Abstract group 52488.mx: $C_3^5.S_3^3$

• Label: 52488.mx
• Order: \( 2^{3} \cdot 3^{8} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{9} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^5.S_3^3$
Order:	\(52488\)\(\medspace = 2^{3} \cdot 3^{8} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.C_3.C_6^3.C_2^2$, of order \(629856\)\(\medspace = 2^{5} \cdot 3^{9} \)
Composition factors:	$C_2$ x 3, $C_3$ x 8
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6	9	18
Elements	1	1143	2186	31662	4374	13122	52488
Conjugacy classes	1	7	61	84	57	24	234
Divisions	1	7	42	56	38	16	160
Autjugacy classes	1	5	41	55	18	12	132

Dimension	1	2	4	6	8	12	16	18	24	36	48	72
Irr. complex chars.	24	48	36	12	18	42	0	8	36	8	0	2	234
Irr. rational chars.	8	24	28	4	18	14	6	8	28	8	12	2	160

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

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:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{6}=d^{3}=e^{3}=f^{3}=g^{9}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,3,9,16,4,5,13,15,14)(2,6,8,17,7,12,18,10,11)(20,23)(21,22), (1,2,5,6,3,8) \!\cdots\! \rangle$
Transitive group:	36T16556				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^6.D_6)$ $\,\rtimes\,$ $S_3$	$(C_3^5.S_3^2)$ $\,\rtimes\,$ $S_3$	$(C_3^5.S_3)$ $\,\rtimes\,$ $S_3^2$	$(C_3^6.S_3^2)$ $\,\rtimes\,$ $C_2$	all 24
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^5$ . $S_3^3$	$(C_3^6:D_6)$ . $S_3$	$C_3^6$ . $(S_3\times D_6)$	$(C_3^6.C_6)$ . $D_6$ (2)	all 42

Elements of the group are displayed as permutations of degree 24.

Homology

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

Subgroups

There are 1097410 subgroups in 15137 conjugacy classes, 131 normal (57 characteristic).

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

Special subgroups

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

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

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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 81 (order at least 648) are shown, as well as all normal subgroups of any index.

Order 52488: $C_3^5.S_3^3$

Order 26244: $C_3^3.C_3^5.C_2^2$ x 2, $C_3^6.S_3^2$ x 2, $C_3^3.C_3^5.C_2^2$, $C_3^6.(C_6\times S_3)$, $C_3^6.(C_6\times S_3)$

Order 17496: $C_3^4.S_3^3$ x 2, $C_3^4.S_3^3$, $C_3^4.S_3^3$, $C_3^4:S_3^3$

Order 13122: $C_3^3.C_3^5.C_2$ x 2, $C_3^3.C_3^5.C_2$ x 2, $C_3^3.C_3^5.C_2$, $C_3^6.C_3.C_6$, $C_3^3.C_3^5.C_2$

Order 8748: $C_3^3.C_3^4.C_2^2$ x 4, $C_3^3.C_3^4.C_2^2$ x 4, $C_3^3.C_3^4.C_2^2$ x 2, $C_3.C_3^5.C_6.C_2$ x 2, $C_3^3.C_3^4.C_2^2$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^3.C_3^4.C_2^2$ x 2, $C_3.C_3^5.C_6.C_2$ x 2, $C_3^3.C_3^3.D_6$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^5.S_3^2$ x 2, $C_3^5.S_3^2$ x 2, $C_3^6.D_6$ x 2, $C_3^4.C_3^3.C_2^2$, $C_3^5.C_6^2$, $C_3^4.C_3^3.C_2^2$, $C_3^3.C_3^3.D_6$, $C_3^5.C_6^2$, $C_9.C_3^5.C_2^2$, $C_3^6.D_6$, $C_3^6:D_6$, $C_3^5.C_6^2$

Order 6561: $C_3^2\times C_3^4.C_3^2$

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

Order 4374: $C_3^3.C_3^4.C_2$ x 12, $C_3^3.C_3^4.C_2$ x 8, $C_3^3.C_3^4.C_2$ x 8, $C_3^3.C_3^4.C_2$ x 6, $C_3^3.C_3^4.C_2$ x 4, $C_3^3.C_3^4.C_2$ x 4, $C_3.C_3^5.C_6$ x 4, $C_3^3.C_3^4.C_2$ x 4, $C_3^5.C_3.C_6$ x 4, $C_3^5.C_3.C_6$ x 3, $C_3^6.C_6$ x 2, $C_3.C_3^5.C_6$ x 2, $C_3^3.C_3^4.C_2$ x 2, $C_3^6:C_6$ x 2, $C_9.C_3^5.C_2$, $C_3^3.C_3^4.C_2$, $C_3^5.C_3.C_6$, $C_3^3.C_3^4.C_2$, $C_3^6.C_6$, $C_3^2.C_3^5.C_2$, $C_3^5.C_3.C_6$, $C_9.C_3^5.C_2$, $C_3^3.C_3^4.C_2$, $C_3^6.C_6$, $C_3^6.C_6$, $C_3^4.C_3^3.C_2$

