Abstract group 34992.mp: $C_3^5.(S_3\times S_4)$

• Label: 34992.mp
• Order: \( 2^{4} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $18$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^5.(S_3\times S_4)$
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^3.(C_9\times A_4).C_6.C_2^2$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Composition factors:	$C_2$ x 4, $C_3$ x 7
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	9	12	18	36
Elements	1	855	890	648	14454	5670	5184	4374	2916	34992
Conjugacy classes	1	5	20	2	42	29	8	24	3	134
Divisions	1	5	15	2	29	20	5	16	2	95
Autjugacy classes	1	5	19	2	40	22	8	18	3	118

Dimension	1	2	3	4	6	8	12	16	18	24	32	36	48	72
Irr. complex chars.	12	12	12	3	20	18	28	9	2	10	0	4	2	2	134
Irr. rational chars.	4	8	4	5	12	7	16	9	2	15	3	4	4	2	95

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$1872$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid b^{6}=c^{18}=d^{6}=e^{3}=f^{3}=[a,e]=[c,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(1,2,4,3)(5,6)(7,8)(10,11,12,13,14,16,15,18,17), (1,3,6,7,8,9)(2,5,4)(11,12,14,17,18,16)(13,15)\rangle$
Transitive group:	36T14206				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^5.S_3)$ $\,\rtimes\,$ $S_4$	$(C_3^3.S_3^3)$ $\,\rtimes\,$ $S_3$	$(C_3^4.C_6^2)$ $\,\rtimes\,$ $D_6$	$C_9$ $\,\rtimes\,$ $(C_3\times S_3\wr S_3)$	all 23
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^5:S_4)$ . $S_3$	$C_3^5$ . $(S_3\times S_4)$	$(C_3^5:A_4)$ . $D_6$	$C_3^4$ . $(C_6^2:D_6)$	all 10

Elements of the group are displayed as permutations of degree 18.

Homology

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

Subgroups

There are 220488 subgroups in 2435 conjugacy classes, 35 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^5.(S_3\times S_4)$
Commutator:	$G' \simeq$ $C_9\times C_3^3:A_4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3^4:(C_6\times S_4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_9:C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_3^5.(S_3\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_6^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5.C_3^2$

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

Order 34992: $C_3^5.(S_3\times S_4)$

Order 17496: $C_9:C_3\wr S_4$, $C_9:C_3^4:S_4$, $C_3^5.(S_3\times A_4)$

Order 11664: $C_3^3:(D_9\times S_4)$, $C_3^5.(S_3\times D_4)$, $C_3^4:(C_6\times S_4)$

Order 8748: $C_3^3.C_3^4.C_2^2$, $C_9:C_3\wr A_4$

Order 5832: $C_9\times C_3^3:S_4$ x 2, $C_3.C_3^5.C_2^3$, $C_3^5:S_4$, $(C_3^3\times C_9):S_4$, $C_9:S_3\wr C_3$, $C_3^3:(A_4\times D_9)$, $(C_3^3\times S_3^2).S_3$, $C_9:C_3\wr D_4$, $C_3^2.C_3\wr D_4$, $C_3^5.D_{12}$, $C_3^4.(S_3\times C_{12})$, $C_3^3.S_3^3$, $C_3^5:S_4$, $C_3^4:(C_6\times A_4)$, $C_3^5.S_4$

Order 4374: $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_9.C_3^4.C_6$

Order 3888: $S_3^2:(S_3\times D_9)$, $(D_9\times S_3^2):C_6$, $C_3\times S_3^2:S_3^2$, $C_3\times S_3\wr S_3$, $C_3^3:(S_3\times S_4)$

Order 2916: $\He_3:C_9.C_6.C_2$ x 2, $(C_3^3\times C_9):A_4$ x 2, $C_9\times C_3^3:A_4$ x 2, $C_9.C_3^4.C_2^2$, $C_3.C_3^5.C_2^2$, $C_3\times (C_3^2\times C_9:C_3):C_2^2$, $(C_9\times \He_3).C_6.C_2$, $C_9.C_3^4.C_2^2$, $\He_3:C_9.D_6$, $C_3.C_3^5.C_2^2$, $C_3^5:D_6$, $C_3^5:A_4$, $(C_3^3\times C_9):C_{12}$, $C_3^5.D_6$, $C_3^4.C_6^2$, $C_3^5.C_{12}$, $C_3^5.A_4$

