Abstract group 38880.s: $C_3^3:C_4\times A_6$

• Label: 38880.s
• Order: \( 2^{5} \cdot 3^{5} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3 \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{5} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$C_3^3:C_4\times A_6$
Order:	\(38880\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \)
Exponent:	\(60\)\(\medspace = 2^{2} \cdot 3 \cdot 5 \)
Automorphism group:	$(S_3\times F_9).C_2.A_6.C_2^2$, of order \(1244160\)\(\medspace = 2^{10} \cdot 3^{5} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 3, $A_6$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	10	12	15	20	30
Elements	1	459	2186	8244	144	4158	1296	8280	3744	7776	2592	38880
Conjugacy classes	1	3	23	8	2	13	2	12	14	4	2	84
Divisions	1	3	17	5	1	11	1	8	5	1	1	54
Autjugacy classes	1	3	7	5	1	7	1	5	3	1	1	35

Dimension	1	2	4	5	8	9	10	16	18	20	32	36	40	64	72	80	128
Irr. complex chars.	4	2	6	8	8	4	8	4	2	14	12	6	6	0	0	0	0	84
Irr. rational chars.	2	3	2	4	2	2	8	2	3	7	3	2	6	2	2	2	2	54

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	$318$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	40	40
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $15$ $\langle(10,11,12)(13,14,15), (10,12,11), (7,8)(10,14,11,13)(12,15), (7,9,8)(10,11,12) \!\cdots\! \rangle$
Direct product:	$(C_3^3:C_4)$ $\, \times\, $ $A_6$
Semidirect product:	$(C_3^3\times A_6)$ $\,\rtimes\,$ $C_4$	$C_3^3$ $\,\rtimes\,$ $(C_4\times A_6)$	$C_3^2$ $\,\rtimes\,$ $(C_3:C_4\times A_6)$	$C_3$ $\,\rtimes\,$ $(C_3^2:C_4\times A_6)$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3:S_3\times A_6)$ . $S_3$	$(C_3:S_3)$ . $(S_3\times A_6)$	$(C_3^2:C_6\times A_6)$ . $C_2$	$(C_3^2:C_6)$ . $(C_2\times A_6)$	more information

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

Homology

Abelianization:	$C_{4} $
Schur multiplier:	$C_{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 201036 subgroups in 1060 conjugacy classes, 14 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:C_4\times A_6$
Commutator:	$G' \simeq$ $C_3^3\times A_6$	$G/G' \simeq$ $C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^3:C_4\times A_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3$	$G/\operatorname{Fit} \simeq$ $C_4\times A_6$
Radical:	$R \simeq$ $C_3^3:C_4$	$G/R \simeq$ $A_6$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times A_6$	$G/\operatorname{soc} \simeq$ $C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5$
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: $C_3^3:C_4\times A_6$

Order 19440: $C_3^2:C_6\times A_6$

Order 12960: $C_3^2:C_4\times A_6$

Order 9720: $C_3^3\times A_6$

Order 6480: $A_5\times C_3^3:C_4$ x 2, $C_3\times S_3\times A_6$ x 2, $C_3:S_3\times A_6$

Order 4320: $C_3:C_4\times A_6$

Order 3888: $C_3^5:C_4^2$

Order 3240: $C_3^2\times A_6$ x 5, $C_3:S_3\times \GL(2,4)$ x 2

Order 2592: $C_3^3:C_4\times S_4$ x 2

Order 2160: $C_3^2:C_4\times A_5$ x 2, $S_3\times A_6$ x 2, $C_6\times A_6$

Order 1944: $C_3^5:(C_2\times C_4)$, $C_3\wr C_4:S_3$, $C_3^3:(S_3\times C_{12})$

Order 1620: $C_3^3\times A_5$ x 2

Order 1440: $C_4\times A_6$

Order 1296: $C_6^2:S_3^2$ x 2, $A_4\times C_3^3:C_4$ x 2, $(A_4\times C_3^3):C_4$ x 2, $C_3^4:C_4^2$

Order 1080: $C_3\times A_6$ x 5, $S_3\times \GL(2,4)$ x 4, $\GL(2,4):S_3$ x 2, $C_3^3:C_4\times D_5$

Order 972: $C_3^5:C_4$ x 2, $C_3^3:S_3^2$, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 864: $C_3^2:C_4\times S_4$ x 2, $C_3^3:C_4\times D_4$

Order 720: $\GL(2,4):C_4$ x 2, $C_2\times A_6$

Order 648: $C_3\times C_6^2:C_6$ x 2, $C_3^3:S_4$ x 2, $S_4\times C_3^3$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:C_4\times S_3$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3^4:(C_2\times C_4)$