Order 2916: $C_3^4.S_3^2$ x 10, $C_3^5.C_6.C_2$ x 5, $C_3^4:C_6^2$ x 5, $C_3^2:C_9.S_3^2$ x 4, $C_3^3.C_3^3.C_2^2$ x 4, $C_9.C_3^4.C_2^2$ x 4, $C_3^2:C_9:C_3.D_6$ x 4, $C_9.C_3^4.C_2^2$ x 4, $C_3^4.(C_6\times S_3)$ x 4, $C_3^5.D_6$ x 4, $C_3^5:D_6$ x 4, $C_3^5.D_6$ x 4, $C_3.C_3^5.C_2^2$ x 3, $C_3^6.C_2^2$ x 3, $C_3^4.C_6^2$ x 3, $C_3^5.D_6$ x 3, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^2.C_3^4.C_2^2$ x 2, $C_3^2:C_9.S_3^2$ x 2, $C_3^2.C_3^4.C_2^2$ x 2, $C_3^2:C_9.S_3^2$ x 2, $C_3^2.C_3^4.C_2^2$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3.C_3^5.C_2^2$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^2.C_3^4.C_2^2$ x 2, $C_3^5.D_6$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4.S_3^2$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5.D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3^4.C_6^2$ x 2, $C_3^3.C_3^3.C_2^2$, $C_3^2.C_3^4.C_2^2$, $C_3.C_3^5.C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_2\times (C_3^4.C_3^2):C_2$, $(C_3^3\times C_9).D_6$, $C_3^5.C_6.C_2$, $(C_3^4\times C_9).C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3^4:C_6^2$, $C_3^4:C_6^2$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^3:(S_3\times C_{18})$, $C_3^4.(C_6\times S_3)$, $C_3^4.(C_6\times S_3)$

Order 2187: $C_3\times C_3^4.C_3^2$ x 26, $C_3^3.C_3^4$ x 4, $C_3^4\times C_9:C_3$, $C_3^2\times C_3^4.C_3$, $C_3^2\times C_3^4:C_3$

Order 1944: $C_3^2:S_3^3$ x 4, $C_3^2.S_3^3$ x 3, $C_3^2:S_3^3$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2.S_3^3$ x 2, $C_2\times C_3^3.S_3^2$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_9:S_3^3$, $C_3^2:S_3^3$, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^4:D_6$

Order 1458: $C_9:C_3^3:S_3$ x 29, $C_3^5.S_3$ x 26, $C_3^5.S_3$ x 23, $C_3^4.(C_3\times S_3)$ x 20, $C_3^4.(C_3\times S_3)$ x 20, $(C_3^2\times \He_3).S_3$ x 13, $S_3\times C_3^3.C_3^2$ x 13, $C_3^5.C_6$ x 13, $C_3^5.S_3$ x 13, $C_3^5:S_3$ x 13, $C_3^5:C_6$ x 9, $C_3^4.(C_3\times S_3)$ x 8, $C_3^4.(C_3\times S_3)$ x 8, $C_3^4.(C_3\times S_3)$ x 8, $C_3^4.(C_3\times S_3)$ x 8, $C_3^5.C_6$ x 8, $C_3^5:C_6$ x 8, $C_3^5:C_6$ x 6, $C_3^5.S_3$ x 6, $C_3^5.S_3$ x 6, $C_3^5:S_3$ x 6, $C_3^5:S_3$ x 6, $C_3^3\wr C_2$ x 4, $C_3^5.S_3$ x 4, $C_3^5.C_6$ x 4, $C_3^5.S_3$ x 4, $C_3^5.C_6$ x 4, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^4:C_{18}$ x 4, $(C_3^2\times \He_3).S_3$ x 3, $D_9:C_3^4$ x 3, $C_3^5:C_6$ x 3, $C_3^5.C_6$ x 3, $C_3^5:C_6$ x 3, $C_3^4:C_{18}$ x 3, $C_3^5:C_6$ x 2, $C_3^5:S_3$ x 2, $C_3^4:D_9$ x 2, $C_3^4:D_9$ x 2, $C_3^4:D_9$ x 2, $C_3^5:C_6$, $C_3^4:D_9$, $C_3^4:D_9$, $C_3^4:C_{18}$, $C_2\times C_3^3.C_3^3$

