Abstract group 58320.bb: $C_3\wr C_3\times S_6$

• Label: 58320.bb
• Order: \( 2^{4} \cdot 3^{6} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{6} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_3\wr C_3\times S_6$
Order:	\(58320\)\(\medspace = 2^{4} \cdot 3^{6} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$(C_3\times C_3.S_3^2).A_6.C_2^2$, of order \(466560\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 5 \)
Composition factors:	$C_2$, $C_3$ x 4, $A_6$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	9	12	15	18	36	45
Elements	1	75	3644	180	144	14100	2916	7920	6336	11340	6480	5184	58320
Conjugacy classes	1	3	38	2	1	62	12	24	12	20	8	4	187
Divisions	1	3	20	2	1	32	6	12	6	10	4	2	99
Autjugacy classes	1	2	9	2	1	13	2	8	4	3	2	1	48

Dimension	1	2	3	5	6	9	10	15	16	18	20	27	30	32	48	54	60	96
Irr. complex chars.	18	0	16	36	0	18	18	32	9	0	0	16	16	0	8	0	0	0	187
Irr. rational chars.	2	8	0	4	8	2	18	0	1	8	8	0	16	4	0	8	8	4	99

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$1908$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	15	30	30
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(10,11,12)(13,14,15), (10,12,11), (7,9,8)(10,11,12), (1,2,3,4,5)(7,10,15,9,11,14,8,12,13) \!\cdots\! \rangle$
Direct product:	$(C_3\wr C_3)$ $\, \times\, $ $S_6$
Semidirect product:	$(C_3^3\times S_6)$ $\,\rtimes\,$ $C_3$	$C_3^3$ $\,\rtimes\,$ $(C_3\times S_6)$	$(\He_3\times S_6)$ $\,\rtimes\,$ $C_3$	$\He_3$ $\,\rtimes\,$ $(C_3\times S_6)$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3\times S_6)$ . $\He_3$	$C_3$ . $(\He_3\times S_6)$	$(C_3^2\times S_6)$ . $C_3^2$	$C_3^2$ . $(C_3^2\times S_6)$	all 6
Aut. group:	$\Aut(C_6^3.(S_3\times S_4))$

Elements of the group are displayed as permutations of degree 15.

Homology

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

Subgroups

There are 161190 subgroups in 1702 conjugacy classes, 24 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $\He_3\times S_6$
Commutator:	$G' \simeq$ $C_3^2\times A_6$	$G/G' \simeq$ $C_3\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3^2\times S_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\wr C_3$	$G/\operatorname{Fit} \simeq$ $S_6$
Radical:	$R \simeq$ $C_3\wr C_3$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_3\times A_6$	$G/\operatorname{soc} \simeq$ $C_2\times \He_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 58320: $C_3\wr C_3\times S_6$

Order 29160: $C_3\wr C_3\times A_6$

Order 19440: $C_9:C_3\times S_6$ x 2, $C_3^3\times S_6$, $\He_3\times S_6$

Order 9720: $C_3\wr C_3\times S_5$ x 2, $C_9:C_3\times A_6$ x 2, $C_3^3\times A_6$, $\He_3\times A_6$

Order 6480: $C_3^2\times S_6$ x 6, $C_9\times S_6$ x 2

Order 5832: $(C_3^3\times S_3^2):C_6$

Order 4860: $C_3^3:\GL(2,4)$ x 2

Order 3888: $C_3^3:C_6\times S_4$ x 2

Order 3240: $C_3^2\times A_6$ x 6, $C_9:C_3\times S_5$ x 4, $C_3^3\times S_5$ x 2, $C_9\times A_6$ x 2, $\He_3\times S_5$ x 2

Order 2916: $C_3^4.C_6^2$ x 2, $C_3^5:C_{12}$

Order 2160: $C_3\times S_6$ x 6

Order 1944: $C_3\wr C_3\times S_4$ x 4, $(C_9\times S_3^2):C_6$ x 2, $C_6^3:C_3^2$ x 2, $C_3^5:D_4$, $\He_3\times \SOPlus(4,2)$

