Abstract group 38880.bf: $C_3^2:S_3\times S_6$

• Label: 38880.bf
• Order: \( 2^{5} \cdot 3^{5} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{5} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$C_3^2:S_3\times S_6$
Order:	\(38880\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$\AGL(2,3).A_6.C_2^2$, of order \(622080\)\(\medspace = 2^{9} \cdot 3^{5} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $A_6$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	30
Elements	1	759	2186	1800	144	18438	1296	7920	3744	2592	38880
Conjugacy classes	1	7	20	4	1	52	1	16	6	2	110
Divisions	1	7	17	4	1	39	1	12	5	1	88
Autjugacy classes	1	5	5	4	1	14	1	6	2	1	40

Dimension	1	2	3	5	6	9	10	15	16	18	20	27	30	32	48	54	60	96
Irr. complex chars.	4	8	8	8	0	4	20	16	2	8	8	8	8	4	4	0	0	0	110
Irr. rational chars.	4	8	0	8	4	4	20	0	2	8	8	0	8	4	0	4	4	2	88

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$34169688$

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 $15$ $\langle(7,9,8)(10,11,12), (10,12,11)(13,14,15), (7,14,10)(8,15,11)(9,13,12), (5,6) \!\cdots\! \rangle$
Direct product:	$(C_3^2:S_3)$ $\, \times\, $ $S_6$
Semidirect product:	$(C_3^2\times S_6)$ $\,\rtimes\,$ $S_3$	$C_3^2$ $\,\rtimes\,$ $(S_3\times S_6)$	$(\He_3\times S_6)$ $\,\rtimes\,$ $C_2$	$(\He_3:S_6)$ $\,\rtimes\,$ $C_2$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3$ . $(C_3:S_3\times S_6)$	$(C_3\times S_6)$ . $(C_3:S_3)$	$(C_3\times A_6)$ . $(C_6:S_3)$		more information

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

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 350735 subgroups in 2672 conjugacy classes, 25 normal (13 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3:S_3\times S_6$
Commutator:	$G' \simeq$ $\He_3\times A_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3:S_3\times S_6$
Fitting:	$\operatorname{Fit} \simeq$ $\He_3$	$G/\operatorname{Fit} \simeq$ $C_2\times S_6$
Radical:	$R \simeq$ $C_3^2:S_3$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_3\times A_6$	$G/\operatorname{soc} \simeq$ $C_6:S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\times \He_3$
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 38880: $C_3^2:S_3\times S_6$

Order 19440: $\He_3\times S_6$, $\He_3:S_6$, $C_3^2:S_3\times A_6$

Order 12960: $C_3\times S_3\times S_6$ x 4

Order 9720: $\He_3\times A_6$

Order 6480: $C_3^2\times S_6$ x 4, $C_3^2:S_6$ x 4, $C_3\times S_3\times A_6$ x 4, $C_3^2:S_3\times S_5$ x 2

Order 4320: $S_3\times S_6$ x 4, $C_6\times S_6$

Order 3888: $C_3^4:(S_3\times D_4)$

Order 3240: $C_3^2\times A_6$ x 4, $\He_3\times S_5$ x 2, $C_3^2:S_3\times A_5$ x 2, $\He_3:S_5$ x 2

Order 2592: $C_3^2:D_6\times S_4$ x 2

Order 2160: $\GL(2,4):D_6$ x 8, $C_3\times S_6$ x 6, $C_3:S_6$ x 4, $S_3\times A_6$ x 4, $C_6\times A_6$

Order 1944: $C_3^2:S_3^3$ x 2, $\He_3:\SOPlus(4,2)$ x 2, $C_3^4:D_{12}$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$

Order 1620: $\He_3\times A_5$ x 2

Order 1440: $C_2\times S_6$

Order 1296: $C_6^2:S_3^2$ x 8, $S_3^3:C_6$ x 4, $C_3^2:D_6\times A_4$ x 2, $C_2\times \He_3:S_4$ x 2, $C_2\times S_4\times \He_3$ x 2

Order 1080: $S_3\times \GL(2,4)$ x 8, $C_3^2\times S_5$ x 8, $C_3^2:S_5$ x 8, $C_3\times A_6$ x 5, $C_3^2:S_3\times F_5$

Order 972: $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $\He_3\times S_3^2$ x 2, $C_3^3:S_3^2$, $(C_3^2\times \He_3):C_4$, $C_3^4:C_{12}$