Order 972: $C_3^3:S_3^2$ x 25, $C_3^3.S_3^2$ x 21, $C_3^3.S_3^2$ x 18, $C_3^3:S_3^2$ x 16, $C_3^3:C_6^2$ x 14, $C_3^3.S_3^2$ x 14, $C_3^3.S_3^2$ x 14, $C_3^4:D_6$ x 14, $C_3^3:C_6^2$ x 12, $C_3^3.S_3^2$ x 12, $C_3^3:C_6^2$ x 10, $C_3^3:S_3^2$ x 8, $C_3^3:S_3^2$ x 8, $C_3^3:S_3^2$ x 7, $C_3^3:D_{18}$ x 7, $C_3^3.S_3^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 5, $C_3^3:D_{18}$ x 4, $C_3^3:D_{18}$ x 4, $C_3^3:D_{18}$ x 4, $S_3\times C_3^2:C_{18}$ x 4, $C_3^3:D_{18}$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:D_{18}$ x 3, $C_3^3:D_{18}$ x 3, $C_3^3.C_6^2$ x 3, $\He_3\times S_3^2$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_9\times C_3:S_3^2$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3.(C_6\times S_3)$ x 2, $C_3^3.(C_6\times S_3)$ x 2, $C_3^4.D_6$, $C_3^3:S_3^2$, $C_3^4.D_6$, $C_3^4:D_6$, $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^3:C_6^2$, $C_3^3.C_6^2$, $(C_6\times \He_3).S_3$

Order 729: $C_3^3.C_3^3$ x 116, $C_9:C_3^4$ x 63, $C_3^3.C_3^3$ x 59, $C_3^5.C_3$ x 19, $C_3^5:C_3$ x 19, $C_3^3\times \He_3$ x 4, $C_3^4:C_9$ x 3, $C_3^6$, $C_3^4\times C_9$

Order 648: $C_3\times S_3^3$ x 8, $C_3:S_3^3$ x 4, $C_3:S_3^3$ x 3, $C_{18}:C_6\times S_3$ x 3, $S_3\times C_3^2:D_6$ x 3, $D_9\times S_3^2$ x 3, $C_3^4:C_2^3$ x 2, $C_3:S_3^3$ x 2, $C_3.S_3^3$ x 2, $C_2\times C_3^2:S_3^2$, $C_2\times C_3^2:D_{18}$

Order 486: $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^3:D_9$ x 2, $(C_3\times \He_3).S_3$

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

Order 216: $C_6\times S_3^2$, $S_3^3$

Order 162: $C_3^2\wr C_2$

Order 108: $C_3\times S_3^2$ x 13

Order 81: $C_3^4$ x 6, $C_3^2:C_9$, $C_3\times \He_3$

Order 72: $S_3\times D_6$ x 3, $C_6\times D_6$ x 2

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

Order 36: $C_6\times S_3$ x 31, $S_3^2$ x 29

Order 27: $C_3^3$ x 6

Order 24: $C_2\times D_6$ x 3, $C_2^2\times C_6$

Order 18: $C_3\times S_3$ x 97, $C_3:S_3$

Order 12: $D_6$ x 27, $C_2\times C_6$ x 11

Order 9: $C_3^2$ x 6

Order 8: $C_2^3$

Order 6: $S_3$ x 37, $C_6$ x 27

Order 4: $C_2^2$ x 7

Order 3: $C_3$ x 22

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 81 (order at least 648) are shown, as well as all normal subgroups of any index.

Order 52488: $C_3^5.S_3^3$

Order 26244: $C_3^3.C_3^5.C_2^2$, $C_3^3.C_3^5.C_2^2$, $C_3^6.(C_6\times S_3)$, $C_3^6.(C_6\times S_3)$, $C_3^6.S_3^2$

Order 17496: $C_3^4.S_3^3$, $C_3^4.S_3^3$, $C_3^4.S_3^3$, $C_3^4:S_3^3$

Order 13122: $C_3^3.C_3^5.C_2$, $C_3^6.C_3.C_6$, $C_3^3.C_3^5.C_2$, $C_3^3.C_3^5.C_2$, $C_3^3.C_3^5.C_2$

