Abstract group 116640.v: $C_3\wr S_3\times S_6$

• Label: 116640.v
• Order: \( 2^{5} \cdot 3^{6} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{6} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_3\wr S_3\times S_6$
Order:	\(116640\)\(\medspace = 2^{5} \cdot 3^{6} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_3\times C_3.S_3^2.A_6.C_2^2$, of order \(466560\)\(\medspace = 2^{7} \cdot 3^{6} \cdot 5 \)
Composition factors:	$C_2$ 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	759	3644	1800	144	45492	2916	1296	20880	6336	11340	10368	6480	5184	116640
Conjugacy classes	1	7	32	4	1	120	6	1	36	10	10	8	4	2	242
Divisions	1	7	20	4	1	68	3	1	20	6	5	4	2	1	143
Autjugacy classes	1	5	9	4	1	25	2	1	12	4	3	2	2	1	72

Dimension	1	2	3	4	5	6	9	10	15	16	18	20	27	30	32	36	40	48	54	60	64	96
Irr. complex chars.	12	6	24	0	24	2	12	24	48	6	6	6	24	28	3	0	0	12	2	2	0	1	242
Irr. rational chars.	4	6	0	2	8	14	4	16	0	2	6	10	0	28	3	2	2	0	14	14	1	7	143

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$3816$

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 $15$ $\langle(10,11,12)(13,14,15), (1,2,3,4,5)(7,8,9)(10,13)(11,14)(12,15), (7,11,8,12,9,10) \!\cdots\! \rangle$
Direct product:	$(C_3\wr S_3)$ $\, \times\, $ $S_6$
Semidirect product:	$(C_3^3\times S_6)$ $\,\rtimes\,$ $S_3$	$C_3^3$ $\,\rtimes\,$ $(S_3\times S_6)$	$(\He_3\times S_6)$ $\,\rtimes\,$ $C_6$	$(\He_3:S_6)$ $\,\rtimes\,$ $C_6$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^2$ . $(C_3\times S_3\times S_6)$	$C_3$ . $(C_3^2:C_6\times S_6)$	$(C_3^2\times S_6)$ . $(C_3\times S_3)$	$(C_3^2\times A_6)$ . $(C_6\times S_3)$	all 6

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

Homology

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

Subgroups

There are 814050 subgroups in 4254 conjugacy classes, 26 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^2:C_6\times S_6$
Commutator:	$G' \simeq$ $\He_3\times A_6$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3\times S_3\times S_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3\wr C_3$	$G/\operatorname{Fit} \simeq$ $C_2\times S_6$
Radical:	$R \simeq$ $C_3\wr S_3$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_3\times A_6$	$G/\operatorname{soc} \simeq$ $C_3^2:D_6$
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: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 116640: $C_3\wr S_3\times S_6$

Order 58320: $A_6\times C_3\wr S_3$, $C_3\wr C_3\times S_6$, $C_3\wr C_3:S_6$

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

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

Order 19440: $C_3\wr S_3\times S_5$ x 2, $C_9:C_3\times S_6$, $C_3^3\times S_6$, $C_3^3:S_6$, $S_3\times C_3^2\times A_6$, $\He_3\times S_6$, $\He_3:S_6$, $C_3^2:S_3\times A_6$

Order 12960: $C_3\times S_3\times S_6$ x 5, $A_6:C_6^2$

Order 11664: $C_3^5:(S_3\times D_4)$

Order 9720: $C_3\wr C_3\times S_5$ x 2, $A_5\times C_3\wr S_3$ x 2, $C_3\wr C_3:S_5$ x 2, $C_3^3\times A_6$, $\He_3\times A_6$, $C_9:C_3\times A_6$

Order 7776: $C_6^3:S_3^2$ x 2

Order 6480: $C_3^2\times S_6$ x 7, $C_3^2:S_6$ x 5, $C_3\times S_3\times A_6$ x 5, $\GL(2,4):C_6^2$ x 2, $C_3^2:S_3\times S_5$ x 2, $C_3\times C_6\times A_6$, $C_9\times S_6$

Order 5832: $(C_3^3\times S_3^2):S_3$ x 2, $C_3^3:S_3^3$ x 2, $(C_3^3\times S_3^2):C_6$, $C_3^5:D_{12}$, $C_3^5:(C_4\times S_3)$

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

Order 4320: $C_6\times S_6$ x 4, $S_3\times S_6$ x 2

Order 3888: $C_3\wr S_3\times S_4$ x 8, $A_4\times C_3^3:D_6$ x 2, $C_3^3:C_6\times S_4$ x 2, $\He_3:(C_6\times S_4)$ x 2, $C_3\times S_3^3:C_6$, $C_3^4:(S_3\times D_4)$

