Abstract group 11664.li: $C_3^4.D_6^2$

• Label: 11664.li
• Order: \( 2^{4} \cdot 3^{6} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $31$
• Trans deg.: $108$
• Rank: $4$

Group information

Description:	$C_3^4.D_6^2$
Order:	\(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$(C_2\times C_6^3).C_3^4.C_2^3$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \)
Composition factors:	$C_2$ x 4, $C_3$ x 6
Derived length:	$3$

This group is nonabelian and supersolvable (hence solvable and monomial).

Group statistics

Order	1	2	3	6	9	18
Elements	1	543	350	7314	378	3078	11664
Conjugacy classes	1	15	13	95	8	36	168
Divisions	1	15	11	73	6	26	132
Autjugacy classes	1	4	13	27	8	11	64

Dimension	1	2	3	4	6	12	18	24	36
Irr. complex chars.	16	40	32	16	40	12	8	0	4	168
Irr. rational chars.	16	40	0	16	24	20	8	4	4	132

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$108$
Rank:	$4$
Inequivalent generating quadruples:	$16765192080$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{3}=c^{3}=d^{6}=e^{6}=f^{18}=[a,c]= \!\cdots\! \rangle}$
Permutation group:	Degree $31$ $\langle(28,29)(30,31), (28,30)(29,31), (1,2,5,11,14,15,4,7,13)(3,6,12,22,24,25,10,16,23) \!\cdots\! \rangle$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $(C_3^4.S_3^2)$
Semidirect product:	$(C_3^4.D_6)$ $\,\rtimes\,$ $D_6$	$(C_3^4:D_6)$ $\,\rtimes\,$ $D_6$	$(C_3^3.C_6^2)$ $\,\rtimes\,$ $D_6$	$(C_3^3.C_6^2)$ $\,\rtimes\,$ $D_6$	all 38
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $D_6^2$	$(C_3^3:C_6^2)$ . $D_6$	$C_3^3$ . $(C_3:D_6^2)$	$C_3^3$ . $(C_6^2:D_6)$	all 30
Aut. group:	$\Aut(C_{12}.D_9)$	$\Aut(C_{12}:D_9)$	$\Aut(C_6.D_{18})$	$\Aut((C_3\times C_9):C_{12})$	all 72

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 119112 subgroups in 4696 conjugacy classes, 229 normal (47 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_3^4.S_3^2$
Commutator:	$G' \simeq$ $C_3^3.C_3^3$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_3^3$	$G/\Phi \simeq$ $C_3:D_6^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_3^3.C_3^3$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^4.D_6^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3^3:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3.C_3^3$

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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Subgroup information

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Classes of subgroups up to conjugation

Order 11664: $C_3^4.D_6^2$

Order 5832: $C_2\times C_3^4.S_3^2$ x 12, $C_2^2\times (C_3^3.C_3^3):C_2$, $C_2^2\times (C_3^3.C_3^3):C_2$, $C_2^2\times (C_3^4.C_3^2):C_2$

Order 3888: $C_2^2\times (C_3\times C_9):C_3^2:C_2^2$, $C_2^2\times (C_9.C_3^3):C_2^2$, $C_2^2\times (C_3^4.C_3):C_2^2$, $C_2^2\times C_3^4:C_3:C_2^2$, $C_2^2\times \He_3:C_3^2:C_2^2$

Order 2916: $C_3^4.S_3^2$ x 16, $C_2\times (C_3^4.C_3^2):C_2$ x 6, $C_2\times (C_3^3.C_3^3):C_2$ x 6, $C_2\times (C_3^3.C_3^3):C_2$ x 6, $C_2^2\times C_3^3.C_3^3$

Order 1944: $C_2\times C_3^3.S_3^2$ x 12, $C_2\times C_3^3:S_3^2$ x 12, $C_2\times C_3^3.S_3^2$ x 12, $C_2\times C_3^3.S_3^2$ x 12, $C_2\times C_3^3.S_3^2$ x 12, $(C_3\times C_{18}):C_6^2$ x 2, $\He_3.(C_6\times D_6)$ x 2, $C_3^3:(C_6\times D_6)$ x 2, $C_2\times \He_3.C_6^2$, $C_2\times C_3^4:D_6$, $\He_3:(C_6\times D_6)$, $C_6\times C_3^3:D_6$, $C_2\times C_3^4.D_6$, $C_2\times C_3^4:D_6$, $(C_3\times C_{18}):C_6^2$, $C_2\times \He_3.C_6^2$, $(C_3\times C_{18}):C_6^2$, $C_2\times \He_3.C_6^2$, $(C_3\times C_{18}):C_6^2$, $C_2\times C_3^3.C_6^2$, $C_2^2\times C_3^4:C_6$

