Abstract group 38880.bh: $C_9:C_6\times S_6$

• Label: 38880.bh
• Order: \( 2^{5} \cdot 3^{5} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{5} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: $54$
• Rank: $2$

Group information

Description:	$C_9:C_6\times S_6$
Order:	\(38880\)\(\medspace = 2^{5} \cdot 3^{5} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$D_9:C_3.A_6.C_2^2$, of order \(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \)
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	9	10	12	15	18	30	36	45
Elements	1	759	728	1800	144	12768	1458	1296	4680	1152	5670	2592	3240	2592	38880
Conjugacy classes	1	7	11	4	1	37	9	1	10	3	15	2	6	3	110
Divisions	1	7	8	4	1	24	6	1	6	2	10	1	4	2	77
Autjugacy classes	1	5	7	4	1	22	6	1	10	3	9	2	6	3	80

Dimension	1	2	4	5	6	9	10	16	18	20	30	32	36	40	54	60	64	96
Irr. complex chars.	12	6	0	24	2	12	24	6	6	6	4	3	0	0	2	2	0	1	110
Irr. rational chars.	4	6	2	8	2	4	16	2	6	10	4	3	2	2	2	2	1	1	77

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$54$
Rank:	$2$
Inequivalent generating pairs:	$2544$

Minimal degrees of faithful linear representations

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

Constructions

Permutation group:	Degree $15$ $\langle(7,14,10,9,15,11,8,13,12), (5,6)(8,9)(10,15)(11,14)(12,13), (7,11,8,12,9,10) \!\cdots\! \rangle$
Direct product:	$(C_9:C_6)$ $\, \times\, $ $S_6$
Semidirect product:	$(D_9\times S_6)$ $\,\rtimes\,$ $C_3$	$D_9$ $\,\rtimes\,$ $(C_3\times S_6)$	$(C_9\times S_6)$ $\,\rtimes\,$ $C_6$	$(C_9:S_6)$ $\,\rtimes\,$ $C_6$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^2\times S_6)$ . $S_3$	$C_3^2$ . $(S_3\times S_6)$	$(C_3^2\times A_6)$ . $D_6$	$C_3$ . $(C_3\times S_3\times S_6)$	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 176310 subgroups in 1464 conjugacy classes, 23 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_9:C_6\times S_6$
Commutator:	$G' \simeq$ $C_9\times A_6$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_3\times S_3\times S_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_9:C_3$	$G/\operatorname{Fit} \simeq$ $C_2\times S_6$
Radical:	$R \simeq$ $C_9:C_6$	$G/R \simeq$ $S_6$
Socle:	$\operatorname{soc} \simeq$ $C_3\times A_6$	$G/\operatorname{soc} \simeq$ $C_6\times S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_3^3$
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 38880: $C_9:C_6\times S_6$

Order 19440: $C_9:C_3\times S_6$, $C_9:(C_3\times S_6)$, $C_9:C_6\times A_6$

Order 12960: $D_9\times S_6$, $C_3\times S_3\times S_6$

Order 9720: $C_9:C_3\times A_6$

Order 6480: $C_9\times S_6$ x 2, $C_9:C_6\times S_5$ x 2, $C_9:S_6$, $D_9\times A_6$, $C_3^2\times S_6$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

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

Order 3888: $(D_9\times S_3^2):C_6$

Order 3240: $C_9\times A_6$ x 2, $C_9:C_3\times S_5$ x 2, $D_9:\GL(2,4)$ x 2, $C_9:(C_3\times S_5)$ x 2, $C_3^2\times A_6$

Order 2592: $C_{18}:C_6\times S_4$ x 2

Order 2160: $C_3\times S_6$ x 3, $D_9\times S_5$ x 2, $\GL(2,4):D_6$ x 2, $C_6\times A_6$, $C_3:S_6$, $S_3\times A_6$

Order 1944: $C_3^2.S_3^3$ x 2, $(C_9\times S_3^2):C_6$ x 2, $C_3^4.D_{12}$, $(C_9\times S_3^2):C_6$, $(C_3^2\times D_9):C_{12}$

Order 1620: $C_9:\GL(2,4)$ x 2

Order 1440: $C_2\times S_6$

Order 1296: $C_6^2.S_3^2$ x 8, $A_4\times C_{18}:C_6$ x 2, $C_9:(C_6\times S_4)$ x 2, $C_9:C_6\times S_4$ x 2, $S_3^3:C_6$, $S_3^2:D_{18}$