Order 3240: $C_3^2\times A_6$ x 6, $C_3^3\times S_5$ x 2, $C_3\times S_3\times \GL(2,4)$ x 2, $C_3^3:S_5$ x 2, $C_9:C_3\times S_5$ x 2, $\He_3\times S_5$ x 2, $C_3^2:S_3\times A_5$ x 2, $\He_3:S_5$ x 2, $C_9\times A_6$, $C_3\wr S_3\times F_5$

Order 2916: $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3^5:D_6$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:C_{12}$

Order 2592: $C_3^2:D_6\times S_4$ x 2, $C_6^3:D_6$ x 2, $C_6^3:D_6$

Order 2160: $\GL(2,4):D_6$ x 10, $C_3\times S_6$ x 10, $C_6\times A_6$ x 4, $A_5:C_6^2$ x 2, $C_3:S_6$ x 2, $S_3\times A_6$ x 2

Order 1944: $A_4\times C_3\wr S_3$ x 4, $C_3\wr C_3:S_4$ x 4, $C_3\wr C_3\times S_4$ x 4, $C_3^2\times S_3^3$ x 2, $C_3^5:D_4$ x 2, $C_3^2:S_3^3$ x 2, $\He_3:\SOPlus(4,2)$ x 2, $C_6^3:C_3^2$ x 2, $C_3\wr S_3\times D_6$ x 2, $C_3^4:D_{12}$, $C_3^5:D_4$, $S_3\times C_3^3:C_{12}$, $C_3^4:D_{12}$, $(C_9\times S_3^2):C_6$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$

Order 1620: $C_3^3\times A_5$ x 2, $C_9:\GL(2,4)$ x 2, $\He_3\times A_5$ x 2, $(C_5\times \He_3):C_{12}$, $C_3\wr C_3\times F_5$, $C_3\wr S_3\times D_5$

Order 1458: $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1440: $C_2\times S_6$

Order 1296: $C_6^2:S_3^2$ x 8, $C_6^2:S_3^2$ x 8, $C_3\wr S_3\times D_4$ x 8, $S_3^3:C_6$ x 5, $C_6^3:C_6$ x 2, $C_6^3:S_3$ x 2, $C_6^3:S_3$ x 2, $C_3^2:D_6\times A_4$ x 2, $C_2\times \He_3:S_4$ x 2, $C_9:C_6\times S_4$ x 2, $C_2\times S_4\times \He_3$ x 2, $C_6^3:S_3$ x 2, $(C_3^2\times A_4):D_6$ x 2, $C_6\wr S_3$ x 2, $C_6^3:S_3$ x 2, $S_3^2:C_6^2$, $C_6^3:C_6$, $C_2\times C_3^3:D_{12}$, $C_4\times C_3^3:D_6$

Order 1080: $C_3^2\times S_5$ x 14, $S_3\times \GL(2,4)$ x 10, $C_3^2:S_5$ x 10, $C_3\times A_6$ x 6, $C_6\times \GL(2,4)$ x 2, $C_9\times S_5$ x 2, $S_3\times C_3^2\times F_5$, $C_3^2:S_3\times F_5$

Order 972: $C_3^3:S_3^2$ x 12, $C_3^3\times S_3^2$ x 4, $C_3^3:C_6^2$ x 3, $C_3^3:C_6^2$ 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, $C_3^3.C_6^2$ x 2, $\He_3\times S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3.C_6^2$ x 2, $C_3^4:D_6$ x 2, $C_6^2.C_3^3$ x 2, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $(C_3^2\times \He_3):C_4$, $C_3^4.C_{12}$, $C_3^4:C_{12}$

Order 864: $C_6\times S_3\times S_4$ x 10, $S_4\times C_6^2$ x 2, $D_{12}:C_6^2$, $C_2\times C_6^2:D_6$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 729: $C_3^5:C_3$

Order 720: $C_6\times S_5$ x 8, $S_3\times S_5$ x 4, $S_6$ x 2, $C_2\times A_6$

Order 648: $C_2^2\times C_3\wr S_3$ x 11, $C_3\times S_3^3$ x 10, $C_3\wr D_4$ x 10, $C_3^4:D_4$ x 9, $(C_3\times C_6^2):S_3$ x 8, $C_3^3:D_{12}$ x 5, $S_3\times C_3^2:C_{12}$ x 5, $C_3^4:C_2^3$ x 4, $S_3\times C_3^2\times A_4$ x 4, $C_3^3:S_4$ x 4, $S_4\times C_3^3$ x 4, $A_4\times C_3^2:S_3$ x 4, $\He_3:S_4$ x 4, $C_9:C_3\times S_4$ x 4, $S_4\times \He_3$ x 4, $(C_3^2\times A_4):S_3$ x 4, $C_3^3:S_4$ x 4, $C_3\wr C_3\times D_4$ x 4, $C_3^3:D_{12}$ x 4, $C_4\times C_3\wr S_3$ x 4, $C_6^3:C_3$ x 2, $C_9:C_6\times A_4$ x 2, $C_2\times A_4\times \He_3$ x 2, $S_3\times C_3^2:D_6$ x 2, $C_6^3:C_3$ x 2, $C_6\wr C_3$ x 2, $(C_3^2\times A_4):C_6$ x 2, $C_2\times \He_3:A_4$ x 2, $C_6\times C_3^2:C_{12}$, $S_3^2:C_{18}$, $C_2\times C_3^3:C_{12}$, $C_2\times \He_3:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 12, $C_9\times A_5$ x 2, $C_{15}:C_6^2$, $C_3^3:F_5$, $F_5\times C_3^3$, $\He_3:D_{10}$, $\He_3:F_5$, $C_{45}:C_{12}$, $F_5\times \He_3$

