:C_2^2$ x 10, $C_6^3:C_2^2$ x 10, $C_6^2:S_4$ x 9, $C_6^2.S_4$ x 9, $C_2^5:C_3^3$ x 8, $A_4^2:S_3$ x 8, $C_6^2.D_{12}$ x 8, $C_2^2\times C_6^3$ x 7, $C_6^2:S_4$ x 6, $C_6^2.S_4$ x 6, $C_3.A_4\wr C_2$ x 6, $C_6^2:S_4$ x 4, $C_6^2:D_{12}$ x 4, $C_6^2.D_{12}$ x 4, $C_6^2.D_{12}$ x 4, $(C_2^2\times C_6^2):C_6$ x 3, $C_6^2:S_4$ x 3, $A_4^2:S_3$ x 3, $A_4^2:C_6$ x 3, $(C_6\times D_6):A_4$ x 3, $C_6^2:S_4$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2:S_4$, $C_6^2.S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_6^2:S_4$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6^2):A_4$

Order 729: $\He_3^2$

Order 432: $C_2\times C_6^3$ x 2, $C_6^2:A_4$ x 2

Order 288: $C_2^3\times C_6^2$

Order 162: $C_3^3\times C_6$

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

Order 128: $C_2^4:D_4$

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 72: $C_3:D_{12}$ x 3, $C_6\wr C_2$ x 2

Order 54: $C_3^2\times C_6$ x 2

Order 48: $C_2^3\times C_6$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 24: $C_3:D_4$ x 3, $C_3\times D_4$ x 2, $D_{12}$

Order 18: $C_3\times S_3$ x 15, $C_3\times C_6$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $D_4$ x 2

Order 6: $S_3$ x 12, $C_6$ x 8

Order 3: $C_3$ x 14

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 729) are shown, as well as all normal subgroups of any index.

Order 93312: $C_6^4:D_6:S_3$

Order 46656: $C_2^4.(C_6\times \He_3).C_3.C_6$, $C_2^4.C_3^4.C_6^2$, $C_6^4:(C_6\times S_3)$

Order 31104: $(C_2\times C_6^3).C_6.C_6.C_2$, $C_2^4.C_3^4.C_6.C_2^2$, $(C_2^2\times C_6^3).S_3^2$, $C_6^4:D_{12}$

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

Order 15552: $C_2^4.C_3^4.C_6.C_2$ x 2, $C_2^4.C_3^4:C_3.C_2^2$ x 2, $C_2^4.C_3^4:C_6.C_2$ x 2, $C_2^4.C_3^4:C_3.C_2^2$ x 2, $C_2^4.(C_6\times \He_3).C_6$, $C_2^4.(C_6\times \He_3).C_6$, $C_6^4.C_6.C_2$, $(C_3\times C_6\times A_4).C_6^2.C_2$, $C_2^4.(C_6\times \He_3).C_6$, $C_2^4.C_3^4.C_6.C_2$, $(C_2\times C_6^4).C_6$, $C_3^3:(A_4^2:C_4)$, $C_6^4:D_6$, $(C_3^2\times A_4^2):C_{12}$

Order 11664: $(C_2\times C_6^3).C_3^3$, $(C_3^3\times A_4).C_6^2$

Order 10368: $C_6^4.C_2^3$, $(C_2\times C_6^3):D_{12}$, $(C_3\times C_6^2).\GL(2,\mathbb{Z}/4)$, $(C_3\times A_4^2):D_{12}$, $(C_2\times C_6^3):D_{12}$, $(D_6\times C_6^2):S_4$

