Abstract group 7776.jr: $C_6^3:S_3^2$

• Label: 7776.jr
• Order: \( 2^{5} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_6^3:S_3^2$
Order:	\(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3\times (A_4\times \He_3).C_2^5$, of order \(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Composition factors:	$C_2$ x 5, $C_3$ x 5
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	199	404	120	3932	324	1392	972	432	7776
Conjugacy classes	1	11	21	4	127	4	36	12	4	220
Divisions	1	11	13	4	71	2	20	6	2	130
Autjugacy classes	1	7	9	2	30	2	6	5	1	63

Dimension	1	2	3	4	6	8	9	12	18
Irr. complex chars.	24	24	72	6	40	0	48	2	4	220
Irr. rational chars.	8	16	8	10	40	2	0	18	28	130

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$3538080$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	9	18	18
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e \mid b^{6}=c^{6}=d^{6}=e^{6}=[a,e]=[b,e]=[c,e]=[d,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(10,11,12)(13,14,15), (5,6)(7,8,9)(10,15)(11,14)(12,13), (3,6)(4,5)(7,11,8,12,9,10) \!\cdots\! \rangle$
Direct product:	$C_2$ $\, \times\, $ $S_4$ $\, \times\, $ $(C_3\wr S_3)$
Semidirect product:	$C_6^3$ $\,\rtimes\,$ $S_3^2$	$(C_6^3:S_3)$ $\,\rtimes\,$ $S_3$	$(C_6^3:S_3)$ $\,\rtimes\,$ $S_3$	$(S_4\times C_3^3)$ $\,\rtimes\,$ $D_6$	all 44
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^2$ . $(C_6\times S_3^2)$	$C_6$ . $(C_6^2:S_3^2)$	$C_3^2$ . $(C_6\times S_3\times S_4)$	$C_3$ . $(C_3^2:D_6\times S_4)$	all 16

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

Homology

Abelianization:	$C_{2}^{2} \times C_{6} \simeq C_{2}^{3} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 43814 subgroups in 2480 conjugacy classes, 88 normal (52 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_6^2:S_3^2$
Commutator:	$G' \simeq$ $A_4\times \He_3$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_6\times S_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^3:C_3$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_6^3:S_3^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3^2:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

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

Order 7776: $C_6^3:S_3^2$

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

Order 2592: $C_3^2:D_6\times S_4$, $C_6^3:D_6$, $C_6^3:D_6$

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

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

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

Order 864: $C_6\times S_3\times S_4$ x 5, $S_4\times C_6^2$, $D_{12}:C_6^2$, $C_2\times C_6^2:D_6$

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

Order 486: $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$

Order 432: $C_6^2:D_6$ x 20, $A_4:C_6^2$ x 12, $C_6\times S_3\times A_4$ x 9, $C_{12}:C_6^2$ x 8, $C_6^2:D_6$ x 8, $C_6^2:D_6$ x 5, $D_6\times C_6^2$ 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, $A_4\times C_6^2$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_{18}\times S_4$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$

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

Order 288: $C_6^2:C_2^3$ x 5, $C_2\times C_6\times S_4$ x 4, $D_6\times S_4$ x 2, $D_4\times C_6^2$

Order 243: $C_3^4:C_3$

Order 216: $C_6^2:C_6$ x 18, $S_3\times C_6^2$ x 16, $C_3^2\times S_4$ x 16, $C_6^2:S_3$ x 15, $C_6^2:C_6$ x 13, $C_3^2:S_4$ x 10, $C_6^2:S_3$ x 8, $C_6^2:C_6$ x 8, $C_6\times S_3^2$ x 5, $C_3^2:S_4$ x 4, $C_{36}:C_6$ x 4, $D_4\times \He_3$ x 4, $C_3^2:D_{12}$ x 4, $C_4\times C_3^2:S_3$ x 4, $D_4\times C_3^3$ x 4, $C_3^2\times D_{12}$ x 4, $C_6.C_6^2$ x 4, $C_9\times S_4$ x 2, $C_6^3$ x 2, $C_6.C_6^2$ x 2, $C_2^3\times \He_3$ x 2, $C_6^2:C_6$ x 2, $C_{18}:C_{12}$, $C_6.C_6^2$, $C_6^2.S_3$, $C_3\times C_6\times C_{12}$, $C_6.C_6^2$, $C_6^2.C_6$, $C_{18}:A_4$, $A_4\times C_{18}$