Order 1080: $C_9\times S_5$ x 4, $S_3\times \GL(2,4)$ x 2, $C_3^2\times S_5$ x 2, $C_3^2:S_5$ x 2, $C_3\times A_6$ x 2, $D_9\times A_5$ x 2, $C_9:S_5$ x 2, $D_{45}:C_{12}$

Order 972: $C_3^3.S_3^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.C_6^2$ x 2, $C_3^3.S_3^2$ x 2, $C_3^3.S_3^2$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$

Order 864: $C_6\times S_3\times S_4$ x 2, $S_4\times D_{18}$ x 2, $C_2\times D_{36}:C_6$

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: $A_4\times C_9:C_6$ x 4, $C_9:(C_3\times S_4)$ x 4, $C_9:C_3\times S_4$ x 4, $C_3\times S_3^3$ x 2, $C_3\wr D_4$ x 2, $C_9:C_6\times A_4$ x 2, $C_{18}:C_6\times S_3$ x 2, $D_9\times S_3^2$ x 2, $S_3^2:D_9$ x 2, $S_3^2:C_{18}$ x 2, $C_3^3:D_{12}$, $C_3^4:D_4$, $S_3\times C_3^2:C_{12}$, $C_3^2:D_{36}$, $C_3^2:C_4\times D_9$

Order 540: $C_9\times A_5$ x 4, $C_3\times \GL(2,4)$ x 2, $D_{45}:C_6$, $C_{45}:C_{12}$, $C_{45}:C_{12}$

Order 486: $C_3^4.S_3$ x 2, $C_3^4.C_6$ x 2, $C_3^4.S_3$, $C_3^4.C_6$, $D_9:C_3^3$

Order 432: $C_6^2:D_6$ x 8, $D_9\times S_4$ x 8, $D_{36}:C_6$ x 8, $C_{18}\times S_4$ x 4, $C_6\times S_3\times A_4$ x 2, $C_6^2:D_6$ x 2, $A_4:C_6^2$ x 2, $C_6^2.D_6$ x 2, $A_4\times D_{18}$ x 2, $D_{18}:A_4$ x 2, $C_{18}:S_4$ x 2, $C_6^2.D_6$ x 2, $C_6^2.D_6$ x 2, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $C_{12}.C_6^2$, $D_{36}:C_6$, $D_{18}:C_{12}$

Order 360: $C_3\times S_5$ x 6, $C_6\times A_5$ x 2, $S_3\times A_5$ x 2, $C_3:S_5$ x 2, $D_9\times F_5$, $D_{15}:C_{12}$, $A_6$

Order 324: $C_3^2.S_3^2$ x 12, $C_3^2\times S_3^2$ x 4, $C_9\times S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3\wr C_2^2$ x 2, $C_3^3.D_6$ x 2, $C_3^2.C_6^2$ x 2, $C_9:C_6^2$ x 2, $C_6^2.C_3^2$ x 2, $C_9:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:D_{18}$ x 2, $C_3^2:C_{36}$ x 2, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$

Order 288: $C_2\times C_6\times S_4$ x 2, $D_6\times S_4$ x 2, $C_6^2:C_2^3$, $D_4\times D_{18}$

Order 270: $C_{45}:C_6$, $C_{45}:C_6$, $C_9:C_{30}$

Order 243: $C_9:C_3^3$

Order 240: $C_2\times S_5$ x 2

Order 216: $D_{18}:C_6$ x 11, $C_9\times S_4$ x 8, $D_{18}:C_6$ x 8, $S_3^2:C_6$ x 5, $A_4\times D_9$ x 4, $D_9:A_4$ x 4, $C_9:S_4$ x 4, $C_3^2.S_4$ x 4, $C_{36}:C_6$ x 4, $C_{36}:C_6$ x 4, $C_{36}:C_6$ x 4, $C_6\times S_3^2$ x 4, $C_6^2:C_6$ x 4, $C_3^2:S_4$ x 4, $C_3^2\times S_4$ x 4, $C_{18}:A_4$ x 4, $A_4\times C_{18}$ x 4, $C_6^2:C_6$ x 2, $S_3^3$ x 2, $S_3^2:S_3$ x 2, $C_6.C_6^2$ x 2, $C_6^2.C_6$ x 2, $S_3\times D_{18}$ x 2, $C_{18}:C_{12}$, $C_{18}:C_{12}$, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$, $C_3^2:C_4\times S_3$

