Abstract group 116640.x: $C_3^3:A_4\times A_6$

• Label: 116640.x
• Order: \( 2^{5} \cdot 3^{6} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{6} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_3^3:A_4\times A_6$
Order:	\(116640\)\(\medspace = 2^{5} \cdot 3^{6} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_3^3:C_2^2.D_6.A_6.C_2^2$, of order \(1866240\)\(\medspace = 2^{9} \cdot 3^{6} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 4, $A_6$
Derived length:	$3$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	9	10	12	15	18	30	36	45
Elements	1	1287	8018	2520	144	13374	11664	3888	13680	14112	6480	7776	12960	20736	116640
Conjugacy classes	1	3	20	2	2	12	12	2	7	12	4	2	4	8	91
Divisions	1	3	14	2	1	10	6	1	5	4	2	1	2	2	54
Autjugacy classes	1	3	9	2	1	8	2	1	5	4	1	1	1	1	40

Dimension	1	2	3	4	5	6	8	9	10	12	15	16	18	20	24	27	30	32	36	40	48	54	60	72	80	96	108	120	128	192
Irr. complex chars.	3	0	1	6	6	2	6	3	3	1	2	0	0	12	2	1	5	12	6	6	4	2	4	0	0	2	1	1	0	0	91
Irr. rational chars.	1	1	1	0	2	2	3	1	3	1	2	1	1	1	0	1	5	1	0	6	1	2	4	3	3	2	1	1	3	1	54

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$2756$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	30	30
Arbitrary	9	11	11

Constructions

Permutation group:	Degree $15$ $\langle(1,5)(2,6,3,4)(7,12,15,8,11,13,9,10,14), (7,14,11)(8,13,10)(9,15,12), (3,4) \!\cdots\! \rangle$
Direct product:	$(C_3^3:A_4)$ $\, \times\, $ $A_6$
Semidirect product:	$(C_3^3\times A_6)$ $\,\rtimes\,$ $A_4$	$C_3^3$ $\,\rtimes\,$ $(A_4\times A_6)$	$(C_3:S_3^2\times A_6)$ $\,\rtimes\,$ $C_3$	$(C_3:S_3^2)$ $\,\rtimes\,$ $(C_3\times A_6)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

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

Homology

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

Subgroups

There are 679784 subgroups in 1368 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:A_4\times A_6$
Commutator:	$G' \simeq$ $C_3:S_3^2\times A_6$	$G/G' \simeq$ $C_3$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^3:A_4\times A_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3$	$G/\operatorname{Fit} \simeq$ $A_4\times A_6$
Radical:	$R \simeq$ $C_3^3:A_4$	$G/R \simeq$ $A_6$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times A_6$	$G/\operatorname{soc} \simeq$ $A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
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

No diagram available

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 116640: $C_3^3:A_4\times A_6$

Order 38880: $C_3:S_3^2\times A_6$

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

Order 19440: $A_5\times C_3^3:A_4$ x 2, $C_3^2:C_6\times A_6$

Order 12960: $S_3^2\times A_6$

Order 11664: $C_3^5:(C_4\times A_4)$

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

Order 7776: $C_3^3:A_4\times S_4$ x 2

Order 6480: $C_3\times S_3\times A_6$ x 3, $\GL(2,4):S_3^2$ x 2, $C_3:S_3\times A_6$

Order 5832: $C_3^4:(S_3\times A_4)$

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

Order 4320: $D_6\times A_6$, $A_4\times A_6$

Order 3888: $C_3^3:A_4^2$ x 2, $C_3^3:(C_4\times S_3^2)$

Order 3240: $C_3^2\times A_6$ x 4, $C_9\times A_6$ x 2, $C_3:S_3\times \GL(2,4)$ x 2, $D_5\times C_3^3:A_4$

Order 2916: $C_3^5:A_4$, $C_3^5:C_{12}$

Order 2592: $S_4\times C_3:S_3^2$ x 2, $D_4\times C_3^3:A_4$

Order 2160: $S_3\times A_6$ x 3, $S_3^2\times A_5$ x 2, $C_6\times A_6$

Order 1944: $S_3\times C_3^3:A_4$ x 2, $C_3\wr C_3\times S_4$ x 2, $C_3^2:S_3^3$, $C_3^4:(C_4\times S_3)$, $C_3^3:(S_3\times C_{12})$

Order 1620: $C_9:\GL(2,4)$ x 4, $C_3^3\times A_5$ x 2, $\He_3\times A_5$ x 2, $C_5\times C_3^3:A_4$

