Abstract group 34992.it: $C_3^5:(C_6\times S_4)$

• Label: 34992.it
• Order: \( 2^{4} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $21$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^5:(C_6\times S_4)$
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^5:(C_2^3\times S_4)$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
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
Elements	1	603	2672	324	15192	3888	8424	3888	34992
Conjugacy classes	1	5	47	2	64	12	34	6	171
Divisions	1	5	31	2	41	8	14	4	106
Autjugacy classes	1	5	31	2	41	4	14	2	100

Dimension	1	2	3	4	6	8	12	16	24	32	48	96
Irr. complex chars.	12	12	12	3	42	18	39	9	18	0	6	0	171
Irr. rational chars.	4	8	4	5	14	7	21	9	19	3	10	2	106

Minimal presentations

Permutation degree:	$21$
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	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{3}=d^{6}=e^{6}=f^{3}=g^{3}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $21$ $\langle(1,2,3,5,6,7,8,9,4)(10,11,12,13,15,18)(14,17,16), (2,4)(3,6,7,9)(11,12,14,17,15,18)(13,16)(19,20,21)\rangle$
Transitive group:	36T14036				more information
Direct product:	$C_3$ $\, \times\, $ $(C_3^5:(C_2\times S_4))$
Semidirect product:	$(C_3^5:S_4)$ $\,\rtimes\,$ $S_3$	$(C_3^6:S_4)$ $\,\rtimes\,$ $C_2$	$(C_3^5:S_4)$ $\,\rtimes\,$ $C_6$	$(C_3^5:S_4)$ $\,\rtimes\,$ $C_6$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^5$ . $(S_3\times S_4)$	$C_3^4$ . $(C_6^2:D_6)$	$C_3$ . $(S_3\times C_3^4:S_4)$	$(C_3^3:C_6^2)$ . $S_3^2$	all 6

Elements of the group are displayed as permutations of degree 21.

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 361368 subgroups in 5072 conjugacy classes, 38 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^5:(C_2\times S_4)$
Commutator:	$G' \simeq$ $C_3^5:A_4$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $S_3\times C_3^4:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $C_2\times S_4$
Radical:	$R \simeq$ $C_3^5:(C_6\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^5$	$G/\operatorname{soc} \simeq$ $S_3\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6.C_3$

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

All subgroups of index up to 162 (order at least 216) are shown, as well as all normal subgroups of any index.

Order 34992: $C_3^5:(C_6\times S_4)$

Order 17496: $C_3^3.C_6^2.C_3.C_6$, $C_3^6:S_4$, $C_3^5:(C_6\times A_4)$

Order 11664: $C_3^5.C_6.C_2^3$, $S_3\times C_3^4:S_4$, $C_3^5:(C_2\times S_4)$

Order 8748: $C_3\times C_3^3.C_6^2.C_3$, $C_3^6:D_6$

Order 5832: $C_3^4.C_6^2.C_2$ x 2, $C_3^5.C_6.C_2^2$ x 2, $C_3^5:S_4$ x 2, $C_3\wr A_4:S_3$ x 2, $C_3^5:S_4$ x 2, $C_3\times C_3^5.C_2^3$, $C_3^6.C_2^2.C_2$, $C_3^5:S_4$, $C_3\wr A_4\times S_3$, $C_3^3:S_3^3$, $C_3^5:S_4$

Order 4374: $C_3^3.\He_3.C_6$, $C_3^5.C_3.C_6$, $C_3^6:C_6$

Order 3888: $S_3\times C_3^3:D_{12}$ x 2, $C_6\times C_3^3:S_4$, $C_3^3:(S_3\times D_{12})$, $C_3\times S_3^2:S_3^2$, $S_3\times C_3^3:S_4$

Order 2916: $C_3^5:A_4$ x 4, $C_3^4:C_6^2$ x 3, $C_3^6.C_4$ x 2, $C_3^5:A_4$ x 2, $C_3^5.D_6$ x 2, $C_3\times C_3^5:C_2^2$, $C_3\times C_3^5:C_2^2$, $C_3^4:C_6^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^6.C_3$