Order 540: $C_3\times \GL(2,4)$ x 10, $(C_3^2\times C_{15}):C_4$, $C_3^3:C_{20}$, $C_3^3:D_{10}$

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

Order 432: $C_6^2:D_6$ x 4, $(C_3^2\times A_4):C_4$ x 4, $C_2^2\times C_3^3:C_4$ x 2, $C_6^2:D_6$ x 2, $C_6^2:C_{12}$ x 2, $(C_3\times C_6^2):C_4$ x 2, $C_6^2:D_6$, $(C_3^2\times C_{12}):C_4$, $C_3^3:C_4^2$, $C_3^3:C_4^2$

Order 360: $S_3\times A_5$ x 4, $C_6\times A_5$ x 2, $C_3^2:C_4\times D_5$, $A_6$

Order 324: $C_3^4:C_4$ x 6, $C_3^3:C_{12}$ x 6, $C_3^2:S_3^2$ x 4, $C_3\wr C_4$ x 4, $A_4\times C_3^3$ x 2, $C_3^4:C_4$ x 2, $C_3^2:S_3^2$

Order 288: $C_2^3.S_3^2$ x 2, $C_3^2:C_4\times D_4$

Order 270: $C_3^2:D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

Order 243: $C_3^5$

Order 240: $C_4\times A_5$ x 2

Order 216: $C_3^2\times S_4$ x 10, $C_3^2:S_4$ x 8, $C_2\times C_3^3:C_4$ x 5, $C_6^2:C_6$ x 4, $C_6^2:C_6$ x 4, $C_3^2:C_4\times S_3$ x 4, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_6^2:C_6$ x 2, $C_2\times C_3^2:C_{12}$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_6.S_3^2$

Order 180: $\GL(2,4)$ x 10, $C_{15}:D_6$ x 2, $C_{15}:D_6$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^2\wr C_2$ x 10, $C_3^3:C_6$ x 8, $S_3\times C_3^3$ x 4, $C_3^3:S_3$

Order 144: $S_3\times S_4$ x 4, $C_6^2:C_4$ x 2, $C_6\times S_4$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $C_3^2:C_4^2$ x 2, $A_4\times C_3:C_4$ x 2, $C_6.S_4$ x 2, $C_{12}:D_6$, $(C_3\times C_{12}):C_4$

Order 135: $C_3^2\times C_{15}$

Order 120: $C_2\times A_5$ x 2, $C_6.D_{10}$

Order 108: $C_3^2\times A_4$ x 20, $C_3^3:C_4$ x 15, $C_3^2:C_{12}$ x 8, $C_3^2:D_6$ x 4, $C_3:S_3^2$ x 4, $C_3\times S_3^2$ x 4, $C_3\times C_6^2$ x 2, $C_3^2\times C_{12}$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $C_4\times S_4$ x 2, $C_2^3.D_6$

Order 90: $C_3^2\times D_5$ x 5, $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$

Order 81: $C_3^4$ x 17

Order 72: $C_3\times S_4$ x 16, $S_3\times A_4$ x 8, $C_3:S_4$ x 8, $C_2\times C_3^2:C_4$ x 6, $D_4\times C_3^2$ x 5, $C_6\times D_6$ x 4, $C_6\wr C_2$ x 4, $C_6:D_6$ x 2, $C_6\times A_4$ x 2, $C_6^2:C_2$ x 2, $C_{12}:S_3$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:D_{12}$

Order 60: $S_3\times D_5$ x 2, $A_5$ x 2, $C_{15}:C_4$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 32, $S_3\times C_3^2$ x 20, $C_3^2:S_3$ x 8, $C_3^2\times C_6$ x 2

Order 48: $A_4:C_4$ x 4, $C_2\times S_4$ x 2, $C_6.C_2^3$ x 2, $S_3\times D_4$ x 2, $C_4\times A_4$ x 2, $C_6.D_4$ x 2, $C_6\times D_4$, $C_{12}:C_4$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$ x 5

Order 40: $C_4\times D_5$

Order 36: $C_3\times A_4$ x 42, $C_3^2:C_4$ x 11, $C_6^2$ x 10, $C_6\times S_3$ x 8, $C_3\times C_{12}$ x 6, $C_3^2:C_4$ x 6, $C_3:C_{12}$ x 4, $C_6:S_3$ x 4, $S_3^2$ x 4

Order 32: $C_4\times D_4$

Order 30: $C_3\times D_5$ x 6, $D_{15}$ x 2, $C_5\times S_3$ x 2, $C_{30}$

Order 27: $C_3^3$ x 102

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

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 28, $C_3:S_3$ x 24, $C_3\times C_6$ x 10