Order 1458: $C_3^4:C_3:S_3$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $(C_3\times C_{18}):\He_3$ x 3

Order 1296: $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 2, $C_6^2:S_3^2$, $C_9.D_6^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2:D_{18}$

Order 972: $C_3^3.S_3^2$ x 16, $C_3^3:S_3^2$ x 16, $C_3^3.S_3^2$ x 16, $C_3^3.S_3^2$ x 16, $C_3^3.S_3^2$ x 16, $(C_3\times C_9):C_6^2$ x 12, $\He_3.(C_6\times S_3)$ x 12, $C_3^3:(C_6\times S_3)$ x 12, $\He_3.C_6^2$ x 6, $C_3^4:D_6$ x 6, $\He_3:(C_6\times S_3)$ x 6, $C_3^4:D_6$ x 6, $C_3^4.D_6$ x 6, $C_3^4:D_6$ x 6, $(C_3\times C_9):C_6^2$ x 6, $\He_3.C_6^2$ x 6, $(C_3\times C_9):C_6^2$ x 6, $\He_3.C_6^2$ x 6, $C_3^4.D_6$ x 6, $C_3^3.C_6^2$ x 6, $C_3^3.C_6^2$ x 6, $C_3^3.C_6^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^3.C_6^2$, $\He_3:C_6^2$, $C_3^3:C_6^2$, $C_3^3.C_6^2$, $C_3^3:C_6^2$

Order 729: $C_3^3.C_3^3$

Order 648: $C_6\times C_3^2:D_6$ x 36, $C_{18}:C_6\times S_3$ x 24, $C_3^4:C_2^3$ x 12, $C_{18}.S_3^2$ x 12, $S_3\times C_3^2:D_6$ x 12, $(C_3\times C_{18}):D_6$ x 12, $(C_3\times C_{18}):D_6$ x 12, $C_2\times C_3^2:D_{18}$ x 12, $C_3\times C_6^2:C_6$ x 7, $C_3\times C_6^2:S_3$ x 5, $C_{18}:C_6^2$ x 5, $C_3^4:C_2^3$ x 3, $(C_6\times C_{18}):C_6$ x 3, $C_{18}:C_6\times S_3$ x 3, $C_2\times C_3^3:D_6$ x 3, $(C_6\times C_{18}):S_3$ x 2, $C_{18}:(C_6\times S_3)$ x 2, $(C_6\times C_{18}):S_3$ x 2, $(C_6\times C_{18}):S_3$ x 2, $(C_6\times C_{18}):S_3$ x 2, $C_6^2:C_{18}$ x 2, $C_2\times C_3^3:D_6$, $C_2\times D_6\times \He_3$, $(C_6\times C_{18}):S_3$, $C_6^2:D_9$, $(C_6\times C_{18}):C_6$, $(C_6\times C_{18}):C_6$, $C_6^2:D_9$

Order 486: $C_9.(S_3\times C_3^2)$ x 8, $(C_3\times \He_3).S_3$ x 8, $C_3\wr C_3:C_6$ x 8, $(C_3\times C_{18}):C_3^2$ x 6, $C_2\times C_3^3:C_3^2$ x 6, $C_3^3.(C_3\times S_3)$ x 4, $C_3^4:S_3$ x 4, $(C_3\times \He_3):S_3$ x 4, $C_3^4:S_3$ x 4, $C_3^4.S_3$ x 4, $C_3^4:S_3$ x 4, $C_3^3.(C_3\times S_3)$ x 4, $C_3^3.(C_3\times S_3)$ x 4, $C_3^3.(C_3\times S_3)$ x 4, $C_3^3.(C_3\times S_3)$ x 4, $C_3^4.S_3$ x 4, $C_3^4.C_6$ x 4, $C_3^4:C_6$ x 4, $C_{18}.C_3^3$ x 3, $(C_3\times C_{18}):C_3^2$ x 3, $C_3^4:C_6$ x 3, $C_3^4.C_6$ x 3, $C_3^4:C_6$ x 3