Order 180: $\GL(2,4)$ x 4, $C_9\times F_5$ x 2, $D_5\times D_9$, $C_{45}:C_4$, $C_{15}:D_6$, $C_{15}:C_{12}$, $C_3^2\times F_5$

Order 162: $C_3^3.C_6$ x 10, $C_3^3.S_3$ x 8, $D_9:C_3^2$ x 8, $C_3^2:C_{18}$ x 4, $C_3^2\wr C_2$ x 3, $S_3\times C_3^3$ x 3, $C_{18}:C_3^2$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:D_9$ x 2, $C_3^3:C_6$, $C_3^2:D_9$, $C_3^2\times D_9$

Order 144: $C_6\times S_4$ x 16, $D_4\times D_9$ x 8, $S_3\times S_4$ x 8, $C_{12}:D_6$ x 8, $A_4\times D_6$ x 4, $D_4\times C_{18}$ x 2, $C_{18}:D_4$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2:C_6^2$ x 2, $C_6:S_4$ x 2, $C_6^2:C_2^2$ x 2, $C_2^2\times D_{18}$ x 2, $C_2^2:D_{18}$ x 2, $C_2\times D_{36}$, $C_4\times D_{18}$, $S_3^2:C_2^2$, $C_4:C_6^2$, $C_6\times D_{12}$, $C_{12}\times D_6$

Order 135: $C_9:C_{15}$

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

Order 108: $C_3\times S_3^2$ x 22, $C_{18}:C_6$ x 16, $S_3\times D_9$ x 10, $C_3^2\times D_6$ x 6, $C_{18}:C_6$ x 5, $S_3\times C_{18}$ x 4, $C_9:A_4$ x 4, $C_9\times A_4$ x 4, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 3, $C_3^2:C_{12}$ x 3, $C_9:C_{12}$ x 2, $C_3^2\times A_4$ x 2, $C_3:S_3^2$ x 2, $C_3:D_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^2.A_4$ x 2, $C_9:C_{12}$ x 2, $C_3^3:C_4$

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

Order 90: $C_9\times D_5$ x 2, $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3^2\times D_5$, $D_{45}$, $C_5\times D_9$

Order 81: $C_9:C_3^2$ x 8, $C_3^2\times C_9$ x 2, $C_3^4$

Order 72: $C_3\times S_4$ x 16, $C_6\times D_6$ x 13, $C_6\times A_4$ x 12, $C_2\times D_{18}$ x 11, $C_9:D_4$ x 8, $S_3\times A_4$ x 8, $C_6\wr C_2$ x 8, $D_4\times C_9$ x 8, $D_{36}$ x 4, $C_4\times D_9$ x 4, $S_3\times D_6$ x 4, $C_3:S_4$ x 4, $\SOPlus(4,2)$ x 4, $D_4\times C_3^2$ x 4, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_2^2\times C_{18}$ x 4, $C_2^2:C_{18}$ x 4, $C_2^2:D_9$ x 4, $C_2\times C_{36}$ x 2, $C_2\times C_6^2$ x 2, $C_{18}:C_4$, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $C_6:C_{12}$

Order 60: $C_3\times F_5$ x 3, $A_5$ x 2, $S_3\times D_5$, $C_{15}:C_4$, $C_3\times D_{10}$

Order 54: $S_3\times C_3^2$ x 23, $C_9:C_6$ x 14, $C_3^2:C_6$ x 13, $S_3\times C_9$ x 12, $C_9:C_6$ x 9, $C_3:D_9$ x 6, $C_3\times D_9$ x 6, $C_3\times C_{18}$ x 4, $C_3^2\times C_6$ x 3, $C_3^2:S_3$

Order 48: $C_2\times S_4$ x 14, $C_6\times D_4$ x 13, $S_3\times D_4$ x 8, $C_2^2\times A_4$ x 4, $C_2^3\times C_6$ x 2, $C_2^2\times D_6$ x 2, $C_6:D_4$ x 2, $C_2^2\times C_{12}$, $C_2\times D_{12}$, $C_4\times D_6$

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

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 36, $S_3^2$ x 16, $D_{18}$ x 14, $C_2\times C_{18}$ x 10, $C_3\times A_4$ x 8, $C_6^2$ x 7, $C_2^2:C_9$ x 4, $C_{36}$ x 4, $C_6:S_3$ x 3, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_9:C_4$ 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: $C_9:C_3$ x 11, $C_3^3$ x 8, $C_3\times C_9$ x 8