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

Order 15: $C_{15}$ x 5

Order 12: $C_2\times C_6$ x 13, $A_4$ x 12, $C_3:C_4$ x 9, $D_6$ x 8, $C_{12}$ x 8

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

Order 9: $C_3^2$ x 102

Order 8: $C_2\times C_4$ x 5, $D_4$ x 4, $C_2^3$ x 2

Order 6: $S_3$ x 12, $C_6$ x 11

Order 5: $C_5$

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

Order 3: $C_3$ x 17

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 38880: $C_3^3:C_4\times A_6$

Order 19440: $C_3^2:C_6\times A_6$

Order 12960: $C_3^2:C_4\times A_6$

Order 9720: $C_3^3\times A_6$

Order 6480: $C_3:S_3\times A_6$, $A_5\times C_3^3:C_4$, $C_3\times S_3\times A_6$

Order 4320: $C_3:C_4\times A_6$

Order 3888: $C_3^5:C_4^2$

Order 3240: $C_3^2\times A_6$ x 3, $C_3:S_3\times \GL(2,4)$

Order 2592: $C_3^3:C_4\times S_4$

Order 2160: $C_6\times A_6$, $C_3^2:C_4\times A_5$, $S_3\times A_6$

Order 1944: $C_3^5:(C_2\times C_4)$, $C_3\wr C_4:S_3$, $C_3^3:(S_3\times C_{12})$

Order 1620: $C_3^3\times A_5$

Order 1440: $C_4\times A_6$

Order 1296: $C_6^2:S_3^2$, $A_4\times C_3^3:C_4$, $(A_4\times C_3^3):C_4$, $C_3^4:C_4^2$

Order 1080: $C_3\times A_6$ x 3, $C_3^3:C_4\times D_5$, $S_3\times \GL(2,4)$, $\GL(2,4):S_3$

Order 972: $C_3^3:S_3^2$, $C_3^5:C_4$, $C_3^5:C_4$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 864: $C_3^2:C_4\times S_4$, $C_3^3:C_4\times D_4$

Order 720: $C_2\times A_6$, $\GL(2,4):C_4$

Order 648: $C_3^3:(C_4\times S_3)$ x 2, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $S_4\times C_3^3$, $C_3^4:(C_2\times C_4)$, $C_3^3:C_4\times S_3$, $S_3\times C_3^2:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 3, $(C_3^2\times C_{15}):C_4$, $C_3^3:C_{20}$, $C_3^3:D_{10}$

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

Order 432: $(C_3^2\times A_4):C_4$ x 2, $C_2^2\times C_3^3:C_4$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $C_6^2:D_6$, $(C_3\times C_6^2):C_4$, $(C_3^2\times C_{12}):C_4$, $C_3^3:C_4^2$, $C_3^3:C_4^2$

Order 360: $C_3^2:C_4\times D_5$, $C_6\times A_5$, $S_3\times A_5$, $A_6$

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

Order 288: $C_3^2:C_4\times D_4$, $C_2^3.S_3^2$

Order 270: $C_3^2:D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

Order 243: $C_3^5$

Order 240: $C_4\times A_5$

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

Order 180: $\GL(2,4)$ x 3, $C_{15}:D_6$, $C_{15}:D_6$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$

Order 162: $C_3^2\wr C_2$ x 6, $C_3^3:C_6$ x 3, $S_3\times C_3^3$ x 2, $C_3^3:S_3$

Order 144: $C_3^2:C_4^2$ x 2, $C_6^2:C_4$, $C_6\times S_4$, $S_3\times S_4$, $C_{12}:D_6$, $C_{12}:D_6$, $C_6^2:C_4$, $(C_3\times C_{12}):C_4$, $A_4\times C_3:C_4$, $C_6.S_4$

Order 135: $C_3^2\times C_{15}$

Order 120: $C_6.D_{10}$, $C_2\times A_5$

Order 108: $C_3^3:C_4$ x 8, $C_3^2\times A_4$ x 6, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 2, $C_3\times C_6^2$, $C_3\times S_3^2$, $C_3^2\times C_{12}$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $C_4\times S_4$, $C_2^3.D_6$

Order 90: $C_3^2\times D_5$ x 3, $C_3:D_{15}$, $C_{15}:S_3$, $C_3\times D_{15}$, $S_3\times C_{15}$

Order 81: $C_3^4$ x 7

Order 72: $C_2\times C_3^2:C_4$ x 5, $C_3\times S_4$ x 5, $D_4\times C_3^2$ x 3, $S_3\times A_4$ x 2, $C_3:S_4$ x 2, $C_{12}:S_3$ x 2, $C_6:D_6$, $C_6\times D_6$, $C_6\times A_4$, $C_6^2:C_2$, $C_3:D_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_6.D_6$