Order 1458: $C_3^5:C_6$

Order 1440: $C_2^2\times A_6$

Order 1296: $A_4\times C_3:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $(C_3\times C_6^2):A_4$ x 2, $C_2^2\times C_3^3:A_4$ x 2, $C_3^2:C_4\times S_3^2$, $C_4\times C_3^3:A_4$

Order 1080: $S_3\times \GL(2,4)$ x 6, $C_3\times A_6$ x 4, $\GL(2,4):S_3$ x 2, $D_5\times C_3:S_3^2$

Order 972: $C_3\wr A_4$ x 4, $C_3^4.C_{12}$ x 2, $C_6^2.C_3^3$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4:C_{12}$

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

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

Order 729: $C_3^5:C_3$

Order 720: $D_6\times A_5$ x 2, $A_4\times A_5$ x 2, $C_2\times A_6$

Order 648: $C_9:C_3\times S_4$ x 4, $S_3\times C_3^2:C_{12}$ x 3, $C_3\times C_6^2:C_6$ x 2, $C_3^3:S_4$ x 2, $S_4\times C_3^3$ x 2, $C_3:S_3^3$ x 2, $C_3^3:C_4\times S_3$ x 2, $S_4\times \He_3$ x 2, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^3:A_4$, $C_3\wr C_3\times D_4$

Order 540: $C_3\times \GL(2,4)$ x 8, $C_9\times A_5$ x 4, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$

Order 486: $C_3^4:C_6$ 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: $C_6^2:D_6$ x 6, $A_4:S_3^2$ x 4, $A_4:S_3^2$ x 4, $C_3:D_6^2$ x 2, $A_4\times S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2:D_6$ x 2, $C_3^2:C_4\times D_6$, $C_6^2:C_{12}$, $C_6^2:D_6$, $C_{12}:S_3^2$, $C_{12}:S_3^2$

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

Order 360: $S_3\times A_5$ x 6, $C_6\times A_5$ x 2, $D_5\times S_3^2$, $A_6$

Order 324: $C_3^2:S_3^2$ x 5, $C_3^2:S_3^2$ x 5, $C_3^3:A_4$ x 5, $C_6^2.C_3^2$ x 4, $C_3^3:C_{12}$ x 4, $C_6^2.C_3^2$ x 4, $C_3\wr C_4$ x 3, $C_3^2.C_6^2$ x 2, $C_3^3:A_4$ x 2, $\He_3:A_4$ x 2, $A_4\times C_3^3$ x 2, $C_3^2:S_3^2$ x 2, $C_3\wr C_2^2$ x 2, $A_4\times \He_3$ x 2, $C_3^2:C_{36}$ x 2, $C_3^3:C_{12}$, $C_3^2:S_3^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$

Order 288: $D_6\times S_4$ x 2, $A_4\times S_4$ x 2, $D_4\times S_3^2$

Order 270: $C_{45}:C_6$ x 2, $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

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 240: $C_2^2\times A_5$ x 2

Order 216: $C_3^2:S_4$ x 12, $C_3^2\times S_4$ x 8, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 5, $C_3^2:C_4\times S_3$ x 5, $C_9\times S_4$ x 4, $C_6:S_3^2$ x 3, $C_{36}:C_6$ x 2, $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $S_3^3$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $D_4\times \He_3$, $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $D_4\times C_3^3$, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_6.S_3^2$

Order 180: $\GL(2,4)$ x 8, $C_{15}:D_6$ x 3, $S_3\times D_{15}$ x 2, $C_{15}:D_6$, $C_5\times S_3^2$, $C_{15}:D_6$

Order 162: $C_3^3:C_6$ x 11, $C_3^2\wr C_2$ x 9, $S_3\times C_3^3$ x 5, $C_3^3.C_6$ x 4, $C_3^2:C_{18}$ x 2, $S_3\times \He_3$ x 2, $C_3^3:S_3$, $C_3^3:C_6$

Order 144: $S_3\times S_4$ x 14, $D_6:D_6$ x 4, $C_{12}:D_6$ x 3, $D_6^2$ x 2, $A_4\times D_6$ x 2, $C_6:S_4$ x 2, $C_6\times S_4$ x 2, $A_4^2$ x 2, $D_6:D_6$ x 2, $S_3\times D_{12}$ x 2, $C_6^2:C_4$, $C_{12}:D_6$, $C_{12}:D_6$, $C_4\times S_3^2$