Order 7776: $C_2^4.C_3^4:C_3.C_2$ x 7, $C_2^4.C_3^4:C_3.C_2$ x 3, $(C_3\times C_6\times A_4).C_6^2$ x 2, $C_2^4.C_3^4:C_3.C_2$ x 2, $C_2^4.C_3^4.C_6$ x 2, $C_2^4.C_3^4:C_3.C_2$ x 2, $C_3\times C_2^4:\He_3.C_6$ x 2, $C_3\times C_6^2.C_6^2.C_2$, $C_2^4.C_3^4:C_6$, $(C_3\times C_6\times A_4).C_6^2$, $C_6^4.C_6$, $C_6^2.(C_3\times C_{12}):C_6$, $C_2^4.C_3^4.C_6$, $(C_3^2\times A_4).C_6^2.C_2$, $C_2^4.C_3^4.C_6$, $C_6^2.(C_3\times D_6:S_3)$, $C_2^4.C_3^4:C_3.C_2$, $(C_3^2\times A_4^2):S_3$, $C_6^4:S_3$, $(C_3^2\times A_4^2):C_6$

Order 5832: $C_3^4.C_6^2.C_2$, $C_6^2.C_3^4.C_2$, $C_6^2.C_3^4.C_2$

Order 5184: $C_2^4.C_3^3:S_3.C_2$ x 6, $(C_3^2\times A_4).C_6.C_2^3$ x 3, $C_2^4:\He_3.C_6.C_2$ x 2, $C_2^4.(C_6\times \He_3).C_2$ x 2, $C_6^2.(C_{12}\times A_4)$ x 2, $C_6^3.C_6.C_2^2$ x 2, $(C_3\times A_4):A_4.C_6.C_2$ x 2, $C_6^2:A_4:C_{12}$ x 2, $(C_3\times A_4^2):C_{12}$ x 2, $C_2^4.(C_6\times \He_3).C_2$, $C_2^4.C_3^4.C_2^2$, $(C_3^2\times C_6^2).C_2^4$, $C_6^4.C_2^2$, $C_2^4:\He_3.C_6.C_2$, $C_3\times (C_3\times C_6\times A_4).C_2^3$, $C_6^3.C_6.C_2^2$, $C_3\times C_6\times (C_2^3\times C_6):S_3$, $C_2\times C_6^2.C_6^2.C_2$, $(C_3\times A_4^2).D_6$, $(C_2^2\times C_6^3).C_6$, $C_2^4:\He_3.C_6.C_2$, $(C_6\times A_4).C_6^2.C_2$, $(C_2\times C_6^3).C_6.C_2$, $C_2^4:\He_3.D_6$, $(C_3\times A_4^2).C_6.C_2$, $C_6^2:S_4:C_6$, $C_6\times A_4^2:S_3$, $(C_2^2\times C_6^3):C_6$

Order 3888: $C_2^4.C_3^4:C_3$ x 7, $C_6^2.C_3^3.C_2^2$ x 4, $C_3\times C_6^2.C_6^2$ x 3, $C_2^4.C_3^4:C_3$ x 3, $C_3\times C_2^4:\He_3.C_3$ x 2, $(C_3^3\times A_4).C_6.C_2$ x 2, $C_2^4.C_3^4:C_3$ x 2, $(C_3^2\times A_4).C_6^2$ x 2, $C_6^2.C_3^3.C_2^2$, $C_3\times (C_3\times C_6\times A_4).C_6$, $(C_3\times C_6^2).C_6.C_6$, $(C_3^3\times A_4).C_6.C_2$, $C_6^2.\He_3.C_2^2$, $C_2^4.C_3^4.C_3$, $C_2\times C_3^3.C_6^2.C_2$, $C_2\times (C_3\times A_4).C_3^3.C_2$, $C_2\times C_3^4:S_4$, $(C_3^2\times C_6^2):C_{12}$

Order 3456: $C_2^4:C_3.C_3^2:D_4$, $C_2^4.C_6^2:C_2:C_3$, $(C_2\times C_6^3).C_2^3$, $A_4^2.D_6.C_2$, $C_2^4:\He_3.D_4$, $C_2^4:\He_3.D_4$, $C_3\times C_6^2.C_2^4.C_2$, $(C_2\times C_6^3).C_2^3$, $C_2^4.(C_3\times C_{12}):C_6$, $C_2^4:C_3.C_3^2:D_4$, $C_6^3.C_2^3.C_2$, $A_4^2:D_{12}$