Order 24: $C_3\times D_4$ x 20, $C_2^2\times C_6$ x 15, $C_2\times A_4$ x 14, $C_2\times D_6$ x 13, $S_4$ x 12, $C_3:D_4$ x 8, $C_2\times C_{12}$ x 7, $D_{12}$ x 4, $C_4\times S_3$ x 4, $C_6:C_4$

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

Order 18: $C_3\times S_3$ x 42, $C_3\times C_6$ x 18, $C_{18}$ x 10, $C_3:S_3$ x 8, $D_9$ x 8

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 2

Order 12: $C_2\times C_6$ x 26, $D_6$ x 26, $A_4$ x 6, $C_{12}$ x 6, $C_3:C_4$ x 2

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

Order 9: $C_3^2$ x 16, $C_9$ x 6

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

Order 6: $C_6$ x 24, $S_3$ x 16

Order 5: $C_5$

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

Order 3: $C_3$ x 8

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 38880: $C_9:C_6\times S_6$

Order 19440: $C_9:C_3\times S_6$, $C_9:(C_3\times S_6)$, $C_9:C_6\times A_6$

Order 12960: $D_9\times S_6$, $C_3\times S_3\times S_6$

Order 9720: $C_9:C_3\times A_6$

Order 6480: $C_9\times S_6$ x 2, $C_9:S_6$, $D_9\times A_6$, $C_3^2\times S_6$, $C_9:C_6\times S_5$, $C_3^2:S_6$, $C_3\times S_3\times A_6$

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

Order 3888: $(D_9\times S_3^2):C_6$

Order 3240: $C_9\times A_6$ x 2, $C_3^2\times A_6$, $C_9:C_3\times S_5$, $D_9:\GL(2,4)$, $C_9:(C_3\times S_5)$

Order 2592: $C_{18}:C_6\times S_4$

Order 2160: $C_3\times S_6$ x 3, $D_9\times S_5$, $C_6\times A_6$, $\GL(2,4):D_6$, $C_3:S_6$, $S_3\times A_6$

Order 1944: $C_3^2.S_3^3$, $C_3^4.D_{12}$, $(C_9\times S_3^2):C_6$, $(C_9\times S_3^2):C_6$, $(C_3^2\times D_9):C_{12}$

Order 1620: $C_9:\GL(2,4)$

Order 1440: $C_2\times S_6$

Order 1296: $C_6^2.S_3^2$ x 4, $S_3^3:C_6$, $A_4\times C_{18}:C_6$, $C_9:(C_6\times S_4)$, $C_9:C_6\times S_4$, $S_3^2:D_{18}$

Order 1080: $C_3\times A_6$ x 2, $C_9\times S_5$ x 2, $S_3\times \GL(2,4)$, $C_3^2\times S_5$, $C_3^2:S_5$, $D_{45}:C_{12}$, $D_9\times A_5$, $C_9:S_5$

Order 972: $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.C_6^2$, $C_3^3.S_3^2$, $(C_3^2\times C_9):C_{12}$, $C_3^4.C_{12}$

Order 864: $C_6\times S_3\times S_4$, $S_4\times D_{18}$, $C_2\times D_{36}:C_6$

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

Order 648: $A_4\times C_9:C_6$ x 2, $C_9:(C_3\times S_4)$ x 2, $C_9:C_3\times S_4$ x 2, $S_3^2:C_{18}$ x 2, $C_3\times S_3^3$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^4:D_4$, $S_3\times C_3^2:C_{12}$, $C_9:C_6\times A_4$, $C_{18}:C_6\times S_3$, $D_9\times S_3^2$, $C_3^2:D_{36}$, $S_3^2:D_9$, $C_3^2:C_4\times D_9$

Order 540: $C_9\times A_5$ x 2, $C_3\times \GL(2,4)$, $D_{45}:C_6$, $C_{45}:C_{12}$, $C_{45}:C_{12}$

Order 486: $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4.S_3$, $C_3^4.C_6$, $D_9:C_3^3$

Order 432: $D_{36}:C_6$ x 6, $C_6^2:D_6$ x 4, $D_9\times S_4$ x 4, $C_{18}\times S_4$ x 2, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $A_4:C_6^2$, $C_6\times \SOPlus(4,2)$, $S_3^3:C_2$, $C_6^2.D_6$, $A_4\times D_{18}$, $D_{18}:A_4$, $C_{18}:S_4$, $C_6^2.D_6$, $C_{12}.C_6^2$, $C_6^2.D_6$, $D_{36}:C_6$, $D_{18}:C_{12}$