Order 162: $C_3\wr S_3$ x 10, $C_3^3:C_6$ x 8, $S_3\times C_3^3$ x 4, $C_3^2\wr C_2$ x 2, $C_3^3:S_3$ x 2, $C_3^3:S_3$ x 2, $C_3^3.C_6$ x 2, $S_3\times \He_3$ x 2, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_6\times \He_3$

Order 144: $C_{12}:D_6$ x 40, $C_6\times S_4$ x 31, $C_4:C_6^2$ x 17, $C_6^2:C_2^2$ x 10, $C_6^2:C_2^2$ x 10, $C_2^2:C_6^2$ x 8, $S_3\times S_4$ x 8, $A_4\times D_6$ x 7, $C_6\times D_{12}$ x 5, $C_{12}\times D_6$ x 5, $C_2^2\times C_6^2$ x 2, $C_6:S_4$ x 2, $D_4\times C_{18}$, $C_2\times C_6\times C_{12}$

Order 108: $C_3^2\times D_6$ x 42, $C_3^2:D_6$ x 23, $C_3\times S_3^2$ x 20, $C_3^2\times A_4$ x 11, $C_3\times C_6^2$ x 8, $C_{18}:C_6$ x 7, $C_2^2\times \He_3$ x 7, $C_3^2:D_6$ x 5, $C_3^2\times C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:A_4$ x 2, $C_9:C_{12}$ x 2, $C_4\times \He_3$ x 2, $\He_3:C_4$ x 2, $S_3\times C_{18}$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$

Order 96: $C_{12}:C_2^3$ x 4, $D_4\times D_6$ x 2, $C_2^2\times S_4$

Order 81: $C_3\wr C_3$ x 4, $C_3^4$, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $C_6\times D_6$ x 79, $C_6\wr C_2$ x 40, $C_6\times A_4$ x 39, $D_4\times C_3^2$ x 36, $C_3\times S_4$ x 30, $C_2\times C_6^2$ x 27, $C_3\times D_{12}$ x 20, $S_3\times C_{12}$ x 20, $S_3\times A_4$ x 14, $C_6\times C_{12}$ x 11, $C_6:C_{12}$ x 5, $C_3:S_4$ x 4, $D_4\times C_9$ x 4, $S_3\times D_6$ x 2, $C_2^2\times C_{18}$ x 2, $C_2\times C_{36}$, $C_2^2:C_{18}$

Order 54: $S_3\times C_3^2$ x 40, $C_3^2\times C_6$ x 18, $C_3^2:S_3$ x 10, $C_3^2:C_6$ x 10, $C_9:C_6$ x 7, $C_2\times \He_3$ x 7, $S_3\times C_9$ x 2, $C_3\times C_{18}$

Order 48: $C_6\times D_4$ x 50, $S_3\times D_4$ x 16, $C_2^3\times C_6$ x 8, $C_2\times S_4$ x 8, $C_2^2\times A_4$ x 5, $C_2^2\times D_6$ x 4, $C_2^2\times C_{12}$ x 4, $C_6:D_4$ x 4, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

Order 36: $C_6\times S_3$ x 143, $C_6^2$ x 71, $C_3\times A_4$ x 23, $C_3\times C_{12}$ x 14, $C_3:C_{12}$ x 10, $S_3^2$ x 8, $C_2\times C_{18}$ x 7, $C_{36}$ x 2, $C_6:S_3$ x 2, $C_2^2:C_9$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$ x 12, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3\times C_9$