Order 864: $C_6\times S_3\times S_4$ x 8, $C_2\times C_6^2:D_6$

Order 720: $S_3\times S_5$ x 8, $C_6\times S_5$ x 2, $S_6$ x 2, $C_2\times A_6$

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

Order 540: $C_3\times \GL(2,4)$ x 8, $\He_3:D_{10}$, $\He_3:F_5$, $F_5\times \He_3$

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

Order 432: $C_6^2:D_6$ x 32, $C_6\times S_3\times A_4$ x 8, $C_6^2:D_6$ x 8, $A_4:C_6^2$ x 8, $C_6^2:D_6$ x 8, $C_6^2:D_6$ x 8, $S_3^3:C_2$ x 4, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_6\times \SOPlus(4,2)$, $C_{12}.C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$

Order 360: $C_3\times S_5$ x 12, $S_3\times A_5$ x 8, $C_3:S_5$ x 8, $D_{15}:C_{12}$ x 4, $C_6\times A_5$ x 2, $A_6$

Order 324: $C_3^2\times S_3^2$ x 16, $C_3^2:S_3^2$ x 16, $C_3^2:S_3^2$ x 12, $C_3\wr C_2^2$ x 8, $C_3\wr C_4$ x 4, $C_3^3:C_{12}$ x 4, $C_3^3:D_6$ x 2, $C_3^3:D_6$ x 2, $D_6\times \He_3$ x 2, $A_4\times \He_3$ x 2

Order 288: $D_6\times S_4$ x 8, $C_6^2:C_2^3$ x 4, $C_2\times C_6\times S_4$ x 2

Order 270: $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 243: $C_3^2\times \He_3$

Order 240: $C_2\times S_5$ x 2

Order 216: $C_3^2:S_4$ x 16, $C_6^2:C_6$ x 16, $C_3^2:S_4$ x 16, $C_3^2\times S_4$ x 16, $C_6^2:S_3$ x 11, $C_6\times S_3^2$ x 10, $C_6^2:S_3$ x 8, $C_6^2:C_6$ x 8, $S_3^3$ x 8, $S_3^2:S_3$ x 8, $S_3^2:C_6$ x 8, $C_6^2:C_6$ x 8, $D_4\times \He_3$ x 4, $C_3^2:D_{12}$ x 4, $C_4\times C_3^2:S_3$ x 4, $C_3^2:D_{12}$ x 4, $C_3^2:C_4\times S_3$ x 4, $C_2^3\times \He_3$ x 2, $C_6.C_6^2$, $C_6^2.S_3$, $C_2\times C_3^2:C_{12}$

Order 180: $\GL(2,4)$ x 10, $C_{15}:D_6$ x 4, $C_{15}:C_{12}$ x 4, $C_3^2\times F_5$ x 4

Order 162: $C_3^2\wr C_2$ x 12, $S_3\times C_3^3$ x 12, $C_3^3:S_3$ x 12, $C_3^3:S_3$ x 12, $S_3\times \He_3$ x 12, $C_3^3:C_6$ x 4, $C_6\times \He_3$ x 2

Order 144: $S_3\times S_4$ x 32, $C_{12}:D_6$ x 32, $C_6\times S_4$ x 28, $A_4\times D_6$ x 16, $C_6^2:C_2^2$ x 8, $C_6:S_4$ x 8, $C_6^2:C_2^2$ x 8, $C_4:C_6^2$ x 4, $C_6\times D_{12}$ x 4, $C_{12}\times D_6$ x 4, $C_2^2:C_6^2$ x 2, $S_3^2:C_2^2$

Order 135: $C_5\times \He_3$

Order 120: $S_3\times F_5$ x 4, $S_5$ x 4, $C_2\times A_5$ x 2, $C_6\times F_5$

Order 108: $C_3\times S_3^2$ x 70, $C_3^2:D_6$ x 20, $C_3^2\times D_6$ x 18, $C_3:S_3^2$ x 12, $C_3^2:D_6$ x 9, $C_3^2\times A_4$ x 8, $C_3:S_3^2$ x 8, $C_3^2:A_4$ x 8, $C_3^2:C_{12}$ x 6, $C_2^2\times \He_3$ x 5, $C_3^3:C_4$ x 4, $C_4\times \He_3$ x 2, $\He_3:C_4$ x 2