Order 486: $C_3^4:C_6$ x 10, $C_3^4:S_3$ x 8, $C_3^4:S_3$ x 8, $C_3^4:C_6$ x 3, $S_3\times C_3^4$ x 3, $C_3^4:S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $C_6^2:D_6$ x 40, $A_4:C_6^2$ x 24, $C_6\times S_3\times A_4$ x 18, $C_6^2:D_6$ x 10, $C_{12}:C_6^2$ x 8, $C_6^2:D_6$ x 8, $C_6\times \SOPlus(4,2)$ x 4, $C_6^2:D_6$ x 4, $D_6\times C_6^2$ x 2, $A_4\times C_6^2$ x 2, $S_3^3:C_2$ x 2, $D_6:C_6^2$ x 2, $C_6^2:D_6$ x 2, $C_{18}\times S_4$ x 2, $C_6^2:D_6$ x 2, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_6^2.D_6$, $C_6^2.D_6$

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

Order 360: $C_3\times S_5$ x 20, $C_6\times A_5$ x 8, $D_{15}:C_{12}$ x 5, $S_3\times A_5$ x 4, $C_3:S_5$ x 4, $C_{30}:C_{12}$, $A_6$

Order 324: $C_3^2\times S_3^2$ x 44, $C_3^3:D_6$ x 16, $C_3^2:S_3^2$ x 15, $C_3^2:S_3^2$ x 12, $C_3\wr C_2^2$ x 10, $C_3^3:C_{12}$ x 7, $D_6\times C_3^3$ x 6, $C_3^2.C_6^2$ x 5, $C_3\wr C_4$ x 5, $C_3^2:C_6^2$ x 3, $C_3^3:A_4$ x 2, $C_6^2.C_3^2$ x 2, $\He_3:A_4$ x 2, $C_3^3:C_{12}$ x 2, $A_4\times C_3^3$ x 2, $C_3^3:D_6$ x 2, $C_3^3:D_6$ x 2, $C_3^2.C_6^2$ x 2, $D_6\times \He_3$ x 2, $C_6^2.C_3^2$ x 2, $A_4\times \He_3$ x 2, $\He_3:C_{12}$ x 2, $C_9\times S_3^2$ x 2, $C_3^2:C_{36}$

Order 288: $C_2\times C_6\times S_4$ x 8, $C_6^2:C_2^3$ x 5, $D_6\times S_4$ x 4, $D_4\times C_6^2$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 243: $C_3^4:C_3$ x 8, $C_3^5$, $C_9:C_3^3$, $C_3^2\times \He_3$

Order 240: $C_2\times S_5$ x 2

Order 216: $C_6^2:C_6$ x 36, $C_3^2\times S_4$ x 32, $C_6^2:C_6$ x 26, $C_3^2:S_4$ x 20, $C_6\times S_3^2$ x 18, $S_3^2:C_6$ x 18, $S_3\times C_6^2$ x 13, $C_6^2:S_3$ x 11, $C_3^2:S_4$ x 8, $C_6^2:S_3$ x 8, $C_6^2:C_6$ x 8, $C_9\times S_4$ x 4, $C_{36}:C_6$ x 4, $D_4\times \He_3$ x 4, $C_3^2:D_{12}$ x 4, $C_4\times C_3^2:S_3$ x 4, $C_2\times C_3^2:C_{12}$ x 4, $S_3^3$ x 4, $S_3^2:S_3$ x 4, $D_4\times C_3^3$ x 4, $C_3^2\times D_{12}$ x 4, $C_6.C_6^2$ x 4, $C_6^2:C_6$ x 4, $C_6^3$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_6.C_6^2$ x 2, $C_2^3\times \He_3$ x 2, $C_6^2.C_6$ x 2, $C_{18}:A_4$ x 2, $A_4\times C_{18}$ x 2, $C_{18}:C_{12}$, $C_6.C_6^2$, $C_6^2.S_3$, $C_3\times C_6\times C_{12}$, $C_6.C_6^2$

Order 180: $\GL(2,4)$ x 12, $C_3^2\times F_5$ x 7, $C_{15}:D_6$ x 5, $C_{15}:C_{12}$ x 5, $C_9\times F_5$, $C_3^2\times D_{10}$