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

Order 120: $C_2\times A_5$ x 2, $S_3\times D_{10}$, $D_5\times A_4$

Order 108: $C_3^2\times A_4$ x 15, $C_3:S_3^2$ x 15, $C_3:S_3^2$ x 8, $C_3\times S_3^2$ x 8, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 4, $C_{18}:C_6$ 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, $C_3^3:C_4$ x 3, $C_3\times C_6^2$ x 2, $C_2^2\times \He_3$ x 2, $C_9:C_{12}$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $C_4\times \He_3$

Order 96: $C_2^2\times S_4$ x 2, $D_4\times D_6$, $D_4\times A_4$

Order 90: $C_3^2\times D_5$ x 4, $C_3\times D_{15}$ x 3, $S_3\times C_{15}$ x 3, $C_9\times D_5$ x 2, $C_3:D_{15}$, $C_{15}:S_3$

Order 81: $C_3\wr C_3$ x 21, $C_3^4$ x 12, $C_9:C_3^2$ x 12, $C_3\times \He_3$ x 6, $C_3^2\times C_9$ x 2

Order 72: $C_3\times S_4$ x 16, $S_3\times A_4$ x 14, $C_3:S_4$ x 12, $S_3\times D_6$ x 9, $C_6\times D_6$ x 6, $C_6\wr C_2$ x 6, $D_4\times C_3^2$ x 4, $C_3:D_{12}$ x 4, $C_6:D_6$ x 3, $C_3\times D_{12}$ x 3, $S_3\times C_{12}$ x 3, $C_6\times A_4$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_6^2:C_2$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $D_4\times C_9$ x 2, $C_3:D_{12}$, $C_{12}:S_3$, $C_6.D_6$

Order 60: $S_3\times D_5$ x 5, $A_5$ x 2, $C_5\times A_4$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $C_3^2:C_6$ x 35, $S_3\times C_3^2$ x 23, $C_3^2:S_3$ x 11, $S_3\times C_9$ x 4, $C_3^2\times C_6$ x 2, $C_9:C_6$ x 2, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 9, $C_2\times S_4$ x 6, $C_2^2\times A_4$ x 4, $C_2^2\times D_6$ x 2, $C_2^2:A_4$ x 2, $C_6:D_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_4\times A_4$

Order 45: $C_3\times C_{15}$ x 4, $C_{45}$ x 2

Order 40: $C_2\times D_{10}$

Order 36: $C_3\times A_4$ x 32, $S_3^2$ x 20, $C_6\times S_3$ x 13, $C_6^2$ x 9, $C_6:S_3$ x 9, $C_3\times C_{12}$ x 4, $C_2\times C_{18}$ x 4, $C_2^2:C_9$ x 4, $C_3:C_{12}$ x 3, $C_3^2:C_4$ x 2, $C_{36}$ x 2, $C_3^2:C_4$

Order 32: $C_2^2\times D_4$

Order 30: $C_3\times D_5$ x 5, $D_{15}$ x 3, $C_5\times S_3$ x 3, $C_{30}$

Order 27: $C_3^3$ x 55, $C_9:C_3$ x 18, $\He_3$ x 9, $C_3\times C_9$ x 8

Order 24: $C_2\times D_6$ x 15, $C_3:D_4$ x 10, $S_4$ x 10, $C_3\times D_4$ x 7, $D_{12}$ x 5, $C_4\times S_3$ x 5, $C_2\times A_4$ x 5, $C_2^2\times C_6$ x 2, $C_2\times C_{12}$, $C_6:C_4$

Order 20: $D_{10}$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 34, $C_3:S_3$ x 28, $C_3\times C_6$ x 9, $C_{18}$ x 2

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

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 21, $A_4$ x 15, $C_2\times C_6$ x 13, $C_{12}$ x 5, $C_3:C_4$ x 3

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

Order 9: $C_3^2$ x 61, $C_9$ x 6

Order 8: $D_4$ x 6, $C_2^3$ x 5, $C_2\times C_4$ x 2

Order 6: $S_3$ x 16, $C_6$ x 10

Order 5: $C_5$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 14

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 116640: $C_3^3:A_4\times A_6$

Order 38880: $C_3:S_3^2\times A_6$

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

Order 19440: $C_3^2:C_6\times A_6$, $A_5\times C_3^3:A_4$

Order 12960: $S_3^2\times A_6$

Order 11664: $C_3^5:(C_4\times A_4)$

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

Order 7776: $C_3^3:A_4\times S_4$

Order 6480: $C_3\times S_3\times A_6$ x 2, $C_3:S_3\times A_6$, $\GL(2,4):S_3^2$

Order 5832: $C_3^4:(S_3\times A_4)$

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

Order 4320: $D_6\times A_6$, $A_4\times A_6$

Order 3888: $C_3^3:(C_4\times S_3^2)$, $C_3^3:A_4^2$

Order 3240: $C_3^2\times A_6$ x 4, $C_9\times A_6$, $C_3:S_3\times \GL(2,4)$, $D_5\times C_3^3:A_4$