Order 432: $C_3\times D_6^2$ x 6, $C_6^2:D_6$ x 3, $D_6\times D_{18}$ x 2, $C_3:D_6^2$, $C_6^2:D_6$, $C_6^2.D_6$

Order 324: $C_3^3:D_6$ x 48, $C_3^2:C_6^2$ x 42, $C_3^2.S_3^2$ x 32, $C_3^3:D_6$ x 30, $C_9:C_6^2$ x 30, $C_3^3:D_6$ x 18, $C_3^2:C_6^2$ x 18, $\He_3.D_6$ x 18, $C_3^2.C_6^2$ x 18, $C_3^2.S_3^2$ x 16, $C_3^2.S_3^2$ x 16, $C_3^2:D_{18}$ x 16, $C_3\wr C_2^2$ x 16, $C_9.S_3^2$ x 16, $C_3^2:S_3^2$ x 16, $\He_3.D_6$ x 12, $(C_3\times C_{18}):S_3$ x 12, $(C_3\times C_{18}):S_3$ x 12, $C_2\times C_3^2:C_{18}$ x 12, $(C_3\times C_{18}):S_3$ x 12, $C_3^3.D_6$ x 12, $C_3^2.C_6^2$ x 6, $\He_3:D_6$ x 6, $C_3^2:D_{18}$ x 6, $\He_3.D_6$ x 6, $\He_3.D_6$ x 6, $C_3^2:D_{18}$ x 6, $C_9:C_6^2$ x 6, $C_3^2:C_6^2$ x 6, $C_3^3:D_6$ x 6, $D_6\times \He_3$ x 6, $C_3^2.C_6^2$ x 4, $C_9.C_6^2$ x 4, $C_3^2.C_6^2$ x 2, $C_6^2:C_9$ x 2, $C_3^2\times C_6^2$

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

Order 216: $C_6\times S_3^2$ x 72, $C_6.S_3^2$ x 36, $D_{18}:C_6$ x 25, $S_3\times D_{18}$ x 24, $C_6^2:S_3$ x 19, $S_3\times C_6^2$ x 14, $C_6:S_3^2$ x 12, $C_6^2:C_6$ x 12, $C_6^2:C_6$ x 11, $D_6\times C_{18}$ x 7, $C_6\times D_{18}$ x 5, $C_6:D_{18}$ x 2, $C_6.C_6^2$, $C_2^3\times \He_3$

Order 162: $C_3^3:C_6$ x 28, $C_3^3:S_3$ x 20, $D_9:C_3^2$ x 20, $C_{18}:C_3^2$ x 18, $C_6\times \He_3$ x 18, $C_3^3:C_6$ x 18, $C_3^2\wr C_2$ x 12, $C_6.C_3^3$ x 12, $\He_3.S_3$ x 12, $C_3^3.C_6$ x 12, $C_6.\He_3$ x 12, $C_3^3:S_3$ x 12, $\He_3.C_6$ x 8, $C_3^3.S_3$ x 8, $C_3^2:C_{18}$ x 8, $\He_3.S_3$ x 8, $\He_3.C_6$ x 8, $\He_3.C_6$ x 8, $\He_3:C_6$ x 6, $C_3^2:C_{18}$ x 6, $C_3^2:D_9$ x 4, $C_3^3:S_3$ x 4, $S_3\times \He_3$ x 4, $\He_3:S_3$ x 4, $C_3^2:D_9$ x 4, $\He_3.S_3$ x 4, $\He_3.S_3$ x 4, $C_3^3\times C_6$ x 3

Order 144: $C_6^2:C_2^2$ x 6, $D_6^2$ x 6, $C_2^2\times D_{18}$

Order 108: $C_3\times S_3^2$ x 96, $C_{18}:C_6$ x 94, $C_3^2\times D_6$ x 84, $C_3^2:D_6$ x 72, $C_3^2:D_6$ x 66, $C_3^2:D_6$ x 58, $C_3^2:D_6$ x 48, $S_3\times C_{18}$ x 42, $S_3\times D_9$ x 32, $C_3\times D_{18}$ x 30, $C_{18}:C_6$ x 23, $C_2^2\times \He_3$ x 19, $C_3:S_3^2$ x 16, $C_3:D_{18}$ x 12, $C_3\times C_6^2$ x 11, $C_6\times C_{18}$ x 10