Order 162: $S_3\times C_3^3$ x 47, $C_3^2\wr C_2$ x 31, $C_3\wr S_3$ x 14, $C_3^3:C_6$ x 9, $C_3^3:S_3$ x 8, $C_3^3:S_3$ x 8, $C_3^3.C_6$ x 8, $S_3\times \He_3$ x 8, $C_3^3:C_6$ x 5, $C_3^3\times C_6$ x 3, $C_{18}:C_3^2$ x 2, $C_6\times \He_3$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$

Order 144: $C_6\times S_4$ x 62, $C_{12}:D_6$ x 40, $C_4:C_6^2$ x 17, $C_2^2:C_6^2$ x 16, $S_3\times S_4$ x 16, $A_4\times D_6$ x 14, $C_6^2:C_2^2$ x 10, $C_6^2:C_2^2$ x 10, $C_6\times D_{12}$ x 5, $C_{12}\times D_6$ x 5, $C_6:S_4$ x 4, $C_2^2\times C_6^2$ x 2, $D_4\times C_{18}$, $S_3^2:C_2^2$, $C_2\times C_6\times C_{12}$

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

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

Order 108: $C_3\times S_3^2$ x 92, $C_3^2\times D_6$ x 70, $C_3^2\times A_4$ x 22, $C_3^2:D_6$ x 16, $C_3^2:D_6$ x 14, $C_3^2:C_{12}$ x 10, $C_3\times C_6^2$ x 7, $C_3:S_3^2$ x 6, $C_{18}:C_6$ x 5, $C_2^2\times \He_3$ x 5, $C_3:S_3^2$ x 4, $C_3^2:A_4$ x 4, $C_3^3:C_4$ x 2, $C_3^2\times C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $S_3\times C_{18}$ x 2, $C_3^2.A_4$ x 2, $C_9:A_4$ x 2, $C_9\times A_4$ x 2, $C_9:C_{12}$ x 2, $C_4\times \He_3$ x 2, $\He_3:C_4$ x 2

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

Order 90: $C_3^2\times D_5$ x 7, $C_3\times D_{15}$ x 5, $S_3\times C_{15}$ x 5, $C_9\times D_5$, $C_3\times C_{30}$

Order 81: $C_3^4$ x 18, $C_3\wr C_3$ x 11, $C_9:C_3^2$ x 6, $C_3\times \He_3$ x 6, $C_3^2\times C_9$

Order 72: $C_6\times A_4$ x 78, $C_6\times D_6$ x 63, $C_3\times S_4$ x 60, $C_6\wr C_2$ x 40, $D_4\times C_3^2$ x 36, $S_3\times A_4$ x 28, $C_2\times C_6^2$ x 23, $C_3\times D_{12}$ x 20, $S_3\times C_{12}$ x 20, $C_6\times C_{12}$ x 11, $C_3:S_4$ x 8, $S_3\times D_6$ x 6, $C_6:C_{12}$ x 5, $\SOPlus(4,2)$ x 4, $D_4\times C_9$ x 4, $C_2^2\times C_{18}$ x 2, $C_2^2:C_{18}$ x 2, $C_2\times C_{36}$, $C_2\times C_3^2:C_4$

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

Order 54: $S_3\times C_3^2$ x 170, $C_3^2:C_6$ x 54, $C_3^2\times C_6$ x 41, $C_3^2:S_3$ x 14, $C_9:C_6$ x 7, $C_2\times \He_3$ x 7, $S_3\times C_9$ x 6, $C_3\times C_{18}$ x 2, $C_3^2:S_3$ x 2

Order 48: $C_6\times D_4$ x 50, $C_2\times S_4$ x 16, $S_3\times D_4$ x 16, $C_2^2\times A_4$ x 10, $C_2^3\times C_6$ x 8, $C_2^2\times D_6$ x 4, $C_2^2\times C_{12}$ x 4, $C_6:D_4$ x 4, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

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

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 146, $C_6^2$ x 58, $C_3\times A_4$ x 46, $S_3^2$ x 26, $C_3\times C_{12}$ x 14, $C_3:C_{12}$ x 10, $C_2\times C_{18}$ x 5, $C_6:S_3$ x 5, $C_3^2:C_4$ x 2, $C_2^2:C_9$ x 2, $C_{36}$ x 2

Order 32: $C_2^2\times D_4$

Order 30: $C_3\times D_5$ x 10, $C_{30}$ x 4, $D_{15}$ x 2, $C_5\times S_3$ x 2

Order 27: $C_3^3$ x 83, $C_9:C_3$ x 7, $\He_3$ x 7, $C_3\times C_9$ x 4

Order 24: $C_3\times D_4$ x 72, $C_2^2\times C_6$ x 56, $C_2\times A_4$ x 34, $C_2\times C_{12}$ x 26, $C_2\times D_6$ x 24, $C_3:D_4$ x 16, $S_4$ x 16, $D_{12}$ x 8, $C_4\times S_3$ x 8, $C_6:C_4$ x 2