Order 2916: $(C_2\times C_3^4:C_3).C_6$, $C_2\times (C_3^4.C_3^2):C_2$, $C_2\times (C_3^3.C_3^3):C_2$, $C_6^2.C_3^4$

Order 2592: $C_3\times (C_3\times A_4):A_4.C_2$ x 21, $C_3\times C_2^4:\He_3.C_2$ x 12, $C_3\times C_6^2:C_3.C_2^3$ x 6, $C_2^4.C_3^3:S_3$ x 6, $(C_3\times C_6\times A_4).C_6.C_2$ x 5, $C_2^4:\He_3.C_6$ x 4, $C_3^2\times C_2\times A_4^2$ x 4, $(C_3\times C_6\times A_4).C_6.C_2$ x 3, $C_3^2\times (C_2^3\times C_6):S_3$ x 3, $(C_3^2\times A_4).C_6.C_2^2$ x 3, $(C_3\times A_4):A_4.C_6$ x 3, $(C_3^2\times C_6^2).C_2^3$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $(C_3^2\times C_6^2).C_2^3$ x 2, $C_6^2\times C_6^2:C_2$ x 2, $C_6^2.C_6^2.C_2$ x 2, $C_3\times C_6\times C_3:(C_6.D_4)$ x 2, $C_3^3\times C_2^3\times A_4$ x 2, $C_3\times A_4.C_6^2:C_2$ x 2, $C_2^4:\He_3.C_6$ x 2, $(C_2\times C_6^3):C_6$ x 2, $C_6^3.C_6.C_2$, $C_3\times C_3^2:S_4.C_2^2$, $C_2^4.C_3^4.C_2$, $(C_3\times A_4).C_6^2.C_2$, $(C_3\times A_4^2).C_6$, $C_3\times (C_6\times A_4).(C_2\times C_6)$, $(C_6\times A_4).C_6^2$, $C_2\times C_6^4$, $(C_3\times A_4).D_6:S_3$, $C_2^2\times C_3^4:D_4$, $(C_6\times A_4).C_6^2$, $C_3\times A_4:(C_3^2:C_4).C_2$, $C_2\times C_6^2.C_6^2$, $C_6^2.(C_6\times S_3).C_2$, $(C_2\times C_6^3).C_6$, $C_2^4:\He_3.S_3$, $C_3^3.(C_6\times D_4).C_2$, $(C_3^2\times C_6^2).C_2^3$, $(A_4\times C_6^2):S_3$, $C_6^3.D_6$, $C_6^3.D_6$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $(C_3\times A_4)\wr C_2$, $C_6^3.D_6$

Order 1944: $C_6^3:C_3^2$ x 13, $C_6^3.C_3^2$ x 7, $C_6^3:C_3^2$ x 5, $C_3^4:S_4$ x 4, $C_6^3:C_3^2$ x 2, $C_6^2:(S_3\times C_3^2)$ x 2, $(A_4\times C_3^3):C_6$ x 2, $C_3^2:C_6^3$, $C_2\times C_3^3:C_6^2$, $C_3^4:S_4$, $C_3^3:(C_3\times S_4)$, $C_3^4:S_4$, $C_6\times C_3^3:C_{12}$, $(C_3^2\times C_6^2):C_6$, $C_3^4:S_4$, $(C_3\times \He_3):D_{12}$, $C_3^3:C_6\wr C_2$, $C_3^4:(C_3\times D_4)$, $C_3^4:D_{12}$