Order 2187: $C_3^5.C_3^2$

Order 1944: $C_3^2.S_3^3$ x 3, $(C_9\times S_3^2):C_6$ x 3, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^3:(S_3\times A_4)$ x 2, $C_3^4:S_4$ x 2, $C_3\wr S_4$ x 2, $C_9\times S_3^2:S_3$ x 2, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^3:(S_3\times C_{12})$, $S_3^3:C_3^2$, $C_3^4:S_4$, $C_9:(C_6\times S_3^2)$, $C_9:S_3^3$, $C_9:S_3^3$, $(C_3^2\times C_9):D_{12}$, $C_3^3:D_{36}$, $C_3^4.D_{12}$, $C_3\times S_3^2:D_9$, $(C_9\times S_3^2):C_6$, $C_3^3:(C_4\times D_9)$, $(C_3^2\times D_9):C_{12}$, $C_3^4.S_4$

Order 1458: $(C_9\times \He_3):C_6$ x 3, $(C_9\times \He_3):C_6$ x 2, $C_9:C_3^3.S_3$ x 2, $C_3^4.(C_3\times S_3)$ x 2, $(C_3^2\times \He_3).S_3$ x 2, $C_3^4.(C_3\times S_3)$ x 2, $C_3\wr S_3\times C_9$ x 2, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$, $C_3^5.C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_9^2:(C_3\times C_6)$, $(C_9\times \He_3):C_6$, $(C_3^3\times C_9):C_6$, $C_3^5.S_3$, $(C_3^3\times C_9):S_3$, $C_3^3:(C_3\times D_9)$

Order 1296: $S_3^3:C_6$ x 2, $S_3^2:S_3^2$, $S_3\wr S_3$, $C_6^2.S_3^2$, $S_3^2:D_{18}$, $C_6^2.S_3^2$

Order 972: $C_3^3.S_3^2$ x 14, $C_3^3.C_6^2$ x 5, $C_3\wr A_4$ x 4, $C_3^3.S_3^2$ x 4, $C_3^3.S_3^2$ x 4, $C_3^4:D_6$ x 3, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_9\times C_3:S_3^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.C_6^2$ x 2, $(C_3^2\times C_9):C_{12}$ x 2, $C_3^3:C_{36}$ x 2, $C_3^4.A_4$ x 2, $\He_3:D_{18}$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^3\times S_3^2$, $C_3^4.D_6$, $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:S_3^2$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$, $\He_3:D_{18}$, $C_3^3:S_3^2$

Order 729: $C_3^3.C_3^3$ x 3, $C_3^3.C_3^3$ x 3, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_9:C_3^4$, $C_3^5:C_3$, $C_9^2:C_3^2$, $C_3^2.C_3^4$, $C_3^5.C_3$

Order 648: $C_3\wr D_4$ x 7, $C_3\times S_3^3$ x 5, $C_{18}:C_6\times S_3$ x 4, $D_9\times S_3^2$ x 3, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $S_3\wr C_3$ x 2, $C_3^3:S_4$ x 2, $S_3^2:D_9$ x 2, $S_3^2:C_{18}$ x 2, $C_3^4:C_2^3$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_3^3:S_4$, $C_{18}:(C_6\times S_3)$, $C_{18}:S_3^2$, $A_4\times C_9:C_6$, $C_9:(C_3\times S_4)$, $C_9:C_3\times S_4$, $C_3^2:D_{36}$, $C_3^2:C_4\times D_9$, $(C_6\times C_{18}):C_6$, $C_{18}.C_6^2$, $(S_3\times C_{18}):C_6$, $C_3^3.D_{12}$, $C_9:(S_3\times C_{12})$

Order 486: $C_3^4.C_6$ x 19, $C_3^4.S_3$ x 14, $C_3^4.S_3$ x 10, $C_3^4:S_3$ x 6, $C_3^4:S_3$ x 5, $\He_3:D_9$ x 4, $\He_3:C_{18}$ x 4, $C_3^4.C_6$ x 4, $C_3^4.S_3$ x 4, $C_3^4:C_6$ x 3, $(C_3^2\times C_9).S_3$ x 2, $\He_3:D_9$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^3:C_{18}$ x 2, $D_9:C_3^3$ x 2, $C_3^3:C_{18}$ x 2, $C_3^4:C_6$ x 2, $(C_3^2\times C_9):C_6$ x 2, $C_9^2:C_6$ x 2, $\He_3:C_{18}$ x 2, $\He_3:D_9$ x 2, $C_3^3:D_9$, $C_3^4:C_6$, $C_3^4:C_6$, $S_3\times C_3^4$, $C_{18}:C_3^3$, $C_3^3:D_9$, $C_3^3:D_9$, $C_3^4:S_3$, $C_3^3:D_9$, $C_3^4:S_3$, $C_9^2:C_6$, $C_9^2:C_6$