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

Order 18: $C_3\times S_3$ x 146, $C_3\times C_6$ x 115, $C_3:S_3$ x 14, $C_{18}$ x 5

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 6

Order 12: $C_2\times C_6$ x 88, $D_6$ x 40, $C_{12}$ x 20, $A_4$ x 14, $C_3:C_4$ x 4

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

Order 9: $C_3^2$ x 87, $C_9$ x 3

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

Order 6: $C_6$ x 68, $S_3$ x 24

Order 5: $C_5$

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

Order 3: $C_3$ x 20

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 116640: $C_3\wr S_3\times S_6$

Order 58320: $A_6\times C_3\wr S_3$, $C_3\wr C_3\times S_6$, $C_3\wr C_3:S_6$

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

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

Order 19440: $C_9:C_3\times S_6$, $C_3^3\times S_6$, $C_3^3:S_6$, $S_3\times C_3^2\times A_6$, $C_3\wr S_3\times S_5$, $\He_3\times S_6$, $\He_3:S_6$, $C_3^2:S_3\times A_6$

Order 12960: $C_3\times S_3\times S_6$ x 3, $A_6:C_6^2$

Order 11664: $C_3^5:(S_3\times D_4)$

Order 9720: $C_3\wr C_3\times S_5$, $A_5\times C_3\wr S_3$, $C_3\wr C_3:S_5$, $C_3^3\times A_6$, $\He_3\times A_6$, $C_9:C_3\times A_6$

Order 7776: $C_6^3:S_3^2$

Order 6480: $C_3^2\times S_6$ x 5, $C_3^2:S_6$ x 3, $C_3\times S_3\times A_6$ x 3, $C_3\times C_6\times A_6$, $C_9\times S_6$, $\GL(2,4):C_6^2$, $C_3^2:S_3\times S_5$

Order 5832: $(C_3^3\times S_3^2):C_6$, $(C_3^3\times S_3^2):S_3$, $C_3^5:D_{12}$, $C_3^5:(C_4\times S_3)$, $C_3^3:S_3^3$

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

Order 4320: $C_6\times S_6$ x 2, $S_3\times S_6$ x 2

Order 3888: $C_3\wr S_3\times S_4$ x 4, $A_4\times C_3^3:D_6$, $C_3^3:C_6\times S_4$, $\He_3:(C_6\times S_4)$, $C_3\times S_3^3:C_6$, $C_3^4:(S_3\times D_4)$

Order 3240: $C_3^2\times A_6$ x 4, $C_3^3\times S_5$, $C_9\times A_6$, $C_3\times S_3\times \GL(2,4)$, $C_3^3:S_5$, $C_9:C_3\times S_5$, $\He_3\times S_5$, $C_3^2:S_3\times A_5$, $\He_3:S_5$, $C_3\wr S_3\times F_5$

Order 2916: $C_3^5:D_6$, $C_3^5:D_6$, $(C_3^2\times \He_3):C_{12}$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^5:C_{12}$

Order 2592: $C_3^2:D_6\times S_4$, $C_6^3:D_6$, $C_6^3:D_6$

Order 2160: $C_3\times S_6$ x 6, $\GL(2,4):D_6$ x 3, $C_6\times A_6$ x 2, $C_3:S_6$ x 2, $S_3\times A_6$ x 2, $A_5:C_6^2$

Order 1944: $A_4\times C_3\wr S_3$ x 2, $C_3\wr C_3:S_4$ x 2, $C_3\wr C_3\times S_4$ x 2, $C_3^2\times S_3^3$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3^5:D_4$, $S_3\times C_3^3:C_{12}$, $C_3^2:S_3^3$, $C_3^4:D_{12}$, $\He_3:\SOPlus(4,2)$, $(C_9\times S_3^2):C_6$, $\He_3\times \SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$, $C_6^3:C_3^2$, $C_3\wr S_3\times D_6$

Order 1620: $(C_5\times \He_3):C_{12}$, $C_3\wr C_3\times F_5$, $C_3^3\times A_5$, $C_9:\GL(2,4)$, $\He_3\times A_5$, $C_3\wr S_3\times D_5$

Order 1458: $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1440: $C_2\times S_6$

Order 1296: $C_3\wr S_3\times D_4$ x 6, $C_6^2:S_3^2$ x 4, $C_6^2:S_3^2$ x 4, $S_3^3:C_6$ x 3, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^3:S_3$, $S_3^2:C_6^2$, $C_3^2:D_6\times A_4$, $C_2\times \He_3:S_4$, $C_9:C_6\times S_4$, $C_2\times S_4\times \He_3$, $C_6^3:S_3$, $(C_3^2\times A_4):D_6$, $C_6\wr S_3$, $C_6^3:C_6$, $C_6^3:S_3$, $C_2\times C_3^3:D_{12}$, $C_4\times C_3^3:D_6$