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

Order 72: $C_6\times D_6$ x 109, $S_3\times D_6$ x 72, $C_2\times D_{18}$ x 17, $C_6:D_6$ x 7, $C_2\times C_6^2$ x 6, $C_2^2\times C_{18}$ x 2

Order 54: $S_3\times C_3^2$ x 56, $C_9:C_6$ x 55, $C_9:C_6$ x 52, $C_3^2:C_6$ x 48, $C_3^2:C_6$ x 44, $C_2\times \He_3$ x 43, $C_3^2\times C_6$ x 33, $C_3\times C_{18}$ x 30, $C_3^2:S_3$ x 28, $S_3\times C_9$ x 28, $C_3\times D_9$ x 20, $C_3:D_9$ x 8

Order 48: $C_2^2\times D_6$ x 6, $C_2^3\times C_6$

Order 36: $C_6\times S_3$ x 318, $S_3^2$ x 96, $C_6^2$ x 62, $D_{18}$ x 46, $C_6:S_3$ x 42, $C_2\times C_{18}$ x 18

Order 27: $C_9:C_3$ x 17, $\He_3$ x 13, $C_3^3$ x 11, $C_3\times C_9$ x 10

Order 24: $C_2\times D_6$ x 87, $C_2^2\times C_6$ x 22

Order 18: $C_3\times S_3$ x 148, $C_3\times C_6$ x 102, $C_3:S_3$ x 28, $C_{18}$ x 26, $D_9$ x 20

Order 16: $C_2^4$

Order 12: $D_6$ x 186, $C_2\times C_6$ x 87

Order 9: $C_3^2$ x 26, $C_9$ x 6

Order 8: $C_2^3$ x 15

Order 6: $C_6$ x 73, $S_3$ x 60

Order 4: $C_2^2$ x 35

Order 3: $C_3$ x 11

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11664: $C_3^4.D_6^2$

Order 5832: $C_2^2\times (C_3^3.C_3^3):C_2$, $C_2^2\times (C_3^3.C_3^3):C_2$, $C_2^2\times (C_3^4.C_3^2):C_2$, $C_2\times C_3^4.S_3^2$

Order 3888: $C_2^2\times (C_3\times C_9):C_3^2:C_2^2$, $C_2^2\times (C_9.C_3^3):C_2^2$, $C_2^2\times (C_3^4.C_3):C_2^2$, $C_2^2\times C_3^4:C_3:C_2^2$, $C_2^2\times \He_3:C_3^2:C_2^2$

Order 2916: $C_2\times (C_3^4.C_3^2):C_2$, $C_2^2\times C_3^3.C_3^3$, $C_2\times (C_3^3.C_3^3):C_2$, $C_2\times (C_3^3.C_3^3):C_2$, $C_3^4.S_3^2$

Order 1944: $(C_3\times C_{18}):C_6^2$ x 2, $\He_3.(C_6\times D_6)$ x 2, $C_3^3:(C_6\times D_6)$ x 2, $C_2\times \He_3.C_6^2$, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^4:D_6$, $\He_3:(C_6\times D_6)$, $C_6\times C_3^3:D_6$, $C_2\times C_3^4.D_6$, $C_2\times C_3^4:D_6$, $(C_3\times C_{18}):C_6^2$, $C_2\times \He_3.C_6^2$, $(C_3\times C_{18}):C_6^2$, $C_2\times \He_3.C_6^2$, $(C_3\times C_{18}):C_6^2$, $C_2\times C_3^3.C_6^2$, $C_2^2\times C_3^4:C_6$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^3.S_3^2$

Order 1458: $(C_3\times C_{18}):\He_3$, $C_3^4:C_3:S_3$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$

Order 1296: $C_6^2.S_3^2$ x 3, $C_6^2.S_3^2$ x 2, $C_6^2:S_3^2$, $C_9.D_6^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2.S_3^2$, $C_6^2:D_{18}$

Order 972: $(C_3\times C_9):C_6^2$ x 2, $C_3^3.C_6^2$ x 2, $C_3^3:C_6^2$ x 2, $\He_3.(C_6\times S_3)$ x 2, $C_3^3:(C_6\times S_3)$ x 2, $C_3^3.C_6^2$, $\He_3.C_6^2$, $C_3^3.S_3^2$, $\He_3:C_6^2$, $C_3^3:C_6^2$, $C_3^3.C_6^2$, $C_3^3:C_6^2$, $C_3^4:D_6$, $\He_3:(C_6\times S_3)$, $C_3^4:D_6$, $C_3^4.D_6$, $C_3^4:D_6$, $(C_3\times C_9):C_6^2$, $\He_3.C_6^2$, $(C_3\times C_9):C_6^2$, $\He_3.C_6^2$, $C_3^4.D_6$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$, $C_3^3.S_3^2$