Order 8748: $C_3^3.C_3^4.C_2^2$ x 2, $C_3^3.C_3^4.C_2^2$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^3.C_3^4.C_2^2$, $C_3.C_3^5.C_6.C_2$, $C_3^3.C_3^4.C_2^2$, $C_3^5.(C_6\times S_3)$, $C_3^4.C_3^3.C_2^2$, $C_3^3.C_3^4.C_2^2$, $C_3^5.C_6^2$, $C_3.C_3^5.C_6.C_2$, $C_3^4.C_3^3.C_2^2$, $C_3^3.C_3^3.D_6$, $C_3^3.C_3^3.D_6$, $C_3^5.C_6^2$, $C_9.C_3^5.C_2^2$, $C_3^5.(C_6\times S_3)$, $C_3^5.(C_6\times S_3)$, $C_3^5.S_3^2$, $C_3^5.S_3^2$, $C_3^6.D_6$, $C_3^6.D_6$, $C_3^6:D_6$, $C_3^5.C_6^2$

Order 6561: $C_3^2\times C_3^4.C_3^2$

Order 5832: $C_3^5.C_6.C_2^2$, $C_3^3.S_3^3$, $C_3^3.S_3^3$, $C_3^3.S_3^3$, $C_3^3.S_3^3$, $C_3^3.S_3^3$, $C_3^3:S_3^3$, $C_2\times C_3^4.S_3^2$, $C_3^3:S_3^3$

Order 4374: $C_3^3.C_3^4.C_2$ x 7, $C_3^5.C_3.C_6$ x 4, $C_3^3.C_3^4.C_2$ x 4, $C_3^3.C_3^4.C_2$ x 4, $C_3^3.C_3^4.C_2$ x 4, $C_3^3.C_3^4.C_2$ x 2, $C_3^3.C_3^4.C_2$ x 2, $C_3.C_3^5.C_6$ x 2, $C_3^3.C_3^4.C_2$ x 2, $C_3^5.C_3.C_6$ x 2, $C_9.C_3^5.C_2$, $C_3^3.C_3^4.C_2$, $C_3^5.C_3.C_6$, $C_3^3.C_3^4.C_2$, $C_3^6.C_6$, $C_3^6.C_6$, $C_3.C_3^5.C_6$, $C_3^2.C_3^5.C_2$, $C_3^5.C_3.C_6$, $C_3^3.C_3^4.C_2$, $C_9.C_3^5.C_2$, $C_3^3.C_3^4.C_2$, $C_3^6.C_6$, $C_3^6.C_6$, $C_3^4.C_3^3.C_2$, $C_3^6:C_6$

Order 2916: $C_3^5.C_6.C_2$ x 4, $C_3^4:C_6^2$ x 4, $C_3^4.S_3^2$ x 4, $C_9.C_3^4.C_2^2$ x 3, $C_3^2:C_9.S_3^2$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_9.C_3^4.C_2^2$ x 2, $C_3^6.C_2^2$ x 2, $C_3^2:C_9:C_3.D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3.C_3^5.C_2^2$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^5.D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5.D_6$ x 2, $C_3^5.D_6$ x 2, $C_3^3.C_3^3.C_2^2$, $C_3^4.(C_6\times S_3)$, $C_3^2.C_3^4.C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3^2.C_3^4.C_2^2$, $C_3^2:C_9.S_3^2$, $C_3.C_3^5.C_2^2$, $C_3^2.C_3^4.C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3^2:C_9.S_3^2$, $C_2\times (C_3^4.C_3^2):C_2$, $C_3^2.C_3^4.C_2^2$, $C_3.C_3^5.C_2^2$, $(C_3^3\times C_9).D_6$, $C_3^3.C_3^3.C_2^2$, $C_3^5.C_6.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^3.C_3^3.C_2^2$, $(C_3^4\times C_9).C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3^3.C_3^3.C_2^2$, $C_3^2.C_3^4.C_2^2$, $C_3^5.D_6$, $C_3^4:C_6^2$, $C_3^4:C_6^2$, $C_3^4.(C_6\times S_3)$, $C_3^4.S_3^2$, $C_3^4.S_3^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^4.C_6^2$, $C_3^3:(S_3\times C_{18})$, $C_3^4.(C_6\times S_3)$, $C_3^4.(C_6\times S_3)$

Order 2187: $C_3\times C_3^4.C_3^2$ x 16, $C_3^3.C_3^4$ x 4, $C_3^4\times C_9:C_3$, $C_3^2\times C_3^4.C_3$, $C_3^2\times C_3^4:C_3$

Order 1944: $C_3^2:S_3^3$ x 3, $C_3^2.S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_2\times C_3^3.S_3^2$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_9:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^4:D_6$, $C_3^2.S_3^3$