Order 1080: $C_3^2\times S_5$ x 5, $C_3\times A_6$ x 4, $S_3\times \GL(2,4)$ x 3, $C_3^2:S_5$ x 3, $S_3\times C_3^2\times F_5$, $C_6\times \GL(2,4)$, $C_3^2:S_3\times F_5$, $C_9\times S_5$

Order 972: $C_3^3:S_3^2$ x 6, $C_3^3:C_6^2$ x 2, $C_3^3\times S_3^2$ x 2, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.C_6^2$, $\He_3\times S_3^2$, $(C_3^2\times \He_3):C_4$, $C_3^4.C_{12}$, $C_3^4:C_{12}$, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^4:D_6$, $C_6^2.C_3^3$

Order 864: $C_6\times S_3\times S_4$ x 3, $S_4\times C_6^2$, $D_{12}:C_6^2$, $C_2\times C_6^2:D_6$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 729: $C_3^5:C_3$

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

Order 648: $C_3^4:D_4$ x 6, $C_2^2\times C_3\wr S_3$ x 6, $(C_3\times C_6^2):S_3$ x 4, $C_4\times C_3\wr S_3$ x 4, $C_3\times S_3^3$ x 3, $C_3^3:D_{12}$ x 3, $C_3\wr D_4$ x 3, $S_3\times C_3^2:C_{12}$ x 3, $C_3\wr C_3\times D_4$ x 3, $C_3^3:D_{12}$ x 3, $C_3^4:C_2^3$ x 2, $S_3\times C_3^2\times A_4$ x 2, $C_3^3:S_4$ x 2, $S_4\times C_3^3$ x 2, $A_4\times C_3^2:S_3$ x 2, $\He_3:S_4$ x 2, $C_9:C_3\times S_4$ x 2, $S_4\times \He_3$ x 2, $(C_3^2\times A_4):S_3$ x 2, $C_3^3:S_4$ x 2, $C_6^3:C_3$, $C_6\times C_3^2:C_{12}$, $C_9:C_6\times A_4$, $C_2\times A_4\times \He_3$, $S_3\times C_3^2:D_6$, $S_3^2:C_{18}$, $C_6^3:C_3$, $C_6\wr C_3$, $(C_3^2\times A_4):C_6$, $C_2\times \He_3:A_4$, $C_2\times C_3^3:C_{12}$, $C_2\times \He_3:C_{12}$

Order 540: $C_3\times \GL(2,4)$ x 4, $C_{15}:C_6^2$, $C_3^3:F_5$, $F_5\times C_3^3$, $\He_3:D_{10}$, $\He_3:F_5$, $C_{45}:C_{12}$, $F_5\times \He_3$, $C_9\times A_5$

Order 486: $C_3^4:C_6$ x 5, $C_3^4:S_3$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 2, $S_3\times C_3^4$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $C_6^2:D_6$ x 12, $A_4:C_6^2$ x 10, $C_{12}:C_6^2$ x 6, $C_6^2:D_6$ x 6, $C_6\times S_3\times A_4$ x 5, $C_6^2:D_6$ x 3, $C_6\times \SOPlus(4,2)$ x 2, $S_3^3:C_2$ x 2, $C_6^2:D_6$ x 2, $D_6\times C_6^2$, $A_4\times C_6^2$, $C_{12}:C_6^2$, $D_6:C_6^2$, $C_{12}:C_6^2$, $C_{12}:C_6^2$, $C_6^2:D_6$, $C_{18}\times S_4$, $C_{12}.C_6^2$, $C_{12}.C_6^2$, $C_6^2:D_6$, $C_6^2.D_6$, $C_6^2.D_6$

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

Order 360: $C_3\times S_5$ x 6, $D_{15}:C_{12}$ x 3, $C_6\times A_5$ x 2, $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $C_{30}:C_{12}$, $A_6$

Order 324: $C_3^2\times S_3^2$ x 16, $C_3^3:D_6$ x 9, $C_3^2:S_3^2$ x 6, $C_3^2:S_3^2$ x 6, $C_3^3:C_{12}$ x 5, $C_3^2.C_6^2$ x 3, $D_6\times C_3^3$ x 3, $C_3\wr C_2^2$ x 3, $C_3\wr C_4$ x 3, $C_3^3:C_{12}$ x 2, $C_3^2:C_6^2$ x 2, $\He_3:C_{12}$ x 2, $C_3^3:A_4$, $C_6^2.C_3^2$, $\He_3:A_4$, $A_4\times C_3^3$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $C_6^2.C_3^2$, $A_4\times \He_3$, $C_9\times S_3^2$, $C_3^2:C_{36}$

Order 288: $C_6^2:C_2^3$ x 3, $C_2\times C_6\times S_4$ x 2, $D_6\times S_4$ x 2, $D_4\times C_6^2$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