Order 60: $S_3\times D_5$, $A_5$, $C_{15}:C_4$, $C_3\times D_{10}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 13, $S_3\times C_3^2$ x 6, $C_3^2:S_3$ x 3, $C_3^2\times C_6$ x 2

Order 48: $A_4:C_4$ x 2, $C_2\times S_4$, $C_6\times D_4$, $C_6.C_2^3$, $S_3\times D_4$, $C_4\times A_4$, $C_6.D_4$, $C_{12}:C_4$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$ x 3

Order 40: $C_4\times D_5$

Order 36: $C_3\times A_4$ x 10, $C_3^2:C_4$ x 7, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 4, $C_6^2$ x 3, $C_6:S_3$ x 3, $C_6\times S_3$ x 3, $C_3:C_{12}$ x 2, $S_3^2$

Order 32: $C_4\times D_4$

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

Order 27: $C_3^3$ x 23

Order 24: $C_3\times D_4$ x 5, $C_6:C_4$ x 3, $S_4$ x 3, $C_4\times S_3$ x 2, $C_2\times A_4$ x 2, $C_2\times C_{12}$, $C_3:D_4$, $D_{12}$, $C_2^2\times C_6$, $C_2\times D_6$

Order 20: $D_{10}$, $C_{20}$, $C_5:C_4$

Order 18: $C_3:S_3$ x 9, $C_3\times S_3$ x 9, $C_3\times C_6$ x 6

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

Order 15: $C_{15}$ x 3

Order 12: $C_3:C_4$ x 6, $C_2\times C_6$ x 5, $C_{12}$ x 5, $A_4$ x 4, $D_6$ x 3

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

Order 9: $C_3^2$ x 23

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

Order 6: $C_6$ x 7, $S_3$ x 5

Order 5: $C_5$

Order 4: $C_4$ x 5, $C_2^2$ x 3

Order 3: $C_3$ x 7

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 38880: $C_3^3:C_4\times A_6$ ($C_1$)

Order 19440: $C_3^2:C_6\times A_6$ ($C_2$)

Order 9720: $C_3^3\times A_6$ ($C_4$)

Order 6480: $C_3:S_3\times A_6$ ($S_3$)

Order 3240: $C_3^2\times A_6$ ($C_3:C_4$)

Order 1080: $C_3\times A_6$ ($C_3^2:C_4$)

Order 360: $A_6$ ($C_3^3:C_4$)

Order 108: $C_3^3:C_4$ ($A_6$)

Order 54: $C_3^2:C_6$ ($C_2\times A_6$)

Order 27: $C_3^3$ ($C_4\times A_6$)

Order 18: $C_3:S_3$ ($S_3\times A_6$)

Order 9: $C_3^2$ ($C_3:C_4\times A_6$)

Order 3: $C_3$ ($C_3^2:C_4\times A_6$)

Order 1: $C_1$ ($C_3^3:C_4\times A_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 38880: $C_3^3:C_4\times A_6$ ($C_1$)

Order 19440: $C_3^2:C_6\times A_6$ ($C_2$)

Order 9720: $C_3^3\times A_6$ ($C_4$)

Order 6480: $C_3:S_3\times A_6$ ($S_3$)

Order 3240: $C_3^2\times A_6$ ($C_3:C_4$)

Order 1080: $C_3\times A_6$ ($C_3^2:C_4$)

Order 360: $A_6$ ($C_3^3:C_4$)

Order 108: $C_3^3:C_4$ ($A_6$)

Order 54: $C_3^2:C_6$ ($C_2\times A_6$)

Order 27: $C_3^3$ ($C_4\times A_6$)

Order 18: $C_3:S_3$ ($S_3\times A_6$)

Order 9: $C_3^2$ ($C_3:C_4\times A_6$)

Order 3: $C_3$ ($C_3^2:C_4\times A_6$)

Order 1: $C_1$ ($C_3^3:C_4\times A_6$)

Series

Derived series

$C_3^3:C_4\times A_6$

$\rhd$

$C_3^3:C_4\times A_6$

$\rhd$

$C_3^3\times A_6$

$\rhd$

$C_3^3\times A_6$

$\rhd$

$A_6$

$\rhd$

$A_6$

Chief series

$C_3^3:C_4\times A_6$

$\rhd$

$C_3^3:C_4\times A_6$

$\rhd$

$C_3^3:C_4$

$\rhd$

$C_3^3:C_4$

$\rhd$

$C_3^2:C_6$

$\rhd$

$C_3^2:C_6$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^3:C_4\times A_6$

$\rhd$

$C_3^3:C_4\times A_6$

$\rhd$

$C_3^3\times A_6$

$\rhd$

$C_3^3\times A_6$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 5 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 $84 \times 84$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.