Order 24: $C_2^2\times C_6$ x 72, $C_3\times D_4$ x 72, $C_2\times D_6$ x 31, $C_2\times C_{12}$ x 26, $C_2\times A_4$ x 17, $C_3:D_4$ x 16, $D_{12}$ x 8, $C_4\times S_3$ x 8, $S_4$ x 8, $C_6:C_4$ x 2

Order 18: $C_3\times C_6$ x 75, $C_3\times S_3$ x 74, $C_{18}$ x 6, $C_3:S_3$ x 4

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 131, $D_6$ x 50, $C_{12}$ x 20, $A_4$ x 7, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 29, $C_9$ x 2

Order 8: $C_2^3$ x 17, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $C_6$ x 71, $S_3$ x 20

Order 4: $C_2^2$ x 28, $C_4$ x 4

Order 3: $C_3$ x 13

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 7776: $C_6^3:S_3^2$

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

Order 2592: $C_3^2:D_6\times S_4$, $C_6^3:D_6$, $C_6^3:D_6$

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

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

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

Order 864: $C_6\times S_3\times S_4$ x 3, $S_4\times C_6^2$, $D_{12}:C_6^2$, $C_2\times C_6^2:D_6$

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

Order 486: $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $A_4:C_6^2$ x 7, $C_6\times S_3\times A_4$ x 5, $C_6^2:D_6$ x 3, $C_6^2:D_6$ x 3, $D_6\times C_6^2$ x 2, $D_6:C_6^2$ x 2, $C_{12}:C_6^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $A_4\times C_6^2$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_{18}\times S_4$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$

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

Order 288: $C_6^2:C_2^3$ x 3, $C_2\times C_6\times S_4$ x 2, $D_6\times S_4$ x 2, $D_4\times C_6^2$

Order 243: $C_3^4:C_3$

Order 216: $S_3\times C_6^2$ x 10, $C_6^2:S_3$ x 9, $C_6^2:C_6$ x 8, $C_3^2\times S_4$ x 6, $C_6^2:C_6$ x 5, $C_6^2:S_3$ x 3, $C_6\times S_3^2$ x 3, $C_3^2:S_4$ x 3, $C_6^2:C_6$ x 3, $C_3^2:S_4$ x 2, $C_{36}:C_6$ x 2, $D_4\times \He_3$ x 2, $C_6^3$ x 2, $D_4\times C_3^3$ x 2, $C_6.C_6^2$ x 2, $C_2^3\times \He_3$ x 2, $C_6^2:C_6$ x 2, $C_9\times S_4$, $C_{18}:C_{12}$, $C_6.C_6^2$, $C_6^2.S_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $C_3\times C_6\times C_{12}$, $C_6.C_6^2$, $C_3^2\times D_{12}$, $C_6.C_6^2$, $C_6^2.C_6$, $C_{18}:A_4$, $A_4\times C_{18}$

Order 162: $C_3^3:C_6$ x 7, $C_3\wr S_3$ x 5, $S_3\times C_3^3$ x 2, $C_3^3\times C_6$, $C_3^2\wr C_2$, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^3:S_3$, $C_3^3:S_3$, $C_3^3.C_6$, $S_3\times \He_3$

Order 144: $C_6\times S_4$ x 11, $C_4:C_6^2$ x 9, $C_6^2:C_2^2$ x 6, $C_6^2:C_2^2$ x 6, $C_{12}:D_6$ x 6, $A_4\times D_6$ x 5, $C_2^2:C_6^2$ x 4, $C_6\times D_{12}$ x 3, $C_{12}\times D_6$ x 3, $C_2^2\times C_6^2$ x 2, $C_6:S_4$ x 2, $S_3\times S_4$ x 2, $D_4\times C_{18}$, $C_2\times C_6\times C_{12}$