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: $C_2\times S_5$

Order 216: $C_3^2\times S_4$ x 12, $C_6^2:C_6$ x 10, $C_6^2:C_6$ x 9, $S_3^2:C_6$ x 8, $S_3\times C_6^2$ x 7, $C_3^2:S_4$ x 6, $C_6^2:S_3$ x 6, $C_6\times S_3^2$ x 5, $C_3^2:S_4$ x 4, $C_6^2:S_3$ x 4, $C_4\times C_3^2:S_3$ x 4, $C_6^2:C_6$ x 4, $C_6.C_6^2$ x 4, $C_{36}:C_6$ x 3, $D_4\times \He_3$ x 3, $C_3^2:D_{12}$ x 3, $D_4\times C_3^3$ x 3, $C_3^2\times D_{12}$ x 3, $C_9\times S_4$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $S_3^3$ x 2, $C_3^2:D_{12}$ x 2, $S_3^2:S_3$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_6^2:C_6$ x 2, $C_{18}:C_{12}$, $C_6.C_6^2$, $C_6^2.S_3$, $C_6^3$, $C_3\times C_6\times C_{12}$, $C_6.C_6^2$, $C_6.C_6^2$, $C_2^3\times \He_3$, $C_6^2.C_6$, $C_{18}:A_4$, $A_4\times C_{18}$

Order 180: $C_3^2\times F_5$ x 5, $\GL(2,4)$ x 4, $C_{15}:D_6$ x 3, $C_{15}:C_{12}$ x 3, $C_9\times F_5$, $C_3^2\times D_{10}$

Order 162: $S_3\times C_3^3$ x 18, $C_3^2\wr C_2$ x 13, $C_3\wr S_3$ x 8, $C_3^3:C_6$ x 5, $C_3^3:S_3$ x 4, $C_3^3:S_3$ x 4, $C_3^3.C_6$ x 4, $S_3\times \He_3$ x 4, $C_3^3:C_6$ x 3, $C_3^3\times C_6$ x 2, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^2:C_{18}$, $C_3^2:C_{18}$

Order 144: $C_{12}:D_6$ x 18, $C_6\times S_4$ x 17, $C_4:C_6^2$ x 12, $S_3\times S_4$ x 8, $A_4\times D_6$ x 5, $C_2^2:C_6^2$ x 4, $C_6^2:C_2^2$ x 3, $C_6^2:C_2^2$ x 3, $C_6\times D_{12}$ x 3, $C_{12}\times D_6$ x 3, $C_6:S_4$ x 2, $D_4\times C_{18}$, $C_2^2\times C_6^2$, $S_3^2:C_2^2$, $C_2\times C_6\times C_{12}$

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

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

Order 108: $C_3\times S_3^2$ x 28, $C_3^2\times D_6$ x 26, $C_3^2:D_6$ x 9, $C_3^2\times A_4$ x 7, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 5, $C_3\times C_6^2$ x 4, $C_3:S_3^2$ x 4, $C_{18}:C_6$ x 3, $C_2^2\times \He_3$ x 3, $C_3:S_3^2$ x 2, $C_3^3:C_4$ x 2, $C_3^2\times C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:A_4$ x 2, $C_9:C_{12}$ x 2, $C_4\times \He_3$ x 2, $\He_3:C_4$ x 2, $S_3\times C_{18}$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$

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

Order 90: $C_3^2\times D_5$ x 5, $C_3\times D_{15}$ x 3, $S_3\times C_{15}$ x 3, $C_9\times D_5$, $C_3\times C_{30}$

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 21, $C_6\times D_6$ x 20, $D_4\times C_3^2$ x 19, $C_3\times S_4$ x 18, $C_6\wr C_2$ x 12, $S_3\times C_{12}$ x 12, $C_2\times C_6^2$ x 10, $S_3\times A_4$ x 10, $C_6\times C_{12}$ x 9, $C_3\times D_{12}$ x 9, $C_3:S_4$ x 4, $S_3\times D_6$ x 3, $\SOPlus(4,2)$ x 3, $C_6:C_{12}$ x 3, $D_4\times C_9$ x 3, $C_2\times C_{36}$, $C_2\times C_3^2:C_4$, $C_2^2\times C_{18}$, $C_2^2:C_{18}$

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

Order 54: $S_3\times C_3^2$ x 53, $C_3^2:C_6$ x 20, $C_3^2\times C_6$ x 15, $C_3^2:S_3$ x 8, $C_9:C_6$ x 4, $C_2\times \He_3$ x 4, $S_3\times C_9$ x 3, $C_3^2:S_3$ x 2, $C_3\times C_{18}$

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

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

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 46, $C_6^2$ x 23, $C_3\times A_4$ x 13, $S_3^2$ x 13, $C_3\times C_{12}$ x 10, $C_3:C_{12}$ x 6, $C_2\times C_{18}$ x 3, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $C_{36}$ x 2, $C_2^2:C_9$