Order 1944: $C_3^4:D_{12}$ x 12, $C_3^4:S_4$ x 8, $C_3^2:S_3^3$ x 5, $C_3^3:(S_3\times C_{12})$ x 3, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $S_3\times C_3^3:C_{12}$ x 2, $S_3\times C_3^3:A_4$ x 2, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_2\times C_3\wr A_4$, $C_3^4:S_4$

Order 1458: $C_3^5:S_3$ x 5, $C_3^5:C_6$ x 5, $C_3^3\wr C_2$ x 2, $C_3^5:C_6$ x 2, $C_3^5.S_3$ x 2, $C_3^5.S_3$ x 2, $C_3^5.C_6$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:S_3$

Order 1296: $S_3\times C_3^2:D_{12}$ x 2, $S_3^3:C_6$ x 2, $C_6\times C_3^2:D_{12}$, $S_3^2:S_3^2$, $C_2\times C_3^3:S_4$, $C_6.S_3^3$, $C_6^2.S_3^2$

Order 972: $C_3^3:S_3^2$ x 22, $C_3^3:C_6^2$ x 14, $C_3^4:C_{12}$ x 14, $C_3^3:C_6^2$ x 9, $C_3\wr A_4$ x 9, $C_3^3.S_3^2$ x 6, $C_3^3:S_3^2$ x 4, $C_3^4:D_6$ x 3, $C_3^4:C_{12}$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3\times S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:D_6$, $C_3^3:S_3^2$

Order 729: $C_3^5:C_3$ x 10, $C_3^5:C_3$ x 5, $C_3^5.C_3$ x 2, $C_3^6$, $C_3^5:C_3$

Order 648: $C_3^3:D_{12}$ x 12, $C_3\times S_3^3$ x 8, $C_3^3:D_{12}$ x 8, $C_3\wr D_4$ x 8, $S_3\times C_3^2:C_{12}$ x 6, $C_3:S_3^3$ x 5, $C_3^4:C_2^3$ x 3, $C_3^3:S_4$ x 3, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_2\times C_3^3:A_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:C_2^3$, $C_6\times C_3^2:C_{12}$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_3^2:(C_6\times C_{12})$, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$

Order 486: $C_3^4:C_6$ x 36, $C_3^4:C_6$ x 26, $C_3^4:S_3$ x 21, $C_3^4.S_3$ x 12, $C_3^4.C_6$ x 8, $C_3^4.S_3$ x 6, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 5, $C_3^3:D_9$ x 4, $S_3\times C_3^4$ x 4, $D_9:C_3^3$ x 4, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $C_3^3:C_{18}$ x 2, $C_3^4.S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^3:D_9$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 2, $C_{12}:S_3^2$ x 2, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_6^2:D_6$, $C_6^2:D_6$, $D_6:S_3^2$, $C_6^2:D_6$

Order 324: $C_3\wr C_2^2$ x 50, $C_3^2:S_3^2$ x 45, $C_3^2\times S_3^2$ x 29, $C_3^2:S_3^2$ x 16, $C_3\wr C_4$ x 14, $C_3^3:C_{12}$ x 14, $C_3^2.S_3^2$ x 6, $C_3^2:C_6^2$ x 5, $C_3^3:A_4$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:D_{18}$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:C_{12}$ x 2, $C_9:C_6^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^3:D_6$, $C_3^3:D_6$, $D_6\times C_3^3$, $C_3^2:S_3^2$, $C_3^2:C_6^2$, $C_3^3:A_4$, $C_3^2:S_3^2$, $C_3^3:D_6$

Order 243: $C_3^4:C_3$ x 53, $C_3^5$ x 27, $C_3^4.C_3$ x 16, $C_3^4:C_3$ x 8, $C_9:C_3^3$ x 4, $C_3^2\times \He_3$ x 4, $C_3^3:C_9$ x 4