Order 432: $S_3^3:C_2$ x 2, $C_6\times \SOPlus(4,2)$, $C_6^2:D_6$, $C_6^2:D_6$, $D_9\times S_4$, $D_{36}:C_6$, $D_6:D_{18}$

Order 324: $C_3^2.S_3^2$ x 25, $C_3^2:S_3^2$ x 19, $C_9:S_3^2$ x 13, $C_3^2\times S_3^2$ x 12, $C_3^3.D_6$ x 12, $C_3\wr C_2^2$ x 8, $C_3^2.C_6^2$ x 7, $C_9\times S_3^2$ x 6, $C_3\wr C_4$ x 5, $C_9:C_6^2$ x 4, $C_3^2:D_{18}$ x 4, $C_3^3:D_6$ x 3, $C_3^3:A_4$ x 3, $C_9:S_3^2$ x 3, $C_3^3:C_{12}$ x 2, $C_3^2.C_6^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:C_{36}$ x 2, $C_3^3.C_{12}$, $C_3\times C_9:C_{12}$, $C_3^3:D_6$, $C_3^2:C_6^2$, $D_6\times C_3^3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_9:C_6^2$, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $C_6^2.C_3^2$, $C_3^2:S_3^2$, $C_3^3:D_6$, $(C_3^2\times C_9):C_4$

Order 243: $C_9:C_3^3$ x 16, $C_3^4:C_3$ x 10, $C_3^3.C_3^2$ x 10, $\He_3:C_9$ x 8, $C_3^3:C_9$ x 5, $C_3^2.C_3^3$ x 4, $C_9:\He_3$ x 4, $C_3^4.C_3$ x 4, $C_9^2:C_3$ x 3, $C_3^3\times C_9$ x 2, $C_9^2:C_3$ x 2, $C_9^2:C_3$ x 2, $C_9\times \He_3$ x 2, $C_3^5$, $C_3^2\times \He_3$

Order 216: $C_6\times S_3^2$ x 7, $S_3^2:C_6$ x 7, $S_3^3$ x 5, $S_3^2:S_3$ x 4, $S_3\times D_{18}$ x 4, $D_{18}:C_6$ x 3, $D_{18}:C_6$ x 3, $C_9\times S_4$ x 2, $D_6:C_{18}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $A_4\times D_9$, $D_9:A_4$, $C_9:S_4$, $C_3^2.S_4$, $C_{36}:C_6$, $D_{18}:C_6$, $C_{36}:C_6$, $C_{36}:C_6$, $C_9:D_{12}$, $C_3:D_{36}$, $C_6.D_{18}$, $C_6^2:C_6$, $C_6:S_3^2$, $C_2\times C_3^2:C_{12}$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$, $C_6.S_3^2$, $C_6:D_{18}$

Order 162: $C_3^3.C_6$ x 43, $C_3^3.S_3$ x 42, $D_9:C_3^2$ x 22, $C_3^2:C_{18}$ x 22, $C_3^2\wr C_2$ x 20, $S_3\times C_3^3$ x 19, $C_{18}:C_3^2$ x 17, $C_3^3:C_6$ x 12, $C_3^2:D_9$ x 12, $C_3^2:D_9$ x 12, $C_3^2:D_9$ x 9, $C_3^3:S_3$ x 6, $C_3^3:C_6$ x 6, $C_3^3:C_6$ x 4, $C_3^2:C_{18}$ x 4, $C_3^3:C_6$ x 4, $C_3\wr S_3$ x 4, $C_3^2\times C_{18}$ x 2, $C_3^3:S_3$ x 2, $C_3^2\times D_9$ x 2, $C_3^3\times C_6$, $C_3^3:S_3$, $C_9:D_9$

Order 144: $C_{12}:D_6$ x 2, $D_4\times D_9$, $C_6\times S_4$, $S_3^2:C_2^2$, $S_3\times S_4$, $D_6:D_6$

Order 108: $C_3\times S_3^2$ x 50, $S_3\times D_9$ x 21, $C_{18}:C_6$ x 18, $C_3:S_3^2$ x 17, $C_3^2:D_6$ x 15, $C_3^2\times D_6$ x 14, $C_3:D_{18}$ x 11, $S_3\times C_{18}$ x 8, $C_3:S_3^2$ x 5, $C_3^2:C_{12}$ x 5, $C_{18}:C_6$ x 5, $C_3\times D_{18}$ x 4, $C_9:C_{12}$ x 2, $C_3:C_{36}$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_6\times C_{18}$ x 2, $C_9:A_4$ x 2, $C_9\times A_4$ x 2, $C_9:C_{12}$, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times A_4$, $C_3^2.A_4$, $C_3^2:D_6$, $C_9:C_{12}$