Order 108: $C_3^2\times D_6$ x 23, $C_3^2:D_6$ x 13, $C_3\times C_6^2$ x 7, $C_3^2\times A_4$ x 7, $C_{18}:C_6$ x 6, $C_2^2\times \He_3$ x 6, $C_3^2:D_6$ x 3, $C_3\times S_3^2$ x 3, $C_3^2:A_4$ x 2, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $S_3\times C_{18}$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$, $C_9:C_{12}$, $C_4\times \He_3$, $\He_3:C_4$

Order 96: $C_{12}:C_2^3$ x 2, $D_4\times D_6$ x 2, $C_2^2\times S_4$

Order 81: $C_3\wr C_3$ x 4, $C_3^4$, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $C_6\times D_6$ x 29, $C_2\times C_6^2$ x 17, $C_6\times A_4$ x 17, $D_4\times C_3^2$ x 12, $C_3\times S_4$ x 9, $C_6\wr C_2$ x 9, $C_6\times C_{12}$ x 6, $S_3\times A_4$ x 5, $C_6:C_{12}$ x 3, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $S_3\times D_6$ x 2, $C_3:S_4$ x 2, $C_2^2\times C_{18}$ x 2, $D_4\times C_9$ x 2, $C_2\times C_{36}$, $C_2^2:C_{18}$

Order 54: $S_3\times C_3^2$ x 14, $C_3^2\times C_6$ x 12, $C_9:C_6$ x 6, $C_2\times \He_3$ x 6, $C_3^2:S_3$ x 5, $C_3^2:C_6$ x 3, $C_3\times C_{18}$, $S_3\times C_9$

Order 48: $C_6\times D_4$ x 14, $C_2\times S_4$ x 5, $C_2^3\times C_6$ x 4, $C_2^2\times D_6$ x 4, $C_6:D_4$ x 4, $S_3\times D_4$ x 4, $C_2^2\times A_4$ x 3, $C_2^2\times C_{12}$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

Order 36: $C_6\times S_3$ x 49, $C_6^2$ x 39, $C_3\times A_4$ x 13, $C_2\times C_{18}$ x 6, $C_3\times C_{12}$ x 5, $C_3:C_{12}$ x 3, $C_6:S_3$ x 2, $S_3^2$ x 2, $C_2^2:C_9$, $C_{36}$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$ x 8, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3\times C_9$

Order 24: $C_2^2\times C_6$ x 26, $C_2\times D_6$ x 19, $C_3\times D_4$ x 16, $C_2\times C_{12}$ x 8, $C_2\times A_4$ x 8, $C_3:D_4$ x 6, $S_4$ x 4, $C_6:C_4$ x 2, $D_{12}$ x 2, $C_4\times S_3$ x 2

Order 18: $C_3\times C_6$ x 36, $C_3\times S_3$ x 23, $C_{18}$ x 5, $C_3:S_3$ x 2

Order 16: $C_2\times D_4$ x 6, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $C_2\times C_6$ x 49, $D_6$ x 27, $C_{12}$ x 6, $A_4$ x 5, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 17, $C_9$ x 2

Order 8: $C_2^3$ x 11, $D_4$ x 6, $C_2\times C_4$ x 3

Order 6: $C_6$ x 30, $S_3$ x 10

Order 4: $C_2^2$ x 17, $C_4$ x 2

Order 3: $C_3$ x 9

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7776: $C_6^3:S_3^2$ ($C_1$)

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

Order 2592: $C_3^2:D_6\times S_4$ ($C_3$)

Order 1944: $A_4\times C_3\wr S_3$ ($C_2^2$) x 2, $C_3\wr C_3:S_4$ ($C_2^2$) x 2, $C_3\wr C_3\times S_4$ ($C_2^2$) x 2, $C_6^3:C_3^2$ ($C_2^2$)

Order 1296: $C_6^2:S_3^2$ ($C_6$) x 4, $C_6^3:S_3$ ($S_3$), $C_3^2:D_6\times A_4$ ($C_6$), $C_2\times \He_3:S_4$ ($C_6$), $C_2\times S_4\times \He_3$ ($C_6$), $C_6^3:S_3$ ($S_3$)

Order 972: $C_6^2.C_3^3$ ($C_2^3$)