Order 1620: $C_9:\GL(2,4)$ x 4, $C_3^3\times A_5$ x 2, $\He_3\times A_5$ x 2, $C_3\wr C_3\times F_5$

Order 1458: $C_3^5:C_6$ x 2, $C_3^5:C_6$

Order 1296: $C_9:C_6\times S_4$ x 4, $C_6^3:S_3$ x 2, $C_2\times S_4\times \He_3$ x 2, $C_6^3:C_6$

Order 1080: $C_3^2\times S_5$ x 12, $C_3\times A_6$ x 6, $C_9\times S_5$ x 4

Order 972: $C_3^3.C_6^2$ x 4, $C_3^3\times S_3^2$ x 2, $\He_3\times S_3^2$ x 2, $C_3^4.C_{12}$ x 2, $C_3^3.C_6^2$ x 2, $C_6^2.C_3^3$ x 2, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 810: $C_3\wr C_3\times D_5$

Order 729: $C_3^5:C_3$

Order 720: $S_6$

Order 648: $C_9:C_3\times S_4$ x 8, $C_3^4:D_4$ x 6, $S_4\times C_3^3$ x 4, $C_9:C_6\times A_4$ x 4, $S_4\times \He_3$ x 4, $(C_3^2\times A_4):C_6$ x 4, $C_3\wr C_3\times D_4$ x 4, $C_6^3:C_3$ x 2, $C_2\times A_4\times \He_3$ x 2, $S_3^2:C_{18}$ x 2, $C_6^3:C_3$ x 2, $C_6\wr C_3$ x 2, $C_2\times \He_3:A_4$ x 2, $C_2\times C_3^3:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 12, $C_9\times A_5$ x 4, $C_{45}:C_{12}$ x 2, $F_5\times C_3^3$, $F_5\times \He_3$

Order 486: $C_3^4:C_6$ x 12, $C_3^4.C_6$ x 4, $S_3\times C_3^4$ x 2, $C_3^4.C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $A_4:C_6^2$ x 12, $C_{18}\times S_4$ x 4, $C_{12}.C_6^2$ x 2, $C_{12}:C_6^2$, $C_{12}.C_6^2$

Order 405: $C_3^3:C_{15}$

Order 360: $C_3\times S_5$ x 12, $A_6$

Order 324: $C_3^2\times S_3^2$ x 12, $C_3^3:C_{12}$ x 6, $C_3^2.C_6^2$ x 5, $C_6^2.C_3^2$ x 4, $C_3^2.C_6^2$ x 4, $C_6^2.C_3^2$ x 4, $C_9\times S_3^2$ x 4, $C_3^3:A_4$ x 2, $\He_3:A_4$ x 2, $C_3^3:C_{12}$ x 2, $D_6\times C_3^3$ x 2, $A_4\times C_3^3$ x 2, $D_6\times \He_3$ x 2, $A_4\times \He_3$ x 2, $C_3^2:C_{36}$ x 2

Order 270: $C_{45}:C_6$ x 2, $D_5\times \He_3$, $D_5\times C_3^3$

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

Order 216: $C_3^2\times S_4$ x 24, $C_6^2:C_6$ x 22, $C_9\times S_4$ x 8, $C_{36}:C_6$ x 8, $S_3^2:C_6$ x 6, $D_4\times \He_3$ x 4, $D_4\times C_3^3$ x 4, $C_6.C_6^2$ x 4, $C_6^2:C_6$ x 4, $C_6^2.C_6$ x 4, $C_{18}:A_4$ x 4, $A_4\times C_{18}$ x 4, $C_{18}:C_{12}$ x 2, $C_6^3$ x 2, $C_2^3\times \He_3$ x 2, $C_6.C_6^2$, $C_3\times C_6\times C_{12}$