Order 81: $C_9:C_3^2$ x 71, $C_3^2:C_9$ x 18, $C_3^2\times C_9$ x 17, $C_3\wr C_3$ x 14, $C_3^4$ x 13, $C_9:C_9$ x 12, $C_3\times \He_3$ x 6, $C_9^2$ x 2

Order 72: $S_3\times D_6$ x 7, $C_6\wr C_2$ x 7, $C_6\times D_6$ x 6, $C_3\times S_4$ x 4, $\SOPlus(4,2)$ x 4, $C_2\times D_{18}$ x 3, $C_9:D_4$ x 2, $S_3\times A_4$ x 2, $D_4\times C_3^2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_3:D_{12}$ x 2, $D_4\times C_9$ x 2, $D_{36}$, $C_4\times D_9$, $C_6:D_6$, $C_6\times A_4$, $C_2\times C_3^2:C_4$, $C_3:S_4$, $C_6.D_6$, $C_2^2:D_9$

Order 54: $S_3\times C_3^2$ x 76, $C_3^2:C_6$ x 60, $C_9:C_6$ x 39, $C_9:C_6$ x 31, $C_3:D_9$ x 29, $S_3\times C_9$ x 28, $C_3\times C_{18}$ x 18, $C_3\times D_9$ x 17, $C_3^2\times C_6$ x 15, $C_3^2:C_6$ x 10, $C_3^2:S_3$ x 10, $C_3^2:S_3$

Order 48: $S_3\times D_4$ x 2, $C_2\times S_4$, $C_6\times D_4$

Order 36: $C_6\times S_3$ x 49, $S_3^2$ x 33, $D_{18}$ x 14, $C_6:S_3$ x 13, $C_6^2$ x 8, $C_2\times C_{18}$ x 6, $C_3:C_{12}$ x 5, $C_3\times A_4$ x 4, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_2^2:C_9$ x 2, $C_{36}$ x 2, $C_9:C_4$

Order 27: $C_9:C_3$ x 57, $C_3^3$ x 52, $C_3\times C_9$ x 43, $\He_3$ x 9

Order 24: $C_3\times D_4$ x 7, $C_2\times D_6$ x 6, $C_3:D_4$ x 4, $S_4$ x 3, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 92, $C_3\times C_6$ x 41, $C_3:S_3$ x 32, $D_9$ x 19, $C_{18}$ x 16

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 29, $C_2\times C_6$ x 16, $C_{12}$ x 5, $A_4$ x 3, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 57, $C_9$ x 20

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 29, $S_3$ x 26

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 15

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 34992: $C_3^5.(S_3\times S_4)$

Order 17496: $C_9:C_3\wr S_4$, $C_9:C_3^4:S_4$, $C_3^5.(S_3\times A_4)$

Order 11664: $C_3^3:(D_9\times S_4)$, $C_3^5.(S_3\times D_4)$, $C_3^4:(C_6\times S_4)$

Order 8748: $C_3^3.C_3^4.C_2^2$, $C_9:C_3\wr A_4$

Order 5832: $C_9\times C_3^3:S_4$ x 2, $C_3.C_3^5.C_2^3$, $C_3^5:S_4$, $(C_3^3\times C_9):S_4$, $C_9:S_3\wr C_3$, $C_3^3:(A_4\times D_9)$, $(C_3^3\times S_3^2).S_3$, $C_9:C_3\wr D_4$, $C_3^2.C_3\wr D_4$, $C_3^5.D_{12}$, $C_3^4.(S_3\times C_{12})$, $C_3^3.S_3^3$, $C_3^5:S_4$, $C_3^4:(C_6\times A_4)$, $C_3^5.S_4$

Order 4374: $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_9.C_3^4.C_6$

Order 3888: $S_3^2:(S_3\times D_9)$, $(D_9\times S_3^2):C_6$, $C_3\times S_3^2:S_3^2$, $C_3\times S_3\wr S_3$, $C_3^3:(S_3\times S_4)$

Order 2916: $(C_3^3\times C_9):A_4$ x 2, $C_9\times C_3^3:A_4$ x 2, $C_9.C_3^4.C_2^2$, $\He_3:C_9.C_6.C_2$, $C_3.C_3^5.C_2^2$, $C_3\times (C_3^2\times C_9:C_3):C_2^2$, $(C_9\times \He_3).C_6.C_2$, $C_9.C_3^4.C_2^2$, $\He_3:C_9.D_6$, $C_3.C_3^5.C_2^2$, $C_3^5:D_6$, $C_3^5:A_4$, $(C_3^3\times C_9):C_{12}$, $C_3^5.D_6$, $C_3^4.C_6^2$, $C_3^5.C_{12}$, $C_3^5.A_4$