Order 2916: $C_3^5:A_4$, $C_3^5:C_{12}$

Order 2592: $S_4\times C_3:S_3^2$, $D_4\times C_3^3:A_4$

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

Order 1944: $C_3^2:S_3^3$, $C_3^4:(C_4\times S_3)$, $C_3^3:(S_3\times C_{12})$, $S_3\times C_3^3:A_4$, $C_3\wr C_3\times S_4$

Order 1620: $C_5\times C_3^3:A_4$, $C_3^3\times A_5$, $C_9:\GL(2,4)$, $\He_3\times A_5$

Order 1458: $C_3^5:C_6$

Order 1440: $C_2^2\times A_6$

Order 1296: $A_4\times C_3:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $(C_3\times C_6^2):A_4$, $C_2^2\times C_3^3:A_4$, $C_3^2:C_4\times S_3^2$, $C_4\times C_3^3:A_4$

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

Order 972: $C_3\wr A_4$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4.C_{12}$, $C_3^4:C_{12}$, $C_6^2.C_3^3$

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

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

Order 729: $C_3^5:C_3$

Order 720: $D_6\times A_5$, $A_4\times A_5$, $C_2\times A_6$

Order 648: $S_3\times C_3^2:C_{12}$ x 2, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $S_4\times C_3^3$, $C_3:S_3^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_3^3:C_4\times S_3$, $C_2\times C_3^3:A_4$, $C_9:C_3\times S_4$, $S_4\times \He_3$, $C_3\wr C_3\times D_4$

Order 540: $C_3\times \GL(2,4)$ x 4, $C_9\times A_5$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_3^3:D_{10}$

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

Order 432: $A_4:S_3^2$ x 2, $C_6^2:D_6$ x 2, $C_3:D_6^2$, $C_3^2:C_4\times D_6$, $A_4\times S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $C_6^2:C_{12}$, $C_6^2:D_6$, $C_6^2:D_6$, $C_{12}:S_3^2$, $C_{12}:S_3^2$

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

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

Order 324: $C_3^3:C_{12}$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^3:A_4$ x 3, $C_3\wr C_4$ x 2, $C_3^2.C_6^2$, $C_3^3:A_4$, $C_6^2.C_3^2$, $\He_3:A_4$, $C_3^3:C_{12}$, $A_4\times C_3^3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3\wr C_2^2$, $C_3^2\times S_3^2$, $C_3^4:C_4$, $C_6^2.C_3^2$, $A_4\times \He_3$, $C_3^2:C_{36}$

Order 288: $D_4\times S_3^2$, $D_6\times S_4$, $A_4\times S_4$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$

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

Order 240: $C_2^2\times A_5$

Order 216: $C_3^2:S_4$ x 4, $C_3^2\times S_4$ x 4, $C_6^2:C_6$ x 3, $C_3^2:C_4\times S_3$ x 3, $C_6:S_3^2$ x 2, $C_6^2:C_6$ x 2, $C_9\times S_4$, $C_{36}:C_6$, $D_4\times \He_3$, $C_6^2:C_6$, $C_6:S_3^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_3^2:S_4$, $S_3^3$, $D_4\times C_3^3$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_3:S_3\times C_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$

Order 180: $\GL(2,4)$ x 4, $C_{15}:D_6$ x 2, $C_{15}:D_6$, $S_3\times D_{15}$, $C_5\times S_3^2$, $C_{15}:D_6$

Order 162: $C_3^2\wr C_2$ x 7, $C_3^3:C_6$ x 5, $S_3\times C_3^3$ x 3, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$, $C_3^3:C_6$

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

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

Order 120: $S_3\times D_{10}$, $D_5\times A_4$, $C_2\times A_5$

Order 108: $C_3^2\times A_4$ x 8, $C_3:S_3^2$ x 6, $C_3:S_3^2$ x 5, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 3, $C_3\times S_3^2$ x 3, $C_3^3:C_4$ x 2, $C_3^2:A_4$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2\times C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$, $C_9:C_{12}$, $C_4\times \He_3$