Order 32: $C_2^2\times D_4$

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

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 26, $C_2^2\times C_6$ x 16, $C_2\times C_{12}$ x 14, $C_2\times D_6$ x 13, $C_2\times A_4$ x 11, $C_3:D_4$ x 8, $C_4\times S_3$ x 8, $S_4$ x 8, $D_{12}$ x 6, $C_6:C_4$ x 2

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

Order 18: $C_3\times S_3$ x 50, $C_3\times C_6$ x 36, $C_3:S_3$ x 8, $C_{18}$ x 3

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

Order 15: $C_{15}$ x 4

Order 12: $C_2\times C_6$ x 29, $D_6$ x 22, $C_{12}$ x 12, $A_4$ x 5, $C_3:C_4$ x 4

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

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

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

Order 6: $C_6$ x 25, $S_3$ x 14

Order 5: $C_5$

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 116640: $C_3\wr S_3\times S_6$ ($C_1$)

Order 58320: $A_6\times C_3\wr S_3$ ($C_2$), $C_3\wr C_3\times S_6$ ($C_2$), $C_3\wr C_3:S_6$ ($C_2$)

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

Order 29160: $C_3\wr C_3\times A_6$ ($C_2^2$)

Order 19440: $C_3^3\times S_6$ ($S_3$), $\He_3\times S_6$ ($C_6$), $\He_3:S_6$ ($C_6$), $C_3^2:S_3\times A_6$ ($C_6$)

Order 9720: $C_3^3\times A_6$ ($D_6$), $\He_3\times A_6$ ($C_2\times C_6$)

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

Order 3240: $C_3^2\times A_6$ ($C_6\times S_3$)

Order 2160: $C_3\times S_6$ ($C_3^2:C_6$)

Order 1080: $C_3\times A_6$ ($C_3^2:D_6$)

Order 720: $S_6$ ($C_3\wr S_3$)

Order 360: $A_6$ ($C_3^3:D_6$)

Order 162: $C_3\wr S_3$ ($S_6$)

Order 81: $C_3\wr C_3$ ($C_2\times S_6$)

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

Order 27: $C_3^3$ ($S_3\times S_6$), $\He_3$ ($C_6\times S_6$)

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

Order 3: $C_3$ ($C_3^2:C_6\times S_6$)

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

Normal subgroups up to automorphism (quotient in parentheses)

Order 116640: $C_3\wr S_3\times S_6$ ($C_1$)

Order 58320: $A_6\times C_3\wr S_3$ ($C_2$), $C_3\wr C_3\times S_6$ ($C_2$), $C_3\wr C_3:S_6$ ($C_2$)

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

Order 29160: $C_3\wr C_3\times A_6$ ($C_2^2$)

Order 19440: $C_3^3\times S_6$ ($S_3$), $\He_3\times S_6$ ($C_6$), $\He_3:S_6$ ($C_6$), $C_3^2:S_3\times A_6$ ($C_6$)

Order 9720: $C_3^3\times A_6$ ($D_6$), $\He_3\times A_6$ ($C_2\times C_6$)

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

Order 3240: $C_3^2\times A_6$ ($C_6\times S_3$)

Order 2160: $C_3\times S_6$ ($C_3^2:C_6$)

Order 1080: $C_3\times A_6$ ($C_3^2:D_6$)

Order 720: $S_6$ ($C_3\wr S_3$)

Order 360: $A_6$ ($C_3^3:D_6$)

Order 162: $C_3\wr S_3$ ($S_6$)

Order 81: $C_3\wr C_3$ ($C_2\times S_6$)

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

Order 27: $C_3^3$ ($S_3\times S_6$), $\He_3$ ($C_6\times S_6$)

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

Order 3: $C_3$ ($C_3^2:C_6\times S_6$)

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

Series

Derived series

$C_3\wr S_3\times S_6$

$\rhd$

$C_3\wr S_3\times S_6$

$\rhd$

$\He_3\times A_6$

$\rhd$

$\He_3\times A_6$

$\rhd$

$C_3\times A_6$

$\rhd$

$C_3\times A_6$

$\rhd$

$A_6$

$\rhd$

$A_6$

Chief series

$C_3\wr S_3\times S_6$

$\rhd$

$C_3\wr S_3\times S_6$

$\rhd$

$A_6\times C_3\wr S_3$

$\rhd$

$A_6\times C_3\wr S_3$

$\rhd$

$C_3\wr S_3$

$\rhd$

$C_3\wr S_3$

$\rhd$

$C_3\wr C_3$

$\rhd$

$C_3\wr C_3$

$\rhd$

$\He_3$

$\rhd$

$\He_3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3\wr S_3\times S_6$

$\rhd$

$C_3\wr S_3\times S_6$

$\rhd$

$\He_3\times A_6$

$\rhd$

$\He_3\times A_6$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 1 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 $242 \times 242$ 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 $143 \times 143$ rational character table (warning: may be slow to load).