Order 2187: $C_3^5.C_3^2$

Order 1944: $C_3^2.S_3^3$ x 3, $(C_9\times S_3^2):C_6$ x 3, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^3:(S_3\times A_4)$ x 2, $C_3^4:S_4$ x 2, $C_3\wr S_4$ x 2, $C_9\times S_3^2:S_3$ x 2, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^3:(S_3\times C_{12})$, $S_3^3:C_3^2$, $C_3^4:S_4$, $C_9:(C_6\times S_3^2)$, $C_9:S_3^3$, $C_9:S_3^3$, $(C_3^2\times C_9):D_{12}$, $C_3^3:D_{36}$, $C_3^4.D_{12}$, $C_3\times S_3^2:D_9$, $(C_9\times S_3^2):C_6$, $C_3^3:(C_4\times D_9)$, $(C_3^2\times D_9):C_{12}$, $C_3^4.S_4$

Order 1458: $(C_9\times \He_3):C_6$ x 2, $(C_9\times \He_3):C_6$ x 2, $C_3\wr S_3\times C_9$ x 2, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$, $C_3^5.C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_9^2:(C_3\times C_6)$, $(C_9\times \He_3):C_6$, $(C_3^3\times C_9):C_6$, $C_9:C_3^3.S_3$, $C_3^4.(C_3\times S_3)$, $(C_3^2\times \He_3).S_3$, $C_3^4.(C_3\times S_3)$, $C_3^5.S_3$, $(C_3^3\times C_9):S_3$, $C_3^3:(C_3\times D_9)$

Order 1296: $S_3^3:C_6$ x 2, $S_3^2:S_3^2$, $S_3\wr S_3$, $C_6^2.S_3^2$, $S_3^2:D_{18}$, $C_6^2.S_3^2$

Order 972: $C_3^3.S_3^2$ x 12, $C_3^3.C_6^2$ x 5, $C_3\wr A_4$ x 4, $C_3^3.S_3^2$ x 4, $C_3^3.S_3^2$ x 3, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_9\times C_3:S_3^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.C_6^2$ x 2, $(C_3^2\times C_9):C_{12}$ x 2, $C_3^3:C_{36}$ x 2, $C_3^4.A_4$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^3\times S_3^2$, $C_3^4.D_6$, $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:S_3^2$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$, $\He_3:D_{18}$, $C_3^3:S_3^2$, $\He_3:D_{18}$

Order 729: $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_3^3.C_3^3$ x 2, $C_9:C_3^4$, $C_3^5:C_3$, $C_9^2:C_3^2$, $C_3^2.C_3^4$, $C_3^3.C_3^3$, $C_3^3.C_3^3$, $C_3^5.C_3$

Order 648: $C_3\wr D_4$ x 7, $C_3\times S_3^3$ x 5, $C_{18}:C_6\times S_3$ x 3, $D_9\times S_3^2$ x 3, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $S_3\wr C_3$ x 2, $C_3^3:S_4$ x 2, $S_3^2:D_9$ x 2, $S_3^2:C_{18}$ x 2, $C_3^4:C_2^3$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_3^3:S_4$, $C_{18}:(C_6\times S_3)$, $C_{18}:S_3^2$, $A_4\times C_9:C_6$, $C_9:(C_3\times S_4)$, $C_9:C_3\times S_4$, $C_3^2:D_{36}$, $C_3^2:C_4\times D_9$, $(C_6\times C_{18}):C_6$, $C_{18}.C_6^2$, $(S_3\times C_{18}):C_6$, $C_3^3.D_{12}$, $C_9:(S_3\times C_{12})$

Order 486: $C_3^4.C_6$ x 15, $C_3^4.S_3$ x 12, $C_3^4.S_3$ x 9, $C_3^4.C_6$ x 4, $C_3^4:S_3$ x 4, $C_3^4:S_3$ x 4, $C_3^4.S_3$ x 3, $C_3^4:C_6$ x 2, $\He_3:D_9$ x 2, $\He_3:C_{18}$ x 2, $C_3^4:C_6$ x 2, $C_3^3:C_{18}$ x 2, $D_9:C_3^3$ x 2, $C_3^3:C_{18}$ x 2, $C_3^4:C_6$ x 2, $(C_3^2\times C_9):C_6$ x 2, $C_9^2:C_6$ x 2, $\He_3:C_{18}$ x 2, $\He_3:D_9$ x 2, $C_3^4:C_6$ x 2, $(C_3^2\times C_9).S_3$, $\He_3:D_9$, $C_3^3:D_9$, $C_3^4:C_6$, $C_3^4:C_6$, $S_3\times C_3^4$, $C_{18}:C_3^3$, $C_3^3:D_9$, $C_3^3:D_9$, $C_3^4:S_3$, $C_3^3:D_9$, $C_3^4:S_3$, $C_9^2:C_6$, $C_9^2:C_6$