Order 1458: $C_3^5.S_3$ x 15, $C_9:C_3^3:S_3$ x 14, $C_3^4.(C_3\times S_3)$ x 8, $C_3^4.(C_3\times S_3)$ x 8, $(C_3^2\times \He_3).S_3$ x 8, $S_3\times C_3^3.C_3^2$ x 8, $C_3^5.C_6$ x 8, $C_3^5.S_3$ x 8, $C_3^5:S_3$ x 8, $C_3^5.S_3$ x 7, $C_3^5:C_6$ x 6, $C_3^5:C_6$ x 5, $C_3^4.(C_3\times S_3)$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $C_3^5.S_3$ x 4, $C_3^5.S_3$ x 4, $C_3^5.C_6$ x 4, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^5:C_6$ x 3, $C_3^5.S_3$ x 3, $C_3^5:S_3$ x 3, $C_3^4:C_{18}$ x 3, $(C_3^2\times \He_3).S_3$ x 2, $C_3^3\wr C_2$ x 2, $C_3^5:C_6$ x 2, $C_3^5.C_6$ x 2, $C_3^5.S_3$ x 2, $C_3^5.C_6$ x 2, $C_3^5.C_6$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^4:D_9$ x 2, $C_3^4:D_9$ x 2, $C_3^4:C_{18}$ x 2, $C_3^5:C_6$, $D_9:C_3^4$, $C_3^5:S_3$, $C_3^4:D_9$, $C_3^4:D_9$, $C_3^4:C_{18}$, $C_2\times C_3^3.C_3^3$, $C_3^4:D_9$

Order 972: $C_3^3:S_3^2$ x 15, $C_3^3.S_3^2$ x 11, $C_3^3:C_6^2$ x 9, $C_3^3:C_6^2$ x 9, $C_3^3:S_3^2$ x 9, $C_3^3:S_3^2$ x 6, $C_3^3.S_3^2$ x 6, $C_3^3.S_3^2$ x 6, $C_3^4:D_6$ x 6, $C_3^3:S_3^2$ x 5, $C_3^3.S_3^2$ x 5, $C_3^3:C_6^2$ x 5, $C_3^3.S_3^2$ x 5, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3.S_3^2$ x 4, $C_3^3:D_{18}$ x 4, $C_3^3:D_{18}$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 2, $C_9\times C_3:S_3^2$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3.C_6^2$ x 2, $\He_3\times S_3^2$ x 2, $C_3^3:D_{18}$ x 2, $S_3\times C_3^2:C_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3.(C_6\times S_3)$ x 2, $C_3^3.(C_6\times S_3)$ x 2, $C_3^3:S_3^2$, $C_3^4.D_6$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:D_{18}$, $C_3^4.D_6$, $C_3^4:D_6$, $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^3:C_6^2$, $C_3^3.C_6^2$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $(C_6\times \He_3).S_3$

Order 729: $C_3^3.C_3^3$ x 45, $C_3^3.C_3^3$ x 34, $C_9:C_3^4$ x 25, $C_3^5.C_3$ x 12, $C_3^5:C_3$ x 12, $C_3^3\times \He_3$ x 4, $C_3^4:C_9$ x 3, $C_3^6$, $C_3^4\times C_9$

Order 648: $C_3\times S_3^3$ x 5, $C_3:S_3^3$ x 3, $C_3^4:C_2^3$ x 2, $C_3:S_3^3$ x 2, $C_3:S_3^3$ x 2, $C_{18}:C_6\times S_3$ x 2, $S_3\times C_3^2:D_6$ x 2, $D_9\times S_3^2$ x 2, $C_2\times C_3^2:S_3^2$, $C_3.S_3^3$, $C_2\times C_3^2:D_{18}$

Order 486: $C_3^4:C_6$ x 2, $(C_3\times \He_3).S_3$, $C_3^4:S_3$, $C_3^3:D_9$

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

Order 216: $C_6\times S_3^2$, $S_3^3$

Order 162: $C_3^2\wr C_2$

Order 108: $C_3\times S_3^2$ x 7

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

Order 72: $S_3\times D_6$ x 2, $C_6\times D_6$

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

Order 36: $C_6\times S_3$ x 15, $S_3^2$ x 14

Order 27: $C_3^3$ x 5

Order 24: $C_2\times D_6$ x 2, $C_2^2\times C_6$

Order 18: $C_3\times S_3$ x 46, $C_3:S_3$

Order 12: $D_6$ x 15, $C_2\times C_6$ x 8

Order 9: $C_3^2$ x 4

Order 8: $C_2^3$

Order 6: $S_3$ x 22, $C_6$ x 18

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 15

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 52488: $C_3^5.S_3^3$ ($C_1$)