Order 729: $C_3^3.C_3^3$

Order 648: $C_3\times C_6^2:C_6$ x 7, $C_3\times C_6^2:S_3$ x 5, $C_{18}:C_6^2$ x 5, $C_3^4:C_2^3$ x 3, $(C_6\times C_{18}):C_6$ x 3, $C_{18}:C_6\times S_3$ x 3, $C_6\times C_3^2:D_6$ x 3, $C_2\times C_3^3:D_6$ x 3, $(C_6\times C_{18}):S_3$ x 2, $C_{18}:(C_6\times S_3)$ x 2, $C_{18}:C_6\times S_3$ x 2, $(C_6\times C_{18}):S_3$ x 2, $(C_6\times C_{18}):S_3$ x 2, $(C_6\times C_{18}):S_3$ x 2, $C_6^2:C_{18}$ x 2, $C_3^4:C_2^3$, $C_2\times C_3^3:D_6$, $C_2\times D_6\times \He_3$, $C_{18}.S_3^2$, $S_3\times C_3^2:D_6$, $(C_6\times C_{18}):S_3$, $C_6^2:D_9$, $(C_6\times C_{18}):C_6$, $(C_6\times C_{18}):C_6$, $C_6^2:D_9$, $(C_3\times C_{18}):D_6$, $(C_3\times C_{18}):D_6$, $C_2\times C_3^2:D_{18}$

Order 486: $C_9.(S_3\times C_3^2)$ x 2, $(C_3\times C_{18}):C_3^2$ x 2, $C_2\times C_3^3:C_3^2$ x 2, $(C_3\times \He_3).S_3$ x 2, $C_3\wr C_3:C_6$ x 2, $C_{18}.C_3^3$, $C_3^3.(C_3\times S_3)$, $(C_3\times C_{18}):C_3^2$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:S_3$, $(C_3\times \He_3):S_3$, $C_3^4:S_3$, $C_3^4.S_3$, $C_3^4:S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3^4.S_3$, $C_3^4.C_6$, $C_3^4:C_6$

Order 432: $C_3\times D_6^2$ x 6, $C_6^2:D_6$ x 3, $D_6\times D_{18}$ x 2, $C_3:D_6^2$, $C_6^2:D_6$, $C_6^2.D_6$

Order 324: $C_3^2:C_6^2$ x 7, $C_3^2.C_6^2$ x 6, $C_9:C_6^2$ x 6, $C_3^2:C_6^2$ x 6, $C_3^3:D_6$ x 5, $C_9:C_6^2$ x 5, $C_3^2.C_6^2$ x 4, $C_9.C_6^2$ x 4, $C_3^3:D_6$ x 3, $C_3^2:C_6^2$ x 3, $\He_3.D_6$ x 3, $C_3^2.C_6^2$ x 3, $C_3^3:D_6$ x 3, $C_3^2.C_6^2$ x 2, $C_6^2:C_9$ x 2, $\He_3.D_6$ x 2, $(C_3\times C_{18}):S_3$ x 2, $(C_3\times C_{18}):S_3$ x 2, $C_2\times C_3^2:C_{18}$ x 2, $(C_3\times C_{18}):S_3$ x 2, $C_3^3.D_6$ x 2, $C_3^2.S_3^2$ x 2, $\He_3:D_6$, $C_3^2:D_{18}$, $\He_3.D_6$, $\He_3.D_6$, $C_3^2:D_{18}$, $C_3^2.S_3^2$, $C_3^2.S_3^2$, $C_3^2:D_{18}$, $C_3^2\times C_6^2$, $C_3\wr C_2^2$, $C_3^3:D_6$, $D_6\times \He_3$, $C_9.S_3^2$, $C_3^2:S_3^2$

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

Order 216: $S_3\times C_6^2$ x 14, $D_{18}:C_6$ x 14, $C_6^2:C_6$ x 12, $C_6^2:C_6$ x 11, $C_6^2:S_3$ x 8, $D_6\times C_{18}$ x 7, $C_6\times S_3^2$ x 6, $C_6\times D_{18}$ x 5, $C_6.S_3^2$ x 3, $C_6:D_{18}$ x 2, $S_3\times D_{18}$ x 2, $C_6:S_3^2$, $C_6.C_6^2$, $C_2^3\times \He_3$