Order 180: $\GL(2,4)$ x 12, $C_3^2\times F_5$ x 6, $C_9\times F_5$ x 2

Order 162: $S_3\times C_3^3$ x 26, $C_3^3.C_6$ x 16, $C_3^3:C_6$ x 11, $S_3\times \He_3$ x 8, $C_3^2\wr C_2$ x 6, $C_{18}:C_3^2$ x 4, $C_3^2:C_{18}$ x 4, $C_3^3\times C_6$ x 2, $C_6\times \He_3$ x 2, $C_3^2:C_{18}$ x 2

Order 144: $C_6\times S_4$ x 12, $C_4:C_6^2$ x 6, $D_4\times C_{18}$ x 2

Order 135: $C_9:C_{15}$ x 2, $C_3^2\times C_{15}$, $C_5\times \He_3$

Order 120: $S_5$ x 2

Order 108: $C_3^2\times A_4$ x 22, $C_3^2\times D_6$ x 12, $C_3\times S_3^2$ x 12, $C_{18}:C_6$ x 10, $C_3^2:C_{12}$ x 6, $C_3\times C_6^2$ x 5, $C_2^2\times \He_3$ x 5, $S_3\times C_{18}$ x 4, $C_3^2:A_4$ x 4, $C_3^2.A_4$ x 4, $C_9:A_4$ x 4, $C_9\times A_4$ x 4, $C_9:C_{12}$ x 4, $C_3^2\times C_{12}$ x 2, $C_4\times \He_3$ x 2

Order 90: $C_3^2\times D_5$ x 6, $C_9\times D_5$ x 2

Order 81: $C_3^4$ x 18, $C_3\wr C_3$ x 15, $C_9:C_3^2$ x 12, $C_3\times \He_3$ x 6, $C_3^2\times C_9$ x 2

Order 72: $C_6\times A_4$ x 52, $C_3\times S_4$ x 24, $D_4\times C_3^2$ x 24, $C_2\times C_6^2$ x 12, $D_4\times C_9$ x 8, $C_6\times C_{12}$ x 6, $C_2^2\times C_{18}$ x 4, $C_2^2:C_{18}$ x 4, $C_2\times C_{36}$ x 2, $\SOPlus(4,2)$

Order 60: $C_3\times F_5$ x 6, $A_5$ x 2

Order 54: $S_3\times C_3^2$ x 76, $C_3^2\times C_6$ x 25, $C_9:C_6$ x 14, $S_3\times C_9$ x 12, $C_2\times \He_3$ x 7, $C_3^2:C_6$ x 6, $C_3\times C_{18}$ x 4

Order 48: $C_6\times D_4$ x 6, $C_2\times S_4$ x 2

Order 45: $C_3\times C_{15}$ x 6, $C_{45}$ x 2

Order 36: $C_3\times A_4$ x 52, $C_6^2$ x 30, $C_3\times C_{12}$ x 12, $C_6\times S_3$ x 12, $C_2\times C_{18}$ x 10, $C_2^2:C_9$ x 4, $C_{36}$ x 4, $S_3^2$ x 2, $C_3^2:C_4$

Order 30: $C_3\times D_5$ x 6

Order 27: $C_3^3$ x 95, $C_9:C_3$ x 14, $C_3\times C_9$ x 8, $\He_3$ x 7

Order 24: $C_3\times D_4$ x 24, $C_2\times A_4$ x 14, $C_2^2\times C_6$ x 12, $C_2\times C_{12}$ x 6, $S_4$ x 4

Order 20: $F_5$

Order 18: $C_3\times C_6$ x 70, $C_3\times S_3$ x 38, $C_{18}$ x 10, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 6

Order 12: $C_2\times C_6$ x 30, $A_4$ x 14, $C_{12}$ x 12, $D_6$ x 2

Order 10: $D_5$

Order 9: $C_3^2$ x 99, $C_9$ x 6

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

Order 6: $C_6$ x 32, $S_3$ x 4

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 2

Order 3: $C_3$ x 20

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 58320: $C_3\wr C_3\times S_6$