Order 216: $C_3^2:D_{12}$ x 12, $C_6\times S_3^2$ x 11, $S_3^3$ x 8, $C_3^2:D_{12}$ x 8, $S_3^2:C_6$ x 8, $S_3^2:S_3$ x 4, $C_6:S_3^2$ x 3, $C_2\times C_3^2:C_{12}$ x 3, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$ x 2, $C_3^2\times D_{12}$ x 2, $C_6.C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:S_4$, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$, $C_3^2:S_4$

Order 144: $C_6\times S_4$

Order 108: $C_3:S_3^2$, $C_3^2:D_6$

Order 81: $C_3^4$ x 2

Order 72: $C_3\times S_4$ x 6

Order 54: $C_3^2:C_6$

Order 48: $C_2\times S_4$

Order 36: $C_6\times S_3$ x 5

Order 27: $C_3^3$ x 2

Order 24: $S_4$ x 3

Order 18: $C_3\times S_3$ x 51, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 4, $C_2\times C_6$ x 3

Order 9: $C_3^2$ x 2

Order 6: $C_6$ x 15, $S_3$ x 15

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 17

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 162 (order at least 216) are shown, as well as all normal subgroups of any index.

Order 34992: $C_3^5:(C_6\times S_4)$

Order 17496: $C_3^3.C_6^2.C_3.C_6$, $C_3^6:S_4$, $C_3^5:(C_6\times A_4)$

Order 11664: $C_3^5.C_6.C_2^3$, $S_3\times C_3^4:S_4$, $C_3^5:(C_2\times S_4)$

Order 8748: $C_3\times C_3^3.C_6^2.C_3$, $C_3^6:D_6$

Order 5832: $C_3^4.C_6^2.C_2$ x 2, $C_3^5.C_6.C_2^2$ x 2, $C_3^5:S_4$ x 2, $C_3\wr A_4:S_3$ x 2, $C_3^5:S_4$ x 2, $C_3\times C_3^5.C_2^3$, $C_3^6.C_2^2.C_2$, $C_3^5:S_4$, $C_3\wr A_4\times S_3$, $C_3^3:S_3^3$, $C_3^5:S_4$

Order 4374: $C_3^3.\He_3.C_6$, $C_3^5.C_3.C_6$, $C_3^6:C_6$

Order 3888: $S_3\times C_3^3:D_{12}$ x 2, $C_6\times C_3^3:S_4$, $C_3^3:(S_3\times D_{12})$, $C_3\times S_3^2:S_3^2$, $S_3\times C_3^3:S_4$

Order 2916: $C_3^5:A_4$ x 4, $C_3^4:C_6^2$ x 3, $C_3^6.C_4$ x 2, $C_3^5:A_4$ x 2, $C_3\times C_3^5:C_2^2$, $C_3\times C_3^5:C_2^2$, $C_3^4:C_6^2$, $C_3^5:D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^6.C_3$

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

Order 1458: $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^3\wr C_2$ x 2, $C_3^5:C_6$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.S_3$, $C_3^5.S_3$, $C_3^5.C_6$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:S_3$

Order 1296: $S_3\times C_3^2:D_{12}$ x 2, $S_3^3:C_6$ x 2, $C_6\times C_3^2:D_{12}$, $S_3^2:S_3^2$, $C_2\times C_3^3:S_4$, $C_6.S_3^3$, $C_6^2.S_3^2$

Order 972: $C_3^3:S_3^2$ x 22, $C_3^3:C_6^2$ x 14, $C_3^4:C_{12}$ x 14, $C_3^3:C_6^2$ x 9, $C_3\wr A_4$ x 8, $C_3^3:S_3^2$ x 4, $C_3^4:C_{12}$ x 2, $C_3^3.S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3\times S_3^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:C_6^2$, $C_3^4:D_6$, $C_3^3:S_3^2$, $C_3^3:D_{18}$

Order 729: $C_3^5:C_3$ x 8, $C_3^5:C_3$ x 4, $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 648: $C_3^3:D_{12}$ x 12, $C_3\times S_3^3$ x 8, $C_3^3:D_{12}$ x 8, $C_3\wr D_4$ x 8, $S_3\times C_3^2:C_{12}$ x 6, $C_3:S_3^3$ x 5, $C_3^4:C_2^3$ x 3, $C_3^3:S_4$ x 3, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_2\times C_3^3:A_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:C_2^3$, $C_6\times C_3^2:C_{12}$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_3^2:(C_6\times C_{12})$, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$