Order 162: $C_3^3:C_6$ x 7, $C_{18}:C_3^2$ x 6, $C_6\times \He_3$ x 6, $C_3^3:C_6$ x 6, $C_3^3:S_3$ x 5, $D_9:C_3^2$ x 5, $C_6.C_3^3$ x 4, $C_6.\He_3$ x 4, $C_3^2\wr C_2$ x 3, $\He_3.S_3$ x 3, $C_3^3.C_6$ x 3, $C_3^3:S_3$ x 3, $\He_3.C_6$ x 2, $C_3^3.S_3$ x 2, $C_3^2:C_{18}$ x 2, $\He_3:C_6$ x 2, $C_3^2:C_{18}$ x 2, $\He_3.S_3$ x 2, $\He_3.C_6$ x 2, $\He_3.C_6$ x 2, $C_3^3\times C_6$, $C_3^2:D_9$, $C_3^3:S_3$, $S_3\times \He_3$, $\He_3:S_3$, $C_3^2:D_9$, $\He_3.S_3$, $\He_3.S_3$

Order 144: $C_6^2:C_2^2$ x 6, $D_6^2$ x 6, $C_2^2\times D_{18}$

Order 108: $C_{18}:C_6$ x 18, $C_3^2\times D_6$ x 14, $C_2^2\times \He_3$ x 14, $C_{18}:C_6$ x 14, $C_3^2:D_6$ x 12, $C_3\times C_6^2$ x 11, $C_3^2:D_6$ x 11, $C_6\times C_{18}$ x 10, $C_3^2:D_6$ x 8, $S_3\times C_{18}$ x 7, $C_3\times S_3^2$ x 6, $C_3\times D_{18}$ x 5, $C_3^2:D_6$ x 3, $C_3:D_{18}$ x 2, $S_3\times D_9$ x 2, $C_3:S_3^2$

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

Order 72: $C_6\times D_6$ x 43, $C_6:D_6$ x 7, $C_2\times C_6^2$ x 6, $S_3\times D_6$ x 6, $C_2\times D_{18}$ x 6, $C_2^2\times C_{18}$ x 2

Order 54: $C_9:C_6$ x 18, $S_3\times C_3^2$ x 14, $C_2\times \He_3$ x 14, $C_9:C_6$ x 13, $C_3^2:C_6$ x 12, $C_3^2\times C_6$ x 11, $C_3^2:C_6$ x 11, $C_3\times C_{18}$ x 10, $C_3^2:S_3$ x 7, $S_3\times C_9$ x 7, $C_3\times D_9$ x 5, $C_3:D_9$ x 2

Order 48: $C_2^2\times D_6$ x 6, $C_2^3\times C_6$

Order 36: $C_6\times S_3$ x 43, $C_6^2$ x 32, $C_2\times C_{18}$ x 8, $C_6:S_3$ x 7, $D_{18}$ x 6, $S_3^2$ x 6

Order 27: $C_9:C_3$ x 17, $\He_3$ x 13, $C_3^3$ x 11, $C_3\times C_9$ x 10

Order 24: $C_2\times D_6$ x 21, $C_2^2\times C_6$ x 11

Order 18: $C_3\times S_3$ x 37, $C_3\times C_6$ x 32, $C_{18}$ x 8, $C_3:S_3$ x 7, $D_9$ x 5

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 22, $D_6$ x 21

Order 9: $C_3^2$ x 26, $C_9$ x 6

Order 8: $C_2^3$ x 4

Order 6: $C_6$ x 21, $S_3$ x 15

Order 4: $C_2^2$ x 5

Order 3: $C_3$ x 11

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11664: $C_3^4.D_6^2$ ($C_1$)

Order 5832: $C_2\times C_3^4.S_3^2$ ($C_2$) x 12, $C_2^2\times (C_3^3.C_3^3):C_2$ ($C_2$), $C_2^2\times (C_3^3.C_3^3):C_2$ ($C_2$), $C_2^2\times (C_3^4.C_3^2):C_2$ ($C_2$)