Order 360: $C_3\times S_5$ x 3, $D_9\times F_5$, $D_{15}:C_{12}$, $C_6\times A_5$, $S_3\times A_5$, $C_3:S_5$, $A_6$

Order 324: $C_3^2.S_3^2$ x 6, $C_3^2:S_3^2$ x 2, $C_3^2\times S_3^2$ x 2, $C_9\times S_3^2$ x 2, $C_3^2:C_{36}$ x 2, $C_3\wr C_2^2$, $C_3\wr C_4$, $C_3^3:C_{12}$, $C_3^3.D_6$, $C_3^2.C_6^2$, $C_9:C_6^2$, $C_6^2.C_3^2$, $C_9:S_3^2$, $C_9:S_3^2$, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $(C_3^2\times C_9):C_4$

Order 288: $C_6^2:C_2^3$, $D_4\times D_{18}$, $C_2\times C_6\times S_4$, $D_6\times S_4$

Order 270: $C_{45}:C_6$, $C_{45}:C_6$, $C_9:C_{30}$

Order 243: $C_9:C_3^3$

Order 240: $C_2\times S_5$

Order 216: $D_{18}:C_6$ x 6, $C_9\times S_4$ x 4, $D_{18}:C_6$ x 4, $C_{36}:C_6$ x 4, $S_3^2:C_6$ x 4, $C_{36}:C_6$ x 3, $C_{36}:C_6$ x 3, $A_4\times D_9$ x 2, $D_9:A_4$ x 2, $C_9:S_4$ x 2, $C_3^2.S_4$ 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, $C_{18}:A_4$ x 2, $A_4\times C_{18}$ x 2, $C_{18}:C_{12}$, $C_{18}:C_{12}$, $C_6^2:C_6$, $C_2\times C_3^2:C_{12}$, $S_3^3$, $C_3^2:D_{12}$, $S_3^2:S_3$, $C_3^2:C_4\times S_3$, $C_6.C_6^2$, $C_6^2.C_6$, $S_3\times D_{18}$

Order 180: $C_9\times F_5$ x 2, $\GL(2,4)$ x 2, $D_5\times D_9$, $C_{45}:C_4$, $C_{15}:D_6$, $C_{15}:C_{12}$, $C_3^2\times F_5$

Order 162: $C_3^3.C_6$ x 5, $C_3^3.S_3$ x 4, $D_9:C_3^2$ x 4, $C_3^2\wr C_2$ x 2, $S_3\times C_3^3$ x 2, $C_3^2:C_{18}$ x 2, $C_3^2:C_{18}$ x 2, $C_3^3:C_6$, $C_{18}:C_3^2$, $C_3^2:D_9$, $C_3^2:D_9$, $C_3^2\times D_9$

Order 144: $C_6\times S_4$ x 8, $D_4\times D_9$ x 6, $C_{12}:D_6$ x 6, $S_3\times S_4$ x 4, $D_4\times C_{18}$ x 2, $A_4\times D_6$ x 2, $C_{18}:D_4$, $C_2\times D_{36}$, $C_4\times D_{18}$, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $C_6:S_4$, $S_3^2:C_2^2$, $C_4:C_6^2$, $C_6^2:C_2^2$, $C_6\times D_{12}$, $C_{12}\times D_6$, $C_2^2\times D_{18}$, $C_2^2:D_{18}$

Order 135: $C_9:C_{15}$

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, $C_{18}:C_6$ x 9, $S_3\times D_9$ x 5, $C_3^2\times D_6$ x 3, $C_3^2:C_{12}$ x 3, $C_{18}:C_6$ x 3, $C_9:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3:S_3^2$ x 2, $S_3\times C_{18}$ x 2, $C_9:A_4$ x 2, $C_9\times A_4$ x 2, $C_9:C_{12}$ x 2, $C_3^2\times A_4$, $C_3:S_3^2$, $C_3^3:C_4$, $C_3:D_{18}$, $C_3\times D_{18}$, $C_3^2.A_4$

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

Order 90: $C_9\times D_5$ x 2, $C_3\times D_{15}$, $S_3\times C_{15}$, $C_3^2\times D_5$, $D_{45}$, $C_5\times D_9$