Order 29160: $C_3\wr C_3\times A_6$

Order 19440: $C_9:C_3\times S_6$, $C_3^3\times S_6$, $\He_3\times S_6$

Order 9720: $C_3\wr C_3\times S_5$, $C_3^3\times A_6$, $\He_3\times A_6$, $C_9:C_3\times A_6$

Order 6480: $C_3^2\times S_6$ x 4, $C_9\times S_6$

Order 5832: $(C_3^3\times S_3^2):C_6$

Order 4860: $C_3^3:\GL(2,4)$

Order 3888: $C_3^3:C_6\times S_4$

Order 3240: $C_3^2\times A_6$ x 4, $C_3^3\times S_5$, $C_9\times A_6$, $C_9:C_3\times S_5$, $\He_3\times S_5$

Order 2916: $C_3^4.C_6^2$, $C_3^5:C_{12}$

Order 2160: $C_3\times S_6$ x 4

Order 1944: $C_3\wr C_3\times S_4$ x 2, $C_3^5:D_4$, $(C_9\times S_3^2):C_6$, $\He_3\times \SOPlus(4,2)$, $C_6^3:C_3^2$

Order 1620: $C_3\wr C_3\times F_5$, $C_3^3\times A_5$, $C_9:\GL(2,4)$, $\He_3\times A_5$

Order 1458: $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_6^3:S_3$, $C_9:C_6\times S_4$, $C_2\times S_4\times \He_3$, $C_6^3:C_6$

Order 1080: $C_3^2\times S_5$ x 4, $C_3\times A_6$ x 4, $C_9\times S_5$

Order 972: $C_3^3\times S_3^2$, $C_3^4:C_{12}$, $C_3^3.C_6^2$, $\He_3\times S_3^2$, $C_3^4.C_{12}$, $C_3^4:C_{12}$, $C_3^3.C_6^2$, $C_6^2.C_3^3$

Order 810: $C_3\wr C_3\times D_5$

Order 729: $C_3^5:C_3$

Order 720: $S_6$

Order 648: $C_3^4:D_4$ x 4, $C_3\wr C_3\times D_4$ x 3, $S_4\times C_3^3$ x 2, $C_9:C_3\times S_4$ x 2, $S_4\times \He_3$ x 2, $C_6^3:C_3$, $C_9:C_6\times A_4$, $C_2\times A_4\times \He_3$, $S_3^2:C_{18}$, $C_6^3:C_3$, $C_6\wr C_3$, $(C_3^2\times A_4):C_6$, $C_2\times \He_3:A_4$, $C_2\times C_3^3:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 4, $F_5\times C_3^3$, $C_{45}:C_{12}$, $F_5\times \He_3$, $C_9\times A_5$

Order 486: $C_3^4:C_6$ x 5, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $A_4:C_6^2$ x 4, $C_{12}:C_6^2$, $C_{18}\times S_4$, $C_{12}.C_6^2$, $C_{12}.C_6^2$

Order 405: $C_3^3:C_{15}$

Order 360: $C_3\times S_5$ x 4, $A_6$

Order 324: $C_3^2\times S_3^2$ x 4, $C_3^3:C_{12}$ x 4, $C_3^2.C_6^2$ x 3, $C_3^3:C_{12}$ x 2, $C_3^3:A_4$, $C_6^2.C_3^2$, $\He_3:A_4$, $D_6\times C_3^3$, $A_4\times C_3^3$, $C_3^2.C_6^2$, $D_6\times \He_3$, $C_6^2.C_3^2$, $A_4\times \He_3$, $C_9\times S_3^2$, $C_3^2:C_{36}$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $D_5\times C_3^3$

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

Order 216: $C_3^2\times S_4$ x 8, $C_6^2:C_6$ x 7, $S_3^2:C_6$ x 4, $C_{36}:C_6$ x 3, $D_4\times \He_3$ x 3, $D_4\times C_3^3$ x 3, $C_9\times S_4$ x 2, $C_6^2:C_6$ x 2, $C_{18}:C_{12}$, $C_6.C_6^2$, $C_6^3$, $C_3\times C_6\times C_{12}$, $C_6.C_6^2$, $C_2^3\times \He_3$, $C_6^2.C_6$, $C_{18}:A_4$, $A_4\times C_{18}$