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

Order 90: $C_3\times D_{15}$ x 4, $S_3\times C_{15}$ x 4, $C_3^2\times D_5$ x 4

Order 81: $C_3\times \He_3$ x 10, $C_3^4$ x 4

Order 72: $C_6\times D_6$ x 46, $C_3\times S_4$ x 40, $S_3\times A_4$ x 32, $C_6\wr C_2$ x 32, $C_6\times A_4$ x 30, $C_3:S_4$ x 16, $D_4\times C_3^2$ x 16, $C_3\times D_{12}$ x 16, $S_3\times C_{12}$ x 16, $S_3\times D_6$ x 10, $C_2\times C_6^2$ x 8, $\SOPlus(4,2)$ x 4, $C_6\times C_{12}$ x 4, $C_6:C_{12}$ x 4, $C_2\times C_3^2:C_4$

Order 60: $C_3\times F_5$ x 6, $S_3\times D_5$ x 4, $C_{15}:C_4$ x 4, $A_5$ x 2, $C_3\times D_{10}$

Order 54: $S_3\times C_3^2$ x 80, $C_3^2:C_6$ x 46, $C_3^2:S_3$ x 26, $C_2\times \He_3$ x 11, $C_3^2\times C_6$ x 9, $C_3^2:S_3$ x 4

Order 48: $S_3\times D_4$ x 32, $C_2\times S_4$ x 20, $C_6\times D_4$ x 16, $C_2^2\times D_6$ x 8, $C_6:D_4$ x 8, $C_2^2\times A_4$ x 4, $C_2\times D_{12}$ x 4, $C_4\times D_6$ x 4, $C_2^3\times C_6$ x 2, $C_2^2\times C_{12}$

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

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 96, $S_3^2$ x 46, $C_3\times A_4$ x 26, $C_6^2$ x 22, $C_6:S_3$ x 9, $C_3\times C_{12}$ x 8, $C_3:C_{12}$ x 8, $C_3^2:C_4$ x 2

Order 32: $C_2^2\times D_4$

Order 30: $C_3\times D_5$ x 6, $D_{15}$ x 4, $C_5\times S_3$ x 4, $C_{30}$

Order 27: $C_3^3$ x 29, $\He_3$ x 15

Order 24: $C_2\times D_6$ x 46, $C_3:D_4$ x 32, $C_3\times D_4$ x 32, $S_4$ x 24, $C_2^2\times C_6$ x 21, $C_2\times A_4$ x 20, $D_{12}$ x 16, $C_4\times S_3$ x 16, $C_2\times C_{12}$ x 10, $C_6:C_4$ x 4

Order 20: $F_5$ x 2, $D_{10}$

Order 18: $C_3\times S_3$ x 120, $C_3\times C_6$ x 45, $C_3:S_3$ x 26

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

Order 15: $C_{15}$ x 5

Order 12: $D_6$ x 68, $C_2\times C_6$ x 41, $A_4$ x 12, $C_{12}$ x 12, $C_3:C_4$ x 8

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 49

Order 8: $D_4$ x 16, $C_2^3$ x 13, $C_2\times C_4$ x 6

Order 6: $S_3$ x 40, $C_6$ x 39

Order 5: $C_5$

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

Order 3: $C_3$ x 17

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 38880: $C_3^2:S_3\times S_6$

Order 19440: $\He_3\times S_6$, $\He_3:S_6$, $C_3^2:S_3\times A_6$

Order 12960: $C_3\times S_3\times S_6$

Order 9720: $\He_3\times A_6$

Order 6480: $C_3^2\times S_6$, $C_3^2:S_3\times S_5$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

Order 4320: $C_6\times S_6$, $S_3\times S_6$

Order 3888: $C_3^4:(S_3\times D_4)$

Order 3240: $C_3^2\times A_6$, $\He_3\times S_5$, $C_3^2:S_3\times A_5$, $\He_3:S_5$

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

Order 2160: $C_3\times S_6$ x 3, $C_6\times A_6$, $\GL(2,4):D_6$, $C_3:S_6$, $S_3\times A_6$

Order 1944: $C_3^2:S_3^3$, $C_3^4:D_{12}$, $\He_3:\SOPlus(4,2)$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$

Order 1620: $\He_3\times A_5$

Order 1440: $C_2\times S_6$

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

Order 1080: $C_3\times A_6$ x 2, $S_3\times \GL(2,4)$, $C_3^2\times S_5$, $C_3^2:S_5$, $C_3^2:S_3\times F_5$

Order 972: $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $\He_3\times S_3^2$, $(C_3^2\times \He_3):C_4$, $C_3^4:C_{12}$