Order 432: $S_3^3:C_2$ x 2, $C_6\times \SOPlus(4,2)$, $C_6^2:D_6$, $C_6^2:D_6$, $D_9\times S_4$, $D_{36}:C_6$, $D_6:D_{18}$

Order 324: $C_3^2.S_3^2$ x 18, $C_3^2:S_3^2$ x 17, $C_3^2\times S_3^2$ x 11, $C_9:S_3^2$ x 11, $C_3^3.D_6$ x 10, $C_3\wr C_2^2$ x 8, $C_9\times S_3^2$ x 6, $C_3\wr C_4$ x 5, $C_3^2.C_6^2$ x 5, $C_3^3:A_4$ x 3, $C_9:C_6^2$ x 3, $C_9:S_3^2$ x 3, $C_3^2:D_{18}$ x 3, $C_3^3:D_6$ x 2, $C_3^3:C_{12}$ x 2, $C_3^2.C_6^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:C_{36}$ x 2, $C_3^3.C_{12}$, $C_3\times C_9:C_{12}$, $C_3^3:D_6$, $C_3^2:C_6^2$, $D_6\times C_3^3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_9:C_6^2$, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $C_6^2.C_3^2$, $C_3^2:S_3^2$, $C_3^3:D_6$, $(C_3^2\times C_9):C_4$

Order 243: $C_9:C_3^3$ x 14, $C_3^4:C_3$ x 8, $C_3^3.C_3^2$ x 5, $C_9:\He_3$ x 4, $\He_3:C_9$ x 4, $C_3^3:C_9$ x 4, $C_3^2.C_3^3$ x 3, $C_3^4.C_3$ x 3, $C_9^2:C_3$ x 3, $C_3^3\times C_9$ x 2, $C_9^2:C_3$ x 2, $C_9^2:C_3$ x 2, $C_9\times \He_3$ x 2, $C_3^5$, $C_3^2\times \He_3$

Order 216: $S_3^2:C_6$ x 7, $C_6\times S_3^2$ x 6, $S_3^3$ x 5, $S_3^2:S_3$ x 4, $D_{18}:C_6$ x 3, $D_{18}:C_6$ x 3, $S_3\times D_{18}$ x 3, $C_9\times S_4$ x 2, $D_6:C_{18}$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $A_4\times D_9$, $D_9:A_4$, $C_9:S_4$, $C_3^2.S_4$, $C_{36}:C_6$, $D_{18}:C_6$, $C_{36}:C_6$, $C_{36}:C_6$, $C_9:D_{12}$, $C_3:D_{36}$, $C_6.D_{18}$, $C_6^2:C_6$, $C_6:S_3^2$, $C_2\times C_3^2:C_{12}$, $C_6^2:C_6$, $C_3^2:S_4$, $C_3^2\times S_4$, $C_6.S_3^2$, $C_6:D_{18}$

Order 162: $C_3^3.S_3$ x 33, $C_3^3.C_6$ x 29, $C_3^2\wr C_2$ x 18, $C_3^2:C_{18}$ x 18, $S_3\times C_3^3$ x 17, $D_9:C_3^2$ x 16, $C_{18}:C_3^2$ x 13, $C_3^3:C_6$ x 11, $C_3^2:D_9$ x 10, $C_3^2:D_9$ x 8, $C_3^2:D_9$ x 7, $C_3^3:C_6$ x 4, $C_3^2:C_{18}$ x 4, $C_3^3:C_6$ x 4, $C_3^3:S_3$ x 4, $C_3^3:C_6$ x 4, $C_3\wr S_3$ x 3, $C_3^2\times C_{18}$ x 2, $C_3^3:S_3$ x 2, $C_3^2\times D_9$ x 2, $C_3^3\times C_6$, $C_3^3:S_3$, $C_9:D_9$

Order 144: $C_{12}:D_6$ x 2, $D_4\times D_9$, $C_6\times S_4$, $S_3^2:C_2^2$, $S_3\times S_4$, $D_6:D_6$