Order 648: $S_4\times C_3^3$ ($D_6$) x 2, $A_4\times C_3^2:S_3$ ($C_2\times C_6$) x 2, $\He_3:S_4$ ($C_2\times C_6$) x 2, $S_4\times \He_3$ ($C_2\times C_6$) x 2, $C_2^2\times C_3\wr S_3$ ($D_6$) x 2, $C_6^3:C_3$ ($D_6$), $C_2\times A_4\times \He_3$ ($C_2\times C_6$), $C_6^3:C_3$ ($D_6$)

Order 432: $A_4:C_6^2$ ($C_3\times S_3$), $C_6^2:D_6$ ($C_3\times S_3$)

Order 324: $C_3^2.C_6^2$ ($C_2\times D_6$), $C_3^3:D_6$ ($S_4$), $A_4\times C_3^3$ ($C_2\times D_6$), $A_4\times \He_3$ ($C_2^2\times C_6$)

Order 216: $C_3^2\times S_4$ ($C_6\times S_3$) x 2, $C_6^2:S_3$ ($C_6\times S_3$) x 2, $C_6^3$ ($S_3^2$), $C_6^2:C_6$ ($C_6\times S_3$), $C_2^3\times \He_3$ ($C_6\times S_3$)

Order 162: $C_3\wr S_3$ ($C_2\times S_4$) x 2, $C_3^3:C_6$ ($C_2\times S_4$)

Order 144: $C_6\times S_4$ ($C_3^2:C_6$)

Order 108: $C_3\times C_6^2$ ($S_3\times D_6$), $C_3^2\times A_4$ ($C_6\times D_6$), $C_2^2\times \He_3$ ($C_6\times D_6$), $C_3^2:D_6$ ($C_3\times S_4$)

Order 81: $C_3\wr C_3$ ($C_2^2\times S_4$)

Order 72: $C_3\times S_4$ ($C_3^2:D_6$) x 2, $C_2\times C_6^2$ ($C_3\times S_3^2$), $C_6\times A_4$ ($C_3^2:D_6$)

Order 54: $C_3^2:S_3$ ($C_6\times S_4$) x 2, $C_3^2\times C_6$ ($S_3\times S_4$), $C_2\times \He_3$ ($C_6\times S_4$)

Order 48: $C_2\times S_4$ ($C_3\wr S_3$)

Order 36: $C_6^2$ ($C_6\times S_3^2$), $C_3\times A_4$ ($C_6^2:C_6$)

Order 27: $C_3^3$ ($D_6\times S_4$), $\He_3$ ($C_2\times C_6\times S_4$)

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

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

Order 12: $C_2\times C_6$ ($S_3\times C_3^2:D_6$), $A_4$ ($C_2^2\times C_3\wr S_3$)

Order 9: $C_3^2$ ($C_6\times S_3\times S_4$)

Order 8: $C_2^3$ ($C_3^3:S_3^2$)

Order 6: $C_6$ ($C_6^2:S_3^2$)

Order 4: $C_2^2$ ($C_3\wr S_3\times D_6$)

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

Order 2: $C_2$ ($C_3\wr S_3\times S_4$)

Order 1: $C_1$ ($C_6^3:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7776: $C_6^3:S_3^2$ ($C_1$)

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

Order 2592: $C_3^2:D_6\times S_4$ ($C_3$)

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

Order 1296: $C_6^3:S_3$ ($S_3$), $C_3^2:D_6\times A_4$ ($C_6$), $C_2\times \He_3:S_4$ ($C_6$), $C_2\times S_4\times \He_3$ ($C_6$), $C_6^2:S_3^2$ ($C_6$), $C_6^3:S_3$ ($S_3$)

Order 972: $C_6^2.C_3^3$ ($C_2^3$)