Order 180: $C_3^2\times F_5$ x 4, $\GL(2,4)$ x 4, $C_9\times F_5$

Order 162: $S_3\times C_3^3$ x 9, $C_3^3:C_6$ x 5, $C_3^2\wr C_2$ x 4, $C_3^3.C_6$ x 4, $S_3\times \He_3$ x 4, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^2:C_{18}$, $C_3^2:C_{18}$

Order 144: $C_6\times S_4$ x 4, $C_4:C_6^2$ x 4, $D_4\times C_{18}$

Order 135: $C_3^2\times C_{15}$, $C_9:C_{15}$, $C_5\times \He_3$

Order 120: $S_5$

Order 108: $C_3^2\times A_4$ x 7, $C_3^2\times D_6$ x 4, $C_3\times S_3^2$ x 4, $C_3^2:C_{12}$ x 4, $C_3\times C_6^2$ x 3, $C_{18}:C_6$ x 3, $C_2^2\times \He_3$ x 3, $C_3^2\times C_{12}$ x 2, $C_3^2:A_4$ x 2, $C_9:C_{12}$ x 2, $C_4\times \He_3$ x 2, $S_3\times C_{18}$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$

Order 90: $C_3^2\times D_5$ x 4, $C_9\times D_5$

Order 81: $C_3^4$ x 8, $C_3\wr C_3$ x 6, $C_9:C_3^2$ x 3, $C_3\times \He_3$ x 3, $C_3^2\times C_9$

Order 72: $C_6\times A_4$ x 13, $D_4\times C_3^2$ x 12, $C_3\times S_4$ x 8, $C_2\times C_6^2$ x 4, $C_6\times C_{12}$ x 4, $D_4\times C_9$ x 3, $C_2\times C_{36}$, $\SOPlus(4,2)$, $C_2^2\times C_{18}$, $C_2^2:C_{18}$

Order 60: $C_3\times F_5$ x 4, $A_5$

Order 54: $S_3\times C_3^2$ x 21, $C_3^2\times C_6$ x 9, $C_3^2:C_6$ x 4, $C_9:C_6$ x 4, $C_2\times \He_3$ x 4, $S_3\times C_9$ x 3, $C_3\times C_{18}$

Order 48: $C_6\times D_4$ x 4, $C_2\times S_4$

Order 45: $C_3\times C_{15}$ x 4, $C_{45}$

Order 36: $C_3\times A_4$ x 13, $C_6^2$ x 12, $C_3\times C_{12}$ x 8, $C_6\times S_3$ x 4, $C_2\times C_{18}$ x 3, $C_{36}$ x 2, $C_3^2:C_4$, $C_2^2:C_9$, $S_3^2$

Order 30: $C_3\times D_5$ x 4

Order 27: $C_3^3$ x 25, $C_9:C_3$ x 4, $\He_3$ x 4, $C_3\times C_9$ x 2

Order 24: $C_3\times D_4$ x 12, $C_2\times A_4$ x 5, $C_2\times C_{12}$ x 4, $C_2^2\times C_6$ x 4, $S_4$ x 2

Order 20: $F_5$

Order 18: $C_3\times C_6$ x 21, $C_3\times S_3$ x 13, $C_{18}$ x 3, $C_3:S_3$

Order 16: $C_2\times D_4$

Order 15: $C_{15}$ x 4

Order 12: $C_2\times C_6$ x 12, $C_{12}$ x 8, $A_4$ x 5, $D_6$

Order 10: $D_5$

Order 9: $C_3^2$ x 27, $C_9$ x 2

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

Order 6: $C_6$ x 13, $S_3$ x 2

Order 5: $C_5$

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 9

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 58320: $C_3\wr C_3\times S_6$ ($C_1$)