Order 26244: $C_3^3.C_3^5.C_2^2$ ($C_2$) x 2, $C_3^6.S_3^2$ ($C_2$) x 2, $C_3^3.C_3^5.C_2^2$ ($C_2$), $C_3^6.(C_6\times S_3)$ ($C_2$), $C_3^6.(C_6\times S_3)$ ($C_2$)

Order 17496: $C_3^4.S_3^3$ ($C_3$)

Order 13122: $C_3^3.C_3^5.C_2$ ($C_2^2$) x 2, $C_3^3.C_3^5.C_2$ ($C_2^2$) x 2, $C_3^3.C_3^5.C_2$ ($C_2^2$), $C_3^6.C_3.C_6$ ($C_2^2$), $C_3^3.C_3^5.C_2$ ($C_2^2$)

Order 8748: $C_3^3.C_3^4.C_2^2$ ($C_6$) x 2, $C_3^3.C_3^3.D_6$ ($C_6$) x 2, $C_3^5.S_3^2$ ($S_3$) x 2, $C_3^3.C_3^3.D_6$ ($C_6$), $C_3^5.(C_6\times S_3)$ ($C_6$), $C_3^5.(C_6\times S_3)$ ($C_6$), $C_3^6.D_6$ ($S_3$), $C_3^6:D_6$ ($S_3$)

Order 6561: $C_3^2\times C_3^4.C_3^2$ ($C_2^3$)

Order 4374: $C_3^3.C_3^4.C_2$ ($C_2\times C_6$) x 2, $C_3.C_3^5.C_6$ ($D_6$) x 2, $C_3.C_3^5.C_6$ ($C_2\times C_6$) x 2, $C_3^3.C_3^4.C_2$ ($D_6$) x 2, $C_3^3.C_3^4.C_2$ ($D_6$) x 2, $C_3^3.C_3^4.C_2$ ($D_6$) x 2, $C_3^3.C_3^4.C_2$ ($D_6$) x 2, $C_3^3.C_3^4.C_2$ ($C_2\times C_6$), $C_3^3.C_3^4.C_2$ ($C_2\times C_6$), $C_3^6.C_6$ ($D_6$), $C_3^5.C_3.C_6$ ($C_2\times C_6$), $C_3^6.C_6$ ($D_6$)

Order 2916: $C_3^4.S_3^2$ ($C_3\times S_3$) x 2, $C_3.C_3^5.C_2^2$ ($C_3\times S_3$), $(C_3^3\times C_9).D_6$ ($C_3\times S_3$)

Order 2187: $C_3\times C_3^4.C_3^2$ ($C_2\times D_6$) x 2, $C_3^2\times C_3^4.C_3$ ($C_2\times D_6$), $C_3^2\times C_3^4:C_3$ ($C_2\times D_6$), $C_3^3.C_3^4$ ($C_2^2\times C_6$)

Order 1458: $C_3^4.(C_3\times S_3)$ ($C_6\times S_3$) x 2, $(C_3^2\times \He_3).S_3$ ($C_6\times S_3$) x 2, $C_3^4.(C_3\times S_3)$ ($C_6\times S_3$) x 2, $C_3^5:S_3$ ($C_6\times S_3$) x 2, $C_3^5.S_3$ ($S_3^2$) x 2, $C_3^5:S_3$ ($S_3^2$) x 2, $C_3^4:D_9$ ($C_6\times S_3$) x 2, $C_9:C_3^3:S_3$ ($S_3^2$), $C_3^5:C_6$ ($S_3^2$), $C_3^5:C_6$ ($C_6\times S_3$), $C_3^4:C_{18}$ ($C_6\times S_3$)

Order 729: $C_3^5.C_3$ ($S_3\times D_6$) x 2, $C_3^5:C_3$ ($S_3\times D_6$) x 2, $C_3^3.C_3^3$ ($C_6\times D_6$) x 2, $C_3^6$ ($S_3\times D_6$), $C_3^3\times \He_3$ ($C_6\times D_6$), $C_3^4:C_9$ ($C_6\times D_6$), $C_3^3.C_3^3$ ($S_3\times D_6$)

Order 486: $C_3^4:S_3$ ($C_3\times S_3^2$) x 2, $C_3^3:D_9$ ($C_3\times S_3^2$) x 2, $(C_3\times \He_3).S_3$ ($C_3\times S_3^2$), $C_3^4:C_6$ ($C_3\times S_3^2$), $C_3^4:C_6$ ($C_3^2:D_6$)