Order 81: $C_9:C_3^2$ x 4, $C_3^2\times C_9$ 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, $C_2\times D_{18}$ x 6, $D_4\times C_9$ x 6, $C_9:D_4$ x 4, $C_4\times D_9$ x 4, $S_3\times A_4$ x 4, $C_6\wr C_2$ x 4, $S_3\times C_{12}$ x 4, $D_{36}$ x 3, $\SOPlus(4,2)$ x 3, $D_4\times C_3^2$ x 3, $C_3\times D_{12}$ x 3, $C_2\times C_{36}$ x 2, $S_3\times D_6$ x 2, $C_3:S_4$ x 2, $C_2^2\times C_{18}$ x 2, $C_2^2:C_{18}$ x 2, $C_2^2:D_9$ x 2, $C_{18}:C_4$, $C_2\times C_6^2$, $C_2\times C_3^2:C_4$, $C_6\times C_{12}$, $C_6:C_{12}$

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, $C_9:C_6$ x 8, $C_3^2:C_6$ x 8, $S_3\times C_9$ x 6, $C_9:C_6$ x 5, $C_3:D_9$ x 3, $C_3\times D_9$ x 3, $C_3\times C_{18}$ x 2, $C_3^2\times C_6$ 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: $C_{45}$ x 2, $C_3\times C_{15}$

Order 40: $C_2\times F_5$

Order 36: $C_6\times S_3$ x 19, $D_{18}$ x 8, $S_3^2$ x 8, $C_2\times C_{18}$ x 6, $C_{36}$ x 4, $C_6^2$ 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, $C_2^2:C_9$ x 2, $C_6:S_3$ x 2, $C_9:C_4$ 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: $C_9:C_3$ x 6, $C_3^3$ x 5, $C_3\times C_9$ x 4

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, $C_{18}$ x 6, $C_3:S_3$ x 5, $D_9$ 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, $C_9$ x 4

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: $C_9:C_6\times S_6$ ($C_1$)

Order 19440: $C_9:C_3\times S_6$ ($C_2$), $C_9:(C_3\times S_6)$ ($C_2$), $C_9:C_6\times A_6$ ($C_2$)

Order 12960: $D_9\times S_6$ ($C_3$)

Order 9720: $C_9:C_3\times A_6$ ($C_2^2$)

Order 6480: $C_9\times S_6$ ($C_6$), $C_9:S_6$ ($C_6$), $D_9\times A_6$ ($C_6$), $C_3^2\times S_6$ ($S_3$)

Order 3240: $C_9\times A_6$ ($C_2\times C_6$), $C_3^2\times A_6$ ($D_6$)

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

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

Order 720: $S_6$ ($C_9:C_6$)

Order 360: $A_6$ ($C_{18}:C_6$)

Order 54: $C_9:C_6$ ($S_6$)

Order 27: $C_9:C_3$ ($C_2\times S_6$)

Order 18: $D_9$ ($C_3\times S_6$)

Order 9: $C_3^2$ ($S_3\times S_6$), $C_9$ ($C_6\times S_6$)

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

Order 1: $C_1$ ($C_9:C_6\times S_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 38880: $C_9:C_6\times S_6$ ($C_1$)

Order 19440: $C_9:C_3\times S_6$ ($C_2$), $C_9:(C_3\times S_6)$ ($C_2$), $C_9:C_6\times A_6$ ($C_2$)

Order 12960: $D_9\times S_6$ ($C_3$)

Order 9720: $C_9:C_3\times A_6$ ($C_2^2$)

Order 6480: $C_9\times S_6$ ($C_6$), $C_9:S_6$ ($C_6$), $D_9\times A_6$ ($C_6$), $C_3^2\times S_6$ ($S_3$)

Order 3240: $C_9\times A_6$ ($C_2\times C_6$), $C_3^2\times A_6$ ($D_6$)

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

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

Order 720: $S_6$ ($C_9:C_6$)

Order 360: $A_6$ ($C_{18}:C_6$)

Order 54: $C_9:C_6$ ($S_6$)

Order 27: $C_9:C_3$ ($C_2\times S_6$)

Order 18: $D_9$ ($C_3\times S_6$)

Order 9: $C_3^2$ ($S_3\times S_6$), $C_9$ ($C_6\times S_6$)

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

Order 1: $C_1$ ($C_9:C_6\times S_6$)

Series

Derived series

$C_9:C_6\times S_6$

$\rhd$

$C_9\times A_6$

$\rhd$

$A_6$

Chief series

$C_9:C_6\times S_6$

$\rhd$

$C_9:C_6\times A_6$

$\rhd$

$C_9:C_6$

$\rhd$

$C_9:C_3$

$\rhd$

$C_9$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_9:C_6\times S_6$

$\rhd$

$C_9\times A_6$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 2 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 $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $77 \times 77$ rational character table.