Order 1728: $C_2\times C_6^2:S_4$ x 8, $C_6^3:D_4$ x 5, $C_6^3:D_4$ x 5, $C_3\times C_6:\GL(2,\mathbb{Z}/4)$ x 4, $C_6^3.D_4$ x 4, $A_4^2:D_6$ x 3, $C_6.A_4\wr C_2$ x 3, $C_2\times C_6^2:S_4$ x 3, $C_6.A_4\wr C_2$ x 3, $C_2\times C_6^2:S_4$ x 2, $C_2\times C_6^2:S_4$ x 2, $A_4^2:D_6$ x 2, $A_4\wr C_2\times C_6$ x 2, $(C_2\times C_6^2).S_4$ x 2, $(C_3\times A_4^2):C_4$ x 2, $A_4^2:C_{12}$ x 2, $C_6^3:D_4$ x 2, $(C_2^3\times C_6^2):C_6$ x 2, $(C_2\times C_6^2).S_4$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$ x 2, $(C_2^3\times C_6^2):C_6$ x 2, $C_6^3.D_4$ x 2, $(C_2^2\times C_6^2):C_{12}$, $C_2\times C_6^2:S_4$, $C_2\times C_6^2:S_4$, $C_6^3:D_4$, $C_6^3:D_4$, $(C_2\times C_6^2).S_4$, $(C_2^2\times C_6^2):A_4$, $C_6^3.D_4$, $C_6^3.D_4$

Order 1458: $C_2\times \He_3^2$, $\He_3^2:C_2$, $\He_3\wr C_2$

Order 1296: $C_6^3:C_6$ x 30, $C_6^3:S_3$ x 24, $C_3^2.A_4^2$ x 21, $C_6^3:C_6$ x 18, $C_3\times C_6^2:A_4$ x 12, $C_6^3.C_6$ x 10, $C_6^3:C_6$ x 6, $C_6^4$ x 5, $C_6^3.C_6$ x 5, $C_6^3:S_3$ x 5, $C_3^2\times A_4^2$ x 4, $C_3^2:A_4^2$ x 4, $C_6^2.S_3^2$ x 4, $C_6^3:S_3$ 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 2, $C_6^2.S_3^2$ x 2, $C_6^3:C_6$ x 2, $C_6^2.S_3^2$ x 2, $C_6^3.C_6$ x 2, $C_6^3:S_3$, $C_6^3:C_6$, $C_6^3.S_3$, $C_6^3:C_6$, $C_6^3:S_3$, $C_2\times \He_3:S_4$, $C_6^3:C_6$, $C_6^3.C_6$, $C_6^3.S_3$

Order 1152: $C_6^2.C_2^4.C_2$ x 2, $(C_6^2\times D_4).C_2^2$ x 2, $C_6^2.C_2^4.C_2$, $C_3\times C_3:(C_2^4.C_2^3)$, $A_4^2:D_4$, $(C_2^3\times C_6):S_4$, $(C_2^2\times C_{12}):S_4$, $C_2^5.S_3^2$, $C_2^5.S_3^2$

Order 972: $C_6^2:\He_3$ x 13, $C_3^4:A_4$ x 7, $\He_3\times C_6^2$ x 3, $C_3^3:C_6^2$ x 3, $C_3^4:A_4$ x 3, $C_3^4:D_6$ x 3, $C_3^3.C_6^2$ x 3, $(C_3\times \He_3):C_{12}$ x 3, $C_3^4:C_{12}$ x 3, $C_6^2:C_3^3$ x 2, $C_2\times C_3^4:C_6$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:D_6$, $C_3^4:C_{12}$