Order 108: $C_3\times S_3^2$ x 42, $C_3:S_3^2$ x 15, $S_3\times D_9$ x 15, $C_{18}:C_6$ x 14, $C_3^2:D_6$ x 13, $C_3^2\times D_6$ x 12, $C_3:D_{18}$ x 9, $S_3\times C_{18}$ x 6, $C_3:S_3^2$ x 5, $C_3^2:C_{12}$ x 5, $C_{18}:C_6$ x 5, $C_3\times D_{18}$ x 3, $C_9:C_{12}$ x 2, $C_3:C_{36}$ x 2, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_6\times C_{18}$ x 2, $C_9:A_4$ x 2, $C_9\times A_4$ x 2, $C_9:C_{12}$, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times A_4$, $C_3^2.A_4$, $C_3^2:D_6$, $C_9:C_{12}$

Order 81: $C_9:C_3^2$ x 53, $C_3^2\times C_9$ x 15, $C_3^4$ x 12, $C_3^2:C_9$ x 11, $C_3\wr C_3$ x 10, $C_9:C_9$ x 8, $C_3\times \He_3$ x 6, $C_9^2$ x 2

Order 72: $C_6\wr C_2$ x 7, $C_6\times D_6$ x 6, $S_3\times D_6$ x 6, $C_3\times S_4$ x 4, $\SOPlus(4,2)$ x 4, $C_2\times D_{18}$ x 3, $C_9:D_4$ x 2, $S_3\times A_4$ x 2, $D_4\times C_3^2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_3:D_{12}$ x 2, $D_4\times C_9$ x 2, $D_{36}$, $C_4\times D_9$, $C_6:D_6$, $C_6\times A_4$, $C_2\times C_3^2:C_4$, $C_3:S_4$, $C_6.D_6$, $C_2^2:D_9$

Order 54: $S_3\times C_3^2$ x 63, $C_3^2:C_6$ x 50, $C_9:C_6$ x 28, $C_3:D_9$ x 22, $C_9:C_6$ x 21, $S_3\times C_9$ x 20, $C_3\times C_{18}$ x 14, $C_3^2\times C_6$ x 13, $C_3\times D_9$ x 12, $C_3^2:S_3$ x 9, $C_3^2:C_6$ x 8, $C_3^2:S_3$

Order 48: $S_3\times D_4$ x 2, $C_2\times S_4$, $C_6\times D_4$

Order 36: $C_6\times S_3$ x 43, $S_3^2$ x 27, $D_{18}$ x 11, $C_6:S_3$ x 11, $C_6^2$ x 8, $C_2\times C_{18}$ x 6, $C_3:C_{12}$ x 5, $C_3\times A_4$ x 4, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_2^2:C_9$ x 2, $C_{36}$ x 2, $C_9:C_4$

Order 27: $C_3^3$ x 44, $C_9:C_3$ x 40, $C_3\times C_9$ x 32, $\He_3$ x 7

Order 24: $C_3\times D_4$ x 7, $C_2\times D_6$ x 6, $C_3:D_4$ x 4, $S_4$ x 3, $D_{12}$ x 2, $C_4\times S_3$ x 2, $C_2^2\times C_6$ x 2, $C_2\times A_4$ x 2, $C_2\times C_{12}$

Order 18: $C_3\times S_3$ x 77, $C_3\times C_6$ x 35, $C_3:S_3$ x 26, $D_9$ x 14, $C_{18}$ x 12

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 26, $C_2\times C_6$ x 16, $C_{12}$ x 5, $A_4$ x 3, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 48, $C_9$ x 15

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 6: $C_6$ x 27, $S_3$ x 23

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 14

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 34992: $C_3^5.(S_3\times S_4)$ ($C_1$)

Order 17496: $C_9:C_3\wr S_4$ ($C_2$), $C_9:C_3^4:S_4$ ($C_2$), $C_3^5.(S_3\times A_4)$ ($C_2$)

Order 11664: $C_3^3:(D_9\times S_4)$ ($C_3$)

Order 8748: $C_9:C_3\wr A_4$ ($C_2^2$)

Order 5832: $C_3^5:S_4$ ($S_3$), $C_9\times C_3^3:S_4$ ($C_6$), $(C_3^3\times C_9):S_4$ ($C_6$), $C_3^3:(A_4\times D_9)$ ($C_6$), $C_3^3.S_3^3$ ($S_3$)

Order 2916: $C_3^5:A_4$ ($D_6$), $C_9\times C_3^3:A_4$ ($C_2\times C_6$), $C_3^4.C_6^2$ ($D_6$)

Order 1944: $C_3\wr S_4$ ($C_3\times S_3$), $C_9:S_3^3$ ($C_3\times S_3$)

Order 1458: $C_3^5.S_3$ ($S_4$)

Order 972: $C_3^3:C_6^2$ ($S_3^2$), $C_3\wr A_4$ ($C_6\times S_3$), $C_9\times C_3:S_3^2$ ($C_6\times S_3$)

Order 729: $C_9:C_3^4$ ($C_2\times S_4$)