Order 486: $C_3^4:C_6$ x 36, $C_3^4:C_6$ x 26, $C_3^4:S_3$ x 14, $S_3\times C_3^4$ x 4, $C_3^4:C_6$ x 4, $C_3^4.S_3$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4.C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^3:D_9$ x 2, $D_9:C_3^3$ x 2, $C_3^4.S_3$ x 2, $C_3^4:S_3$ x 2, $C_3^3:C_{18}$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^3:D_9$

Order 432: $S_3^3:C_2$ x 2, $C_{12}:S_3^2$ x 2, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_6^2:D_6$, $C_6^2:D_6$, $D_6:S_3^2$, $C_6^2:D_6$

Order 324: $C_3\wr C_2^2$ x 49, $C_3^2:S_3^2$ x 45, $C_3^2\times S_3^2$ x 29, $C_3^2:S_3^2$ x 16, $C_3\wr C_4$ x 14, $C_3^3:C_{12}$ x 14, $C_3^2:C_6^2$ x 5, $C_3^3:A_4$ x 4, $C_3^2:S_3^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^2.S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$, $C_3^2:D_{18}$, $C_3^3:D_6$, $D_6\times C_3^3$, $C_3^2:S_3^2$, $C_9:C_6^2$, $C_3^2:C_6^2$, $C_3^3:A_4$, $C_3^2:S_3^2$, $C_3^3:D_6$, $C_3^2:D_{18}$

Order 243: $C_3^4:C_3$ x 33, $C_3^5$ x 27, $C_3^4.C_3$ x 6, $C_3^4:C_3$ x 6, $C_3^2\times \He_3$ x 4, $C_9:C_3^3$ x 2, $C_3^3:C_9$ x 2

Order 216: $C_3^2:D_{12}$ x 12, $C_6\times S_3^2$ x 11, $S_3^3$ x 8, $C_3^2:D_{12}$ x 8, $S_3^2:C_6$ x 8, $S_3^2:S_3$ x 4, $C_6:S_3^2$ x 3, $C_2\times C_3^2:C_{12}$ x 3, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$ x 2, $C_3^2\times D_{12}$ x 2, $C_6.C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:S_4$, $C_6^2:C_6$, $C_6:S_3^2$, $C_6^2:C_6$, $C_3^2:S_4$

Order 144: $C_6\times S_4$

Order 108: $C_3:S_3^2$, $C_3^2:D_6$

Order 81: $C_3^4$ x 2

Order 72: $C_3\times S_4$ x 5

Order 54: $C_3^2:C_6$

Order 48: $C_2\times S_4$

Order 36: $C_6\times S_3$ x 5

Order 27: $C_3^3$ x 2

Order 24: $S_4$ x 3

Order 18: $C_3\times S_3$ x 43, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 4, $C_2\times C_6$ x 3

Order 9: $C_3^2$ x 2

Order 6: $C_6$ x 15, $S_3$ x 15

Order 4: $C_2^2$ x 2

Order 3: $C_3$ x 17

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 34992: $C_3^5:(C_6\times S_4)$ ($C_1$)

Order 17496: $C_3^3.C_6^2.C_3.C_6$ ($C_2$), $C_3^6:S_4$ ($C_2$), $C_3^5:(C_6\times A_4)$ ($C_2$)

Order 11664: $C_3^5:(C_2\times S_4)$ ($C_3$)

Order 8748: $C_3\times C_3^3.C_6^2.C_3$ ($C_2^2$)

Order 5832: $C_3\times C_3^5.C_2^3$ ($S_3$), $C_3^5:S_4$ ($S_3$), $C_3^5:S_4$ ($C_6$), $C_3\wr A_4:S_3$ ($C_6$), $C_3^5:S_4$ ($C_6$)