Order 864: $C_6.A_4^2$ x 21, $C_2^5:\He_3$ x 16, $C_6^3:C_4$ x 16, $C_6^3:C_4$ x 16, $C_2^2:C_6^3$ x 15, $C_6^3.C_2^2$ x 15, $C_6^3.C_2^2$ x 15, $C_6\times A_4^2$ x 13, $C_6^3:C_2^2$ x 10, $C_6^3:C_2^2$ x 10, $C_2^5:C_3^3$ x 8, $C_6^2:S_4$ x 8, $C_2^5:\He_3$ x 8, $C_6^2:S_4$ x 8, $C_2^2\times C_6^3$ x 7, $C_6^2:S_4$ x 7, $C_6^2:S_4$ x 4, $C_6^2.S_4$ x 4, $C_6^2:D_{12}$ x 4, $C_6^2.D_{12}$ x 4, $A_4^2:S_3$ x 3, $C_6^2:S_4$ x 3, $C_3.A_4\wr C_2$ x 3, $C_6^2.S_4$ x 3, $(C_2^2\times C_6^2):C_6$ x 2, $C_6^2:S_4$ x 2, $C_6^2:S_4$ x 2, $A_4^2:S_3$ x 2, $A_4^2:C_6$ x 2, $C_6^2:S_4$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:D_{12}$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2:D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^3.C_2^2$ x 2, $(C_6\times D_6):A_4$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2:S_4$, $C_6^2.S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_6^2:S_4$, $C_6^2:D_{12}$, $C_6^2:D_{12}$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_6^2):A_4$

Order 729: $\He_3^2$

Order 432: $C_2\times C_6^3$ x 2, $C_6^2:A_4$ x 2

Order 288: $C_2^3\times C_6^2$

Order 162: $C_3^3\times C_6$

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

Order 128: $C_2^4:D_4$

Order 96: $C_2^4\times C_6$ x 2

Order 81: $C_3^4$

Order 72: $C_6\wr C_2$ x 2, $C_3:D_{12}$

Order 54: $C_3^2\times C_6$ x 2

Order 48: $C_2^3\times C_6$ x 2

Order 32: $C_2^5$

Order 27: $C_3^3$ x 2

Order 24: $C_3:D_4$ x 3, $C_3\times D_4$ x 2, $D_{12}$

Order 18: $C_3\times S_3$ x 4, $C_3\times C_6$

Order 16: $C_2^4$

Order 9: $C_3^2$

Order 8: $D_4$ x 2

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

Order 3: $C_3$ x 12

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 93312: $C_6^4:D_6:S_3$ ($C_1$)

Order 46656: $C_2^4.(C_6\times \He_3).C_3.C_6$ ($C_2$), $C_2^4.C_3^4.C_6^2$ ($C_2$), $C_6^4:(C_6\times S_3)$ ($C_2$)

Order 31104: $(C_2^2\times C_6^3).S_3^2$ ($C_3$)

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

Order 15552: $C_2^4.(C_6\times \He_3).C_6$ ($S_3$), $C_2^4.C_3^4.C_6.C_2$ ($C_6$), $C_2^4.C_3^4.C_6.C_2$ ($S_3$), $C_2^4.C_3^4:C_3.C_2^2$ ($C_6$), $C_3^3:(A_4^2:C_4)$ ($C_6$), $C_6^4:D_6$ ($S_3$)

Order 11664: $(C_2\times C_6^3).C_3^3$ ($D_4$)

Order 7776: $C_2^4.C_3^4:C_3.C_2$ ($D_6$) x 2, $C_2^4.C_3^4:C_3.C_2$ ($D_6$), $C_2^4.C_3^4.C_6$ ($C_2\times C_6$)

Order 5184: $(C_3\times A_4^2).D_6$ ($C_3\times S_3$), $C_2^4:\He_3.C_6.C_2$ ($C_3\times S_3$), $C_2^4:\He_3.D_6$ ($C_3\times S_3$)

Order 3888: $C_2^4.C_3^4:C_3$ ($C_3:D_4$), $C_2^4.C_3^4:C_3$ ($D_{12}$), $C_2^4.C_3^4:C_3$ ($C_3:D_4$), $C_2^4.C_3^4.C_3$ ($C_3\times D_4$)

Order 2592: $C_3\times C_2^4:\He_3.C_2$ ($C_6\times S_3$) x 2, $C_3\times C_2^4:\He_3.C_2$ ($S_3^2$) x 2, $C_2\times C_6^4$ ($S_3^2$), $C_3^2\times C_2\times A_4^2$ ($C_6\times S_3$)