Order 648: $C_3^3:S_4$ ($C_9:C_6$)

Order 486: $C_3^3:D_9$ ($C_3\times S_4$)

Order 324: $C_3\wr C_2^2$ ($C_3\times S_3^2$), $C_3^3:A_4$ ($C_{18}:C_6$)

Order 243: $C_3^5$ ($S_3\times S_4$), $C_3^3\times C_9$ ($C_6\times S_4$)

Order 108: $C_3:S_3^2$ ($C_3^2.S_3^2$)

Order 81: $C_3^4$ ($C_6^2:D_6$)

Order 27: $C_3^3$ ($C_6^2.S_3^2$), $C_9:C_3$ ($S_3\wr S_3$)

Order 9: $C_3^2$ ($C_3^3:(S_3\times S_4)$), $C_9$ ($C_3\times S_3\wr S_3$)

Order 3: $C_3$ ($C_3^4:(C_6\times S_4)$)

Order 1: $C_1$ ($C_3^5.(S_3\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 34992: $C_3^5.(S_3\times S_4)$ ($C_1$)

Order 17496: $C_9:C_3\wr S_4$ ($C_2$), $C_9:C_3^4:S_4$ ($C_2$), $C_3^5.(S_3\times A_4)$ ($C_2$)

Order 11664: $C_3^3:(D_9\times S_4)$ ($C_3$)

Order 8748: $C_9:C_3\wr A_4$ ($C_2^2$)

Order 5832: $C_3^5:S_4$ ($S_3$), $C_9\times C_3^3:S_4$ ($C_6$), $(C_3^3\times C_9):S_4$ ($C_6$), $C_3^3:(A_4\times D_9)$ ($C_6$), $C_3^3.S_3^3$ ($S_3$)

Order 2916: $C_3^5:A_4$ ($D_6$), $C_9\times C_3^3:A_4$ ($C_2\times C_6$), $C_3^4.C_6^2$ ($D_6$)

Order 1944: $C_3\wr S_4$ ($C_3\times S_3$), $C_9:S_3^3$ ($C_3\times S_3$)

Order 1458: $C_3^5.S_3$ ($S_4$)

Order 972: $C_3^3:C_6^2$ ($S_3^2$), $C_3\wr A_4$ ($C_6\times S_3$), $C_9\times C_3:S_3^2$ ($C_6\times S_3$)

Order 729: $C_9:C_3^4$ ($C_2\times S_4$)

Order 648: $C_3^3:S_4$ ($C_9:C_6$)

Order 486: $C_3^3:D_9$ ($C_3\times S_4$)

Order 324: $C_3\wr C_2^2$ ($C_3\times S_3^2$), $C_3^3:A_4$ ($C_{18}:C_6$)

Order 243: $C_3^5$ ($S_3\times S_4$), $C_3^3\times C_9$ ($C_6\times S_4$)

Order 108: $C_3:S_3^2$ ($C_3^2.S_3^2$)

Order 81: $C_3^4$ ($C_6^2:D_6$)

Order 27: $C_3^3$ ($C_6^2.S_3^2$), $C_9:C_3$ ($S_3\wr S_3$)

Order 9: $C_3^2$ ($C_3^3:(S_3\times S_4)$), $C_9$ ($C_3\times S_3\wr S_3$)

Order 3: $C_3$ ($C_3^4:(C_6\times S_4)$)

Order 1: $C_1$ ($C_3^5.(S_3\times S_4)$)

Series

Derived series

$C_3^5.(S_3\times S_4)$

$\rhd$

$C_3^5.(S_3\times S_4)$

$\rhd$

$C_9\times C_3^3:A_4$

$\rhd$

$C_9\times C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^5.(S_3\times S_4)$

$\rhd$

$C_3^5.(S_3\times S_4)$

$\rhd$

$C_3^5.(S_3\times A_4)$

$\rhd$

$C_3^5.(S_3\times A_4)$

$\rhd$

$C_9:C_3\wr A_4$

$\rhd$

$C_9:C_3\wr A_4$

$\rhd$

$C_9\times C_3^3:A_4$

$\rhd$

$C_9\times C_3^3:A_4$

$\rhd$

$C_3\wr A_4$

$\rhd$

$C_3\wr A_4$

$\rhd$

$C_9\times C_3:S_3^2$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^5.(S_3\times S_4)$

$\rhd$

$C_3^5.(S_3\times S_4)$

$\rhd$

$C_9\times C_3^3:A_4$

$\rhd$

$C_9\times C_3^3:A_4$

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 1 larger groups in the database.

Character theory

Complex character table

See the $134 \times 134$ 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 $95 \times 95$ rational character table.