Order 864: $C_6\times S_3\times S_4$, $C_2\times C_6^2:D_6$

Order 720: $S_6$ x 2, $C_6\times S_5$, $S_3\times S_5$, $C_2\times A_6$

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

Order 540: $C_3\times \GL(2,4)$, $\He_3:D_{10}$, $\He_3:F_5$, $F_5\times \He_3$

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

Order 432: $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 4, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}.C_6^2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$

Order 360: $C_3\times S_5$ x 3, $D_{15}:C_{12}$, $C_6\times A_5$, $S_3\times A_5$, $C_3:S_5$, $A_6$

Order 324: $C_3^2:S_3^2$ x 5, $C_3^2:S_3^2$ x 2, $C_3^2\times S_3^2$ x 2, $C_3\wr C_2^2$, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^3:D_6$, $D_6\times \He_3$, $A_4\times \He_3$

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

Order 270: $D_5\times \He_3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 243: $C_3^2\times \He_3$

Order 240: $C_2\times S_5$

Order 216: $C_6^2:S_3$ x 6, $C_6^2:S_3$ x 4, $C_4\times C_3^2:S_3$ x 4, $S_3^2:C_6$ x 4, $D_4\times \He_3$ x 3, $C_3^2:D_{12}$ x 3, $C_3^2:S_4$ x 2, $C_6\times S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times S_4$ x 2, $C_6.C_6^2$, $C_6^2.S_3$, $C_6^2:C_6$, $C_2\times C_3^2:C_{12}$, $S_3^3$, $C_3^2:D_{12}$, $S_3^2:S_3$, $C_3^2:C_4\times S_3$, $C_2^3\times \He_3$, $C_6^2:C_6$

Order 180: $\GL(2,4)$ x 2, $C_{15}:D_6$, $C_{15}:C_{12}$, $C_3^2\times F_5$

Order 162: $C_3^3:S_3$ x 3, $C_3^3:S_3$ x 3, $S_3\times \He_3$ x 3, $C_3^2\wr C_2$ x 2, $S_3\times C_3^3$ x 2, $C_3^3:C_6$, $C_6\times \He_3$

Order 144: $C_6\times S_4$ x 8, $C_{12}:D_6$ x 6, $S_3\times S_4$ x 4, $A_4\times D_6$ x 2, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $C_6:S_4$, $S_3^2:C_2^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_{12}\times D_6$

Order 135: $C_5\times \He_3$

Order 120: $S_5$ x 2, $C_6\times F_5$, $S_3\times F_5$, $C_2\times A_5$

Order 108: $C_3\times S_3^2$ x 11, $C_3^2:D_6$ x 8, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_2^2\times \He_3$ x 3, $C_3^2:D_6$ x 2, $C_3:S_3^2$ x 2, $C_4\times \He_3$ x 2, $\He_3:C_4$ x 2, $C_3^2\times A_4$, $C_3:S_3^2$, $C_3^3:C_4$, $C_3^2:A_4$

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

Order 90: $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3^2\times D_5$

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

Order 72: $C_3\times S_4$ x 8, $C_6\times D_6$ x 7, $C_6\times A_4$ x 6, $S_3\times A_4$ x 4, $C_6\wr C_2$ x 4, $S_3\times C_{12}$ x 4, $\SOPlus(4,2)$ x 3, $D_4\times C_3^2$ x 3, $C_3\times D_{12}$ x 3, $S_3\times D_6$ x 2, $C_3:S_4$ x 2, $C_2\times C_6^2$, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 60: $C_3\times F_5$ x 3, $S_3\times D_5$, $C_{15}:C_4$, $A_5$, $C_3\times D_{10}$

Order 54: $S_3\times C_3^2$ x 12, $C_3^2:C_6$ x 8, $C_3^2:S_3$ x 6, $C_2\times \He_3$ x 3, $C_3^2\times C_6$ x 2, $C_3^2:S_3$

Order 48: $C_6\times D_4$ x 10, $C_2\times S_4$ x 7, $S_3\times D_4$ x 6, $C_2^2\times A_4$ x 2, $C_2^3\times C_6$, $C_2^2\times D_6$, $C_2^2\times C_{12}$, $C_6:D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 45: $C_3\times C_{15}$

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 19, $S_3^2$ x 8, $C_6^2$ x 4, $C_3\times A_4$ x 4, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6:S_3$ x 2