Order 1296: $C_6^4$ ($C_3:D_{12}$), $C_3^2\times A_4^2$ ($C_6\wr C_2$), $C_3\times C_6^2:A_4$ ($C_6\wr C_2$), $C_3\times C_6^2:A_4$ ($C_3\times D_{12}$), $C_3\times C_6^2:A_4$ ($C_3:D_{12}$), $C_3\times C_6^2:A_4$ ($D_6:S_3$)

Order 864: $C_2^5:C_3^3$ ($C_3\times S_3^2$) x 2, $C_2^2\times C_6^3$ ($C_3:S_3^2$), $C_2^2\times C_6^3$ ($C_3\times S_3^2$)

Order 432: $C_2\times C_6^3$ ($C_3^2:D_{12}$), $C_2\times C_6^3$ ($C_3^2:D_{12}$), $C_6^2:A_4$ ($C_3^2:D_{12}$), $C_6^2:A_4$ ($C_3^3:D_4$)

Order 288: $C_2^3\times C_6^2$ ($C_3\wr C_2^2$)

Order 162: $C_3^3\times C_6$ ($\POPlus(4,3)$)

Order 144: $C_2^2\times C_6^2$ ($C_3^3:D_{12}$)

Order 96: $C_2^4\times C_6$ ($C_3^3:S_3^2$) x 2

Order 81: $C_3^4$ ($A_4^2:D_4$)

Order 54: $C_3^2\times C_6$ ($A_4^2:D_6$), $C_3^2\times C_6$ ($C_3\times \POPlus(4,3)$)

Order 48: $C_2^3\times C_6$ ($C_3^3:(C_3\times D_{12})$), $C_2^3\times C_6$ ($(C_3\times \He_3):D_{12}$)

Order 32: $C_2^5$ ($C_3^4:S_3^2$)

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

Order 18: $C_3\times C_6$ ($C_3\times A_4^2:D_6$)

Order 16: $C_2^4$ ($\He_3^2:D_4$)

Order 9: $C_3^2$ ($(C_3^2\times A_4^2):D_4$)

Order 6: $C_6$ ($C_2^4:\He_3.(C_6\times S_3)$), $C_6$ ($(C_3\times A_4^2).(C_6\times S_3)$)

Order 3: $C_3$ ($C_2^4:\He_3.C_6.C_6.C_2$), $C_3$ ($(C_3\times A_4^2).C_6^2.C_2$)

Order 2: $C_2$ ($C_2^4.C_3^4.(C_6\times S_3)$)

Order 1: $C_1$ ($C_6^4:D_6:S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 93312: $C_6^4:D_6:S_3$ ($C_1$)

Order 46656: $C_2^4.(C_6\times \He_3).C_3.C_6$ ($C_2$), $C_2^4.C_3^4.C_6^2$ ($C_2$), $C_6^4:(C_6\times S_3)$ ($C_2$)

Order 31104: $(C_2^2\times C_6^3).S_3^2$ ($C_3$)

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

Order 15552: $C_2^4.(C_6\times \He_3).C_6$ ($S_3$), $C_2^4.C_3^4.C_6.C_2$ ($C_6$), $C_2^4.C_3^4.C_6.C_2$ ($S_3$), $C_2^4.C_3^4:C_3.C_2^2$ ($C_6$), $C_3^3:(A_4^2:C_4)$ ($C_6$), $C_6^4:D_6$ ($S_3$)

Order 11664: $(C_2\times C_6^3).C_3^3$ ($D_4$)

Order 7776: $C_2^4.C_3^4:C_3.C_2$ ($D_6$) x 2, $C_2^4.C_3^4:C_3.C_2$ ($D_6$), $C_2^4.C_3^4.C_6$ ($C_2\times C_6$)

Order 5184: $(C_3\times A_4^2).D_6$ ($C_3\times S_3$), $C_2^4:\He_3.C_6.C_2$ ($C_3\times S_3$), $C_2^4:\He_3.D_6$ ($C_3\times S_3$)