Order 243: $C_3^5$ ($S_3^3$) x 2, $C_3^2\times \He_3$ ($C_6\times S_3^2$) x 2, $C_3^3:C_9$ ($C_6\times S_3^2$) x 2, $C_3^5$ ($C_6\times S_3^2$), $C_3^5$ ($C_6.S_3^2$), $C_3^3.C_3^2$ ($C_6\times S_3^2$), $C_3^4.C_3$ ($S_3^3$), $C_3^4:C_3$ ($S_3^3$)

Order 162: $C_3^2\wr C_2$ ($C_3^3:D_6$)

Order 81: $C_3^4$ ($C_3\times S_3^3$) x 2, $C_3^4$ ($C_3.S_3^3$) x 2, $C_3^2:C_9$ ($C_3\times S_3^3$), $C_3^4$ ($C_3:S_3^3$), $C_3^4$ ($C_6\times C_3^2:D_6$), $C_3\times \He_3$ ($C_3\times S_3^3$)

Order 54: $C_3^2:C_6$ ($C_3^3.S_3^2$), $C_3^2:C_6$ ($C_3^3.S_3^2$)

Order 27: $C_3^3$ ($C_3^2.S_3^3$) x 2, $C_3^3$ ($C_3^2:S_3^3$), $C_3^3$ ($C_3^2:S_3^3$), $C_3^3$ ($C_2\times C_3^3.S_3^2$), $C_3^3$ ($C_3^3:(C_6\times D_6)$)

Order 18: $C_3:S_3$ ($C_3^3.C_3^3.C_2^2$)

Order 9: $C_3^2$ ($\He_3.C_3^3.C_2^3$) x 2, $C_3^2$ ($C_3^3.S_3^3$) x 2, $C_3^2$ ($C_3^3.C_3^3.C_2^3$), $C_3^2$ ($C_3^3:S_3^3$)

Order 3: $C_3$ ($C_3^4.S_3^3$) x 2, $C_3$ ($C_3^3.C_3^4.C_2^3$), $C_3$ ($C_3^2.C_3^5.C_2^3$)

Order 1: $C_1$ ($C_3^5.S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 52488: $C_3^5.S_3^3$ ($C_1$)

Order 26244: $C_3^3.C_3^5.C_2^2$ ($C_2$), $C_3^3.C_3^5.C_2^2$ ($C_2$), $C_3^6.(C_6\times S_3)$ ($C_2$), $C_3^6.(C_6\times S_3)$ ($C_2$), $C_3^6.S_3^2$ ($C_2$)

Order 17496: $C_3^4.S_3^3$ ($C_3$)

Order 13122: $C_3^3.C_3^5.C_2$ ($C_2^2$), $C_3^6.C_3.C_6$ ($C_2^2$), $C_3^3.C_3^5.C_2$ ($C_2^2$), $C_3^3.C_3^5.C_2$ ($C_2^2$), $C_3^3.C_3^5.C_2$ ($C_2^2$)

Order 8748: $C_3^3.C_3^4.C_2^2$ ($C_6$), $C_3^3.C_3^3.D_6$ ($C_6$), $C_3^3.C_3^3.D_6$ ($C_6$), $C_3^5.(C_6\times S_3)$ ($C_6$), $C_3^5.(C_6\times S_3)$ ($C_6$), $C_3^5.S_3^2$ ($S_3$), $C_3^6.D_6$ ($S_3$), $C_3^6:D_6$ ($S_3$)

Order 6561: $C_3^2\times C_3^4.C_3^2$ ($C_2^3$)

Order 4374: $C_3^3.C_3^4.C_2$ ($C_2\times C_6$), $C_3^3.C_3^4.C_2$ ($C_2\times C_6$), $C_3^3.C_3^4.C_2$ ($C_2\times C_6$), $C_3^6.C_6$ ($D_6$), $C_3.C_3^5.C_6$ ($D_6$), $C_3.C_3^5.C_6$ ($C_2\times C_6$), $C_3^3.C_3^4.C_2$ ($D_6$), $C_3^5.C_3.C_6$ ($C_2\times C_6$), $C_3^3.C_3^4.C_2$ ($D_6$), $C_3^6.C_6$ ($D_6$), $C_3^3.C_3^4.C_2$ ($D_6$), $C_3^3.C_3^4.C_2$ ($D_6$)

Order 2916: $C_3.C_3^5.C_2^2$ ($C_3\times S_3$), $(C_3^3\times C_9).D_6$ ($C_3\times S_3$), $C_3^4.S_3^2$ ($C_3\times S_3$)