Order 648: $C_6^3:C_3$ ($D_6$), $S_4\times C_3^3$ ($D_6$), $C_2\times A_4\times \He_3$ ($C_2\times C_6$), $A_4\times C_3^2:S_3$ ($C_2\times C_6$), $\He_3:S_4$ ($C_2\times C_6$), $S_4\times \He_3$ ($C_2\times C_6$), $C_6^3:C_3$ ($D_6$), $C_2^2\times C_3\wr S_3$ ($D_6$)

Order 432: $A_4:C_6^2$ ($C_3\times S_3$), $C_6^2:D_6$ ($C_3\times S_3$)

Order 324: $C_3^2.C_6^2$ ($C_2\times D_6$), $C_3^3:D_6$ ($S_4$), $A_4\times C_3^3$ ($C_2\times D_6$), $A_4\times \He_3$ ($C_2^2\times C_6$)

Order 216: $C_6^3$ ($S_3^2$), $C_6^2:C_6$ ($C_6\times S_3$), $C_3^2\times S_4$ ($C_6\times S_3$), $C_2^3\times \He_3$ ($C_6\times S_3$), $C_6^2:S_3$ ($C_6\times S_3$)

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

Order 144: $C_6\times S_4$ ($C_3^2:C_6$)

Order 108: $C_3\times C_6^2$ ($S_3\times D_6$), $C_3^2\times A_4$ ($C_6\times D_6$), $C_2^2\times \He_3$ ($C_6\times D_6$), $C_3^2:D_6$ ($C_3\times S_4$)

Order 81: $C_3\wr C_3$ ($C_2^2\times S_4$)

Order 72: $C_2\times C_6^2$ ($C_3\times S_3^2$), $C_6\times A_4$ ($C_3^2:D_6$), $C_3\times S_4$ ($C_3^2:D_6$)

Order 54: $C_3^2:S_3$ ($C_6\times S_4$), $C_3^2\times C_6$ ($S_3\times S_4$), $C_2\times \He_3$ ($C_6\times S_4$)

Order 48: $C_2\times S_4$ ($C_3\wr S_3$)

Order 36: $C_6^2$ ($C_6\times S_3^2$), $C_3\times A_4$ ($C_6^2:C_6$)

Order 27: $C_3^3$ ($D_6\times S_4$), $\He_3$ ($C_2\times C_6\times S_4$)

Order 24: $C_2^2\times C_6$ ($C_3^2:S_3^2$), $C_2\times A_4$ ($C_3^3:D_6$), $S_4$ ($C_3^3:D_6$)

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

Order 12: $C_2\times C_6$ ($S_3\times C_3^2:D_6$), $A_4$ ($C_2^2\times C_3\wr S_3$)

Order 9: $C_3^2$ ($C_6\times S_3\times S_4$)

Order 8: $C_2^3$ ($C_3^3:S_3^2$)

Order 6: $C_6$ ($C_6^2:S_3^2$)

Order 4: $C_2^2$ ($C_3\wr S_3\times D_6$)

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

Order 2: $C_2$ ($C_3\wr S_3\times S_4$)

Order 1: $C_1$ ($C_6^3:S_3^2$)

Series

Derived series

$C_6^3:S_3^2$

$\rhd$

$C_6^3:S_3^2$

$\rhd$

$A_4\times \He_3$

$\rhd$

$A_4\times \He_3$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_6^3:S_3^2$

$\rhd$

$C_6^3:S_3^2$

$\rhd$

$C_3\wr S_3\times S_4$

$\rhd$

$C_3\wr S_3\times S_4$

$\rhd$

$C_3\wr C_3\times S_4$

$\rhd$

$C_3\wr C_3\times S_4$

$\rhd$

$C_6^2.C_3^3$

$\rhd$

$C_6^2.C_3^3$

$\rhd$

$A_4\times \He_3$

$\rhd$

$A_4\times \He_3$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3^2\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_2^2$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_6^3:S_3^2$

$\rhd$

$C_6^3:S_3^2$

$\rhd$

$A_4\times \He_3$

$\rhd$

$A_4\times \He_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

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

Character theory

Complex character table

See the $220 \times 220$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

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