Order 3888: $C_2^4.C_3^4:C_3$ ($C_3:D_4$), $C_2^4.C_3^4:C_3$ ($D_{12}$), $C_2^4.C_3^4:C_3$ ($C_3:D_4$), $C_2^4.C_3^4.C_3$ ($C_3\times D_4$)

Order 2592: $C_3\times C_2^4:\He_3.C_2$ ($C_6\times S_3$) x 2, $C_3\times C_2^4:\He_3.C_2$ ($S_3^2$) x 2, $C_2\times C_6^4$ ($S_3^2$), $C_3^2\times C_2\times A_4^2$ ($C_6\times S_3$)

Order 1296: $C_6^4$ ($C_3:D_{12}$), $C_3^2\times A_4^2$ ($C_6\wr C_2$), $C_3\times C_6^2:A_4$ ($C_6\wr C_2$), $C_3\times C_6^2:A_4$ ($C_3\times D_{12}$), $C_3\times C_6^2:A_4$ ($C_3:D_{12}$), $C_3\times C_6^2:A_4$ ($D_6:S_3$)

Order 864: $C_2^5:C_3^3$ ($C_3\times S_3^2$) x 2, $C_2^2\times C_6^3$ ($C_3:S_3^2$), $C_2^2\times C_6^3$ ($C_3\times S_3^2$)

Order 432: $C_2\times C_6^3$ ($C_3^2:D_{12}$), $C_2\times C_6^3$ ($C_3^2:D_{12}$), $C_6^2:A_4$ ($C_3^2:D_{12}$), $C_6^2:A_4$ ($C_3^3:D_4$)

Order 288: $C_2^3\times C_6^2$ ($C_3\wr C_2^2$)

Order 162: $C_3^3\times C_6$ ($\POPlus(4,3)$)

Order 144: $C_2^2\times C_6^2$ ($C_3^3:D_{12}$)

Order 96: $C_2^4\times C_6$ ($C_3^3:S_3^2$) x 2

Order 81: $C_3^4$ ($A_4^2:D_4$)

Order 54: $C_3^2\times C_6$ ($A_4^2:D_6$), $C_3^2\times C_6$ ($C_3\times \POPlus(4,3)$)

Order 48: $C_2^3\times C_6$ ($C_3^3:(C_3\times D_{12})$), $C_2^3\times C_6$ ($(C_3\times \He_3):D_{12}$)

Order 32: $C_2^5$ ($C_3^4:S_3^2$)

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

Order 18: $C_3\times C_6$ ($C_3\times A_4^2:D_6$)

Order 16: $C_2^4$ ($\He_3^2:D_4$)

Order 9: $C_3^2$ ($(C_3^2\times A_4^2):D_4$)

Order 6: $C_6$ ($C_2^4:\He_3.(C_6\times S_3)$), $C_6$ ($(C_3\times A_4^2).(C_6\times S_3)$)

Order 3: $C_3$ ($C_2^4:\He_3.C_6.C_6.C_2$), $C_3$ ($(C_3\times A_4^2).C_6^2.C_2$)

Order 2: $C_2$ ($C_2^4.C_3^4.(C_6\times S_3)$)

Order 1: $C_1$ ($C_6^4:D_6:S_3$)

Series

Derived series

$C_6^4:D_6:S_3$

$\rhd$

$C_2^4.C_3^4.C_6$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^4:D_6:S_3$

$\rhd$

$C_2^4.C_3^4.C_6^2$

$\rhd$

$(C_2\times C_6^3).C_3^3.C_2$

$\rhd$

$C_2^4.C_3^4.C_6$

$\rhd$

$C_2^4.C_3^4.C_3$

$\rhd$

$C_3\times C_6^2:A_4$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_2\times C_6^3$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Lower central series

$C_6^4:D_6:S_3$

$\rhd$

$C_2^4.C_3^4.C_6$

$\rhd$

$C_2^4.C_3^4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $314 \times 314$ 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 $186 \times 186$ rational character table (warning: may be slow to load).