Order 29160: $C_3\wr C_3\times A_6$ ($C_2$)

Order 19440: $C_9:C_3\times S_6$ ($C_3$) x 2, $C_3^3\times S_6$ ($C_3$), $\He_3\times S_6$ ($C_3$)

Order 9720: $C_9:C_3\times A_6$ ($C_6$) x 2, $C_3^3\times A_6$ ($C_6$), $\He_3\times A_6$ ($C_6$)

Order 6480: $C_3^2\times S_6$ ($C_3^2$)

Order 3240: $C_3^2\times A_6$ ($C_3\times C_6$)

Order 2160: $C_3\times S_6$ ($\He_3$)

Order 1080: $C_3\times A_6$ ($C_2\times \He_3$)

Order 720: $S_6$ ($C_3\wr C_3$)

Order 360: $A_6$ ($C_3^3:C_6$)

Order 81: $C_3\wr C_3$ ($S_6$)

Order 27: $C_9:C_3$ ($C_3\times S_6$) x 2, $C_3^3$ ($C_3\times S_6$), $\He_3$ ($C_3\times S_6$)

Order 9: $C_3^2$ ($C_3^2\times S_6$)

Order 3: $C_3$ ($\He_3\times S_6$)

Order 1: $C_1$ ($C_3\wr C_3\times S_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 58320: $C_3\wr C_3\times S_6$ ($C_1$)

Order 29160: $C_3\wr C_3\times A_6$ ($C_2$)

Order 19440: $C_9:C_3\times S_6$ ($C_3$), $C_3^3\times S_6$ ($C_3$), $\He_3\times S_6$ ($C_3$)

Order 9720: $C_3^3\times A_6$ ($C_6$), $\He_3\times A_6$ ($C_6$), $C_9:C_3\times A_6$ ($C_6$)

Order 6480: $C_3^2\times S_6$ ($C_3^2$)

Order 3240: $C_3^2\times A_6$ ($C_3\times C_6$)

Order 2160: $C_3\times S_6$ ($\He_3$)

Order 1080: $C_3\times A_6$ ($C_2\times \He_3$)

Order 720: $S_6$ ($C_3\wr C_3$)

Order 360: $A_6$ ($C_3^3:C_6$)

Order 81: $C_3\wr C_3$ ($S_6$)

Order 27: $C_3^3$ ($C_3\times S_6$), $C_9:C_3$ ($C_3\times S_6$), $\He_3$ ($C_3\times S_6$)

Order 9: $C_3^2$ ($C_3^2\times S_6$)

Order 3: $C_3$ ($\He_3\times S_6$)

Order 1: $C_1$ ($C_3\wr C_3\times S_6$)

Series

Derived series

$C_3\wr C_3\times S_6$

$\rhd$

$C_3\wr C_3\times S_6$

$\rhd$

$C_3^2\times A_6$

$\rhd$

$C_3^2\times A_6$

$\rhd$

$A_6$

$\rhd$

$A_6$

Chief series

$C_3\wr C_3\times S_6$

$\rhd$

$C_3\wr C_3\times S_6$

$\rhd$

$C_3\wr C_3\times A_6$

$\rhd$

$C_3\wr C_3\times A_6$

$\rhd$

$C_3\wr C_3$

$\rhd$

$C_3\wr C_3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3\wr C_3\times S_6$

$\rhd$

$C_3\wr C_3\times S_6$

$\rhd$

$C_3^2\times A_6$

$\rhd$

$C_3^2\times A_6$

$\rhd$

$C_3\times A_6$

$\rhd$

$C_3\times A_6$

$\rhd$

$A_6$

$\rhd$

$A_6$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

$\lhd$

$C_3^2$

$\lhd$

$C_3^2$

$\lhd$

$C_3\wr C_3$

$\lhd$

$C_3\wr C_3$

Supergroups

This group is a maximal subgroup of 4 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 $187 \times 187$ 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 $99 \times 99$ rational character table.