Order 2187: $C_3^2\times C_3^4.C_3$ ($C_2\times D_6$), $C_3^2\times C_3^4:C_3$ ($C_2\times D_6$), $C_3\times C_3^4.C_3^2$ ($C_2\times D_6$), $C_3^3.C_3^4$ ($C_2^2\times C_6$)

Order 1458: $C_9:C_3^3:S_3$ ($S_3^2$), $C_3^4.(C_3\times S_3)$ ($C_6\times S_3$), $(C_3^2\times \He_3).S_3$ ($C_6\times S_3$), $C_3^4.(C_3\times S_3)$ ($C_6\times S_3$), $C_3^5:C_6$ ($S_3^2$), $C_3^5:S_3$ ($C_6\times S_3$), $C_3^5:C_6$ ($C_6\times S_3$), $C_3^5.S_3$ ($S_3^2$), $C_3^5:S_3$ ($S_3^2$), $C_3^4:D_9$ ($C_6\times S_3$), $C_3^4:C_{18}$ ($C_6\times S_3$)

Order 729: $C_3^6$ ($S_3\times D_6$), $C_3^3\times \He_3$ ($C_6\times D_6$), $C_3^5.C_3$ ($S_3\times D_6$), $C_3^5:C_3$ ($S_3\times D_6$), $C_3^4:C_9$ ($C_6\times D_6$), $C_3^3.C_3^3$ ($S_3\times D_6$), $C_3^3.C_3^3$ ($C_6\times D_6$)

Order 486: $(C_3\times \He_3).S_3$ ($C_3\times S_3^2$), $C_3^4:C_6$ ($C_3\times S_3^2$), $C_3^4:C_6$ ($C_3^2:D_6$), $C_3^4:S_3$ ($C_3\times S_3^2$), $C_3^3:D_9$ ($C_3\times S_3^2$)

Order 243: $C_3^5$ ($C_6\times S_3^2$), $C_3^5$ ($S_3^3$), $C_3^5$ ($C_6.S_3^2$), $C_3^2\times \He_3$ ($C_6\times S_3^2$), $C_3^3.C_3^2$ ($C_6\times S_3^2$), $C_3^4.C_3$ ($S_3^3$), $C_3^4:C_3$ ($S_3^3$), $C_3^3:C_9$ ($C_6\times S_3^2$)

Order 162: $C_3^2\wr C_2$ ($C_3^3:D_6$)

Order 81: $C_3^2:C_9$ ($C_3\times S_3^3$), $C_3^4$ ($C_3:S_3^3$), $C_3^4$ ($C_3\times S_3^3$), $C_3^4$ ($C_6\times C_3^2:D_6$), $C_3^4$ ($C_3.S_3^3$), $C_3\times \He_3$ ($C_3\times S_3^3$)

Order 54: $C_3^2:C_6$ ($C_3^3.S_3^2$), $C_3^2:C_6$ ($C_3^3.S_3^2$)

Order 27: $C_3^3$ ($C_3^2:S_3^3$), $C_3^3$ ($C_3^2:S_3^3$), $C_3^3$ ($C_3^2.S_3^3$), $C_3^3$ ($C_2\times C_3^3.S_3^2$), $C_3^3$ ($C_3^3:(C_6\times D_6)$)

Order 18: $C_3:S_3$ ($C_3^3.C_3^3.C_2^2$)

Order 9: $C_3^2$ ($C_3^3.C_3^3.C_2^3$), $C_3^2$ ($\He_3.C_3^3.C_2^3$), $C_3^2$ ($C_3^3.S_3^3$), $C_3^2$ ($C_3^3:S_3^3$)

Order 3: $C_3$ ($C_3^3.C_3^4.C_2^3$), $C_3$ ($C_3^2.C_3^5.C_2^3$), $C_3$ ($C_3^4.S_3^3$)

Order 1: $C_1$ ($C_3^5.S_3^3$)

Series

Derived series

$C_3^5.S_3^3$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

Chief series

$C_3^5.S_3^3$

$\rhd$

$C_3^3.C_3^5.C_2^2$

$\rhd$

$C_3^3.C_3^5.C_2$

$\rhd$

$C_3^2\times C_3^4.C_3^2$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3\times \He_3$

$\rhd$

$C_3^2\times \He_3$

$\rhd$

$C_3^3.C_3^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^5.S_3^3$

$\rhd$

$C_3^3.C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 2 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 $234 \times 234$ 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 $160 \times 160$ rational character table (warning: may be slow to load).