Order 96: $C_2^2\times S_4$, $D_4\times D_6$, $D_4\times A_4$

Order 90: $C_3^2\times D_5$ x 4, $C_3\times D_{15}$ x 2, $S_3\times C_{15}$ x 2, $C_3:D_{15}$, $C_{15}:S_3$, $C_9\times D_5$

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: $C_3\times S_4$ x 7, $S_3\times A_4$ x 5, $S_3\times D_6$ x 4, $C_3:S_4$ x 4, $D_4\times C_3^2$ x 4, $C_6:D_6$ x 2, $C_6\times D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_6\times A_4$, $C_6^2:C_2$, $C_3:D_{12}$, $C_{12}:S_3$, $C_3:D_{12}$, $D_6:S_3$, $C_6.D_6$, $C_6.D_6$, $D_4\times C_9$

Order 60: $S_3\times D_5$ x 3, $C_5\times A_4$, $A_5$, $D_{30}$, $S_3\times C_{10}$, $C_3\times D_{10}$

Order 54: $C_3^2:C_6$ x 17, $S_3\times C_3^2$ x 10, $C_3^2:S_3$ x 5, $C_3^2\times C_6$ x 2, $S_3\times C_9$, $C_9:C_6$, $C_2\times \He_3$

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

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

Order 40: $C_2\times D_{10}$

Order 36: $C_3\times A_4$ x 15, $S_3^2$ x 8, $C_6:S_3$ x 6, $C_6\times S_3$ x 6, $C_6^2$ x 5, $C_3\times C_{12}$ x 4, $C_3^2:C_4$ x 2, $C_3:C_{12}$ x 2, $C_3^2:C_4$, $C_2\times C_{18}$, $C_2^2:C_9$, $C_{36}$

Order 32: $C_2^2\times D_4$

Order 30: $C_3\times D_5$ x 5, $D_{15}$ x 2, $C_5\times S_3$ x 2, $C_{30}$

Order 27: $C_3^3$ x 26, $C_9:C_3$ x 4, $\He_3$ x 4, $C_3\times C_9$ x 2

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

Order 20: $D_{10}$ x 2, $C_2\times C_{10}$

Order 18: $C_3\times S_3$ x 15, $C_3:S_3$ x 12, $C_3\times C_6$ x 7, $C_{18}$

Order 16: $C_2\times D_4$ x 3, $C_2^4$, $C_2^2\times C_4$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 11, $A_4$ x 8, $C_2\times C_6$ x 7, $C_{12}$ x 5, $C_3:C_4$ x 2

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

Order 9: $C_3^2$ x 28, $C_9$ x 2

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

Order 6: $C_6$ x 8, $S_3$ x 8

Order 5: $C_5$

Order 4: $C_2^2$ x 6, $C_4$ x 2

Order 3: $C_3$ x 9

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 116640: $C_3^3:A_4\times A_6$ ($C_1$)

Order 38880: $C_3:S_3^2\times A_6$ ($C_3$)

Order 9720: $C_3^3\times A_6$ ($A_4$)

Order 360: $A_6$ ($C_3^3:A_4$)

Order 324: $C_3^3:A_4$ ($A_6$)

Order 108: $C_3:S_3^2$ ($C_3\times A_6$)

Order 27: $C_3^3$ ($A_4\times A_6$)

Order 1: $C_1$ ($C_3^3:A_4\times A_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 116640: $C_3^3:A_4\times A_6$ ($C_1$)

Order 38880: $C_3:S_3^2\times A_6$ ($C_3$)

Order 9720: $C_3^3\times A_6$ ($A_4$)

Order 360: $A_6$ ($C_3^3:A_4$)

Order 324: $C_3^3:A_4$ ($A_6$)

Order 108: $C_3:S_3^2$ ($C_3\times A_6$)

Order 27: $C_3^3$ ($A_4\times A_6$)

Order 1: $C_1$ ($C_3^3:A_4\times A_6$)

Series

Derived series	$C_3^3:A_4\times A_6$	$\rhd$	$C_3:S_3^2\times A_6$	$\rhd$	$C_3^3\times A_6$	$\rhd$	$A_6$
Chief series	$C_3^3:A_4\times A_6$	$\rhd$	$C_3^3:A_4$	$\rhd$	$C_3:S_3^2$	$\rhd$	$C_3^3$	$\rhd$	$C_1$
Lower central series	$C_3^3:A_4\times A_6$	$\rhd$	$C_3:S_3^2\times A_6$
Upper central series	$C_1$

Supergroups

This group is a maximal subgroup of 7 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 $91 \times 91$ 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.