Order 32: $C_2^2\times D_4$

Order 30: $C_3\times D_5$ x 3, $C_{30}$, $D_{15}$, $C_5\times S_3$

Order 27: $C_3^3$ x 5, $\He_3$ x 3

Order 24: $C_3\times D_4$ x 13, $C_2^2\times C_6$ x 8, $C_2\times C_{12}$ x 7, $C_2\times D_6$ x 7, $C_2\times A_4$ x 7, $S_4$ x 6, $C_3:D_4$ x 4, $C_4\times S_3$ x 4, $D_{12}$ x 3, $C_6:C_4$

Order 20: $F_5$ x 2, $D_{10}$

Order 18: $C_3\times S_3$ x 22, $C_3\times C_6$ x 10, $C_3:S_3$ x 5

Order 16: $C_2\times D_4$ x 9, $C_2^4$, $C_2^2\times C_4$

Order 15: $C_{15}$ x 2

Order 12: $C_2\times C_6$ x 15, $D_6$ x 14, $C_{12}$ x 6, $A_4$ x 3, $C_3:C_4$ x 2

Order 10: $D_5$ x 2, $C_{10}$

Order 9: $C_3^2$ x 9

Order 8: $D_4$ x 10, $C_2^3$ x 7, $C_2\times C_4$ x 6

Order 6: $C_6$ x 14, $S_3$ x 9

Order 5: $C_5$

Order 4: $C_2^2$ x 10, $C_4$ x 4

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 38880: $C_3^2:S_3\times S_6$ ($C_1$)

Order 19440: $\He_3\times S_6$ ($C_2$), $\He_3:S_6$ ($C_2$), $C_3^2:S_3\times A_6$ ($C_2$)

Order 9720: $\He_3\times A_6$ ($C_2^2$)

Order 6480: $C_3^2\times S_6$ ($S_3$) x 4

Order 3240: $C_3^2\times A_6$ ($D_6$) x 4

Order 2160: $C_3\times S_6$ ($C_3:S_3$)

Order 1080: $C_3\times A_6$ ($C_6:S_3$)

Order 720: $S_6$ ($C_3^2:S_3$)

Order 360: $A_6$ ($C_3^2:D_6$)

Order 54: $C_3^2:S_3$ ($S_6$)

Order 27: $\He_3$ ($C_2\times S_6$)

Order 9: $C_3^2$ ($S_3\times S_6$) x 4

Order 3: $C_3$ ($C_3:S_3\times S_6$)

Order 1: $C_1$ ($C_3^2:S_3\times S_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 38880: $C_3^2:S_3\times S_6$ ($C_1$)

Order 19440: $\He_3\times S_6$ ($C_2$), $\He_3:S_6$ ($C_2$), $C_3^2:S_3\times A_6$ ($C_2$)

Order 9720: $\He_3\times A_6$ ($C_2^2$)

Order 6480: $C_3^2\times S_6$ ($S_3$)

Order 3240: $C_3^2\times A_6$ ($D_6$)

Order 2160: $C_3\times S_6$ ($C_3:S_3$)

Order 1080: $C_3\times A_6$ ($C_6:S_3$)

Order 720: $S_6$ ($C_3^2:S_3$)

Order 360: $A_6$ ($C_3^2:D_6$)

Order 54: $C_3^2:S_3$ ($S_6$)

Order 27: $\He_3$ ($C_2\times S_6$)

Order 9: $C_3^2$ ($S_3\times S_6$)

Order 3: $C_3$ ($C_3:S_3\times S_6$)

Order 1: $C_1$ ($C_3^2:S_3\times S_6$)

Series

Derived series

$C_3^2:S_3\times S_6$

$\rhd$

$C_3^2:S_3\times S_6$

$\rhd$

$\He_3\times A_6$

$\rhd$

$\He_3\times A_6$

$\rhd$

$C_3\times A_6$

$\rhd$

$C_3\times A_6$

$\rhd$

$A_6$

$\rhd$

$A_6$

Chief series

$C_3^2:S_3\times S_6$

$\rhd$

$C_3^2:S_3\times S_6$

$\rhd$

$C_3^2:S_3\times A_6$

$\rhd$

$C_3^2:S_3\times A_6$

$\rhd$

$C_3^2:S_3$

$\rhd$

$C_3^2:S_3$

$\rhd$

$\He_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^2:S_3\times S_6$

$\rhd$

$C_3^2:S_3\times S_6$

$\rhd$

$\He_3\times A_6$

$\rhd$

$\He_3\times A_6$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

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 $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $88 \times 88$ rational character table.