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

Order 1944: $C_3^2:S_3^3$ ($C_3\times S_3$), $C_3^4:S_4$ ($C_3\times S_3$)

Order 1458: $C_3^5:C_6$ ($S_4$)

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

Order 729: $C_3^6$ ($C_2\times S_4$)

Order 486: $C_3^4:C_6$ ($C_3\times S_4$)

Order 324: $C_3\wr C_2^2$ ($C_3\times S_3^2$), $C_3\wr C_2^2$ ($C_3^2:D_6$)

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

Order 108: $C_3:S_3^2$ ($C_3^3:D_6$)

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

Order 54: $C_3^2:C_6$ ($C_3^3:S_4$)

Order 27: $C_3^3$ ($C_2\times C_3^3:S_4$), $C_3^3$ ($C_6^2.S_3^2$)

Order 18: $C_3:S_3$ ($C_3^4:S_4$)

Order 9: $C_3^2$ ($C_6\times C_3^3:S_4$), $C_3^2$ ($S_3\times C_3^3:S_4$)

Order 3: $C_3$ ($S_3\times C_3^4:S_4$), $C_3$ ($C_3^5:(C_2\times S_4)$)

Order 1: $C_1$ ($C_3^5:(C_6\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 34992: $C_3^5:(C_6\times S_4)$ ($C_1$)

Order 17496: $C_3^3.C_6^2.C_3.C_6$ ($C_2$), $C_3^6:S_4$ ($C_2$), $C_3^5:(C_6\times A_4)$ ($C_2$)

Order 11664: $C_3^5:(C_2\times S_4)$ ($C_3$)

Order 8748: $C_3\times C_3^3.C_6^2.C_3$ ($C_2^2$)

Order 5832: $C_3\times C_3^5.C_2^3$ ($S_3$), $C_3^5:S_4$ ($S_3$), $C_3^5:S_4$ ($C_6$), $C_3\wr A_4:S_3$ ($C_6$), $C_3^5:S_4$ ($C_6$)

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

Order 1944: $C_3^2:S_3^3$ ($C_3\times S_3$), $C_3^4:S_4$ ($C_3\times S_3$)

Order 1458: $C_3^5:C_6$ ($S_4$)

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

Order 729: $C_3^6$ ($C_2\times S_4$)

Order 486: $C_3^4:C_6$ ($C_3\times S_4$)

Order 324: $C_3\wr C_2^2$ ($C_3\times S_3^2$), $C_3\wr C_2^2$ ($C_3^2:D_6$)

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

Order 108: $C_3:S_3^2$ ($C_3^3:D_6$)

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

Order 54: $C_3^2:C_6$ ($C_3^3:S_4$)

Order 27: $C_3^3$ ($C_2\times C_3^3:S_4$), $C_3^3$ ($C_6^2.S_3^2$)

Order 18: $C_3:S_3$ ($C_3^4:S_4$)

Order 9: $C_3^2$ ($C_6\times C_3^3:S_4$), $C_3^2$ ($S_3\times C_3^3:S_4$)

Order 3: $C_3$ ($S_3\times C_3^4:S_4$), $C_3$ ($C_3^5:(C_2\times S_4)$)

Order 1: $C_1$ ($C_3^5:(C_6\times S_4)$)

Series

Derived series

$C_3^5:(C_6\times S_4)$

$\rhd$

$C_3^5:(C_6\times S_4)$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^5:(C_6\times S_4)$

$\rhd$

$C_3^5:(C_6\times S_4)$

$\rhd$

$C_3^5:(C_6\times A_4)$

$\rhd$

$C_3^5:(C_6\times A_4)$

$\rhd$

$C_3\times C_3^3.C_6^2.C_3$

$\rhd$

$C_3\times C_3^3.C_6^2.C_3$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3\wr A_4$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3\wr C_2^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^5:(C_6\times S_4)$

$\rhd$

$C_3^5:(C_6\times S_4)$

$\rhd$

$C_3^5:A_4$

$\rhd$

$C_3^5:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Supergroups

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

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $171 \times 171$ 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 $106 \times 106$ rational character table.