Order 2916: $C_3^4.S_3^2$ ($C_2^2$) x 16, $C_2\times (C_3^4.C_3^2):C_2$ ($C_2^2$) x 6, $C_2\times (C_3^3.C_3^3):C_2$ ($C_2^2$) x 6, $C_2\times (C_3^3.C_3^3):C_2$ ($C_2^2$) x 6, $C_2^2\times C_3^3.C_3^3$ ($C_2^2$)

Order 1944: $(C_3\times C_{18}):C_6^2$ ($S_3$), $C_2\times C_3^4:D_6$ ($S_3$), $(C_3\times C_{18}):C_6^2$ ($S_3$), $(C_3\times C_{18}):C_6^2$ ($S_3$), $(C_3\times C_{18}):C_6^2$ ($S_3$)

Order 1458: $C_3^4:C_3:S_3$ ($C_2^3$) x 4, $C_3^4.(C_3\times S_3)$ ($C_2^3$) x 4, $C_3^4.(C_3\times S_3)$ ($C_2^3$) x 4, $(C_3\times C_{18}):\He_3$ ($C_2^3$) x 3

Order 972: $(C_3\times C_9):C_6^2$ ($D_6$) x 6, $C_3^4:D_6$ ($D_6$) x 6, $(C_3\times C_9):C_6^2$ ($D_6$) x 6, $(C_3\times C_9):C_6^2$ ($D_6$) x 6, $C_3^4.D_6$ ($D_6$) x 6, $C_3^3.C_6^2$ ($D_6$), $\He_3:C_6^2$ ($D_6$), $C_3^3.C_6^2$ ($D_6$), $C_3^3.C_6^2$ ($D_6$), $C_3^3:C_6^2$ ($D_6$)

Order 729: $C_3^3.C_3^3$ ($C_2^4$)

Order 648: $C_{18}:(C_6\times S_3)$ ($C_3:S_3$)

Order 486: $C_9.(S_3\times C_3^2)$ ($C_2\times D_6$) x 4, $C_3^4:S_3$ ($C_2\times D_6$) x 4, $C_3^3.(C_3\times S_3)$ ($C_2\times D_6$) x 4, $C_3^3.(C_3\times S_3)$ ($C_2\times D_6$) x 4, $C_3^4.S_3$ ($C_2\times D_6$) x 4, $C_{18}.C_3^3$ ($C_2\times D_6$) x 3, $(C_3\times C_{18}):C_3^2$ ($C_2\times D_6$) x 3, $(C_3\times C_{18}):C_3^2$ ($C_2\times D_6$) x 3, $C_3^4.C_6$ ($C_2\times D_6$) x 3, $C_3^4:C_6$ ($C_2\times D_6$) x 3

Order 324: $C_3^3.D_6$ ($C_6:S_3$) x 6, $C_3^2:C_6^2$ ($S_3^2$) x 3, $C_3^2\times C_6^2$ ($S_3^2$), $C_9:C_6^2$ ($C_6:S_3$)

Order 243: $C_9.C_3^3$ ($C_2^2\times D_6$), $\He_3:C_3^2$ ($C_2^2\times D_6$), $C_3^2.C_3^3$ ($C_2^2\times D_6$), $C_3^4.C_3$ ($C_2^2\times D_6$), $C_3^4:C_3$ ($C_2^2\times D_6$)

Order 216: $C_6:D_{18}$ ($C_3^2:S_3$)

Order 162: $C_6\times \He_3$ ($S_3\times D_6$) x 9, $C_3^3.S_3$ ($C_6:D_6$) x 4, $C_3^3\times C_6$ ($S_3\times D_6$) x 3, $C_{18}:C_3^2$ ($C_6:D_6$) x 3

Order 108: $C_3:D_{18}$ ($C_3^2:D_6$) x 6, $C_3\times C_6^2$ ($C_3:S_3^2$), $C_3\times C_6^2$ ($C_3^2:D_6$), $C_6\times C_{18}$ ($C_3^2:D_6$)

Order 81: $C_3\times \He_3$ ($D_6^2$) x 3, $C_3^4$ ($D_6^2$), $C_9:C_3^2$ ($C_6^2:C_2^2$)

Order 54: $C_3:D_9$ ($C_6^2:S_3$) x 4, $C_3\times C_{18}$ ($C_6^2:S_3$) x 3, $C_3^2\times C_6$ ($C_6:S_3^2$) x 3, $C_3^2\times C_6$ ($C_6.S_3^2$) x 3

Order 36: $C_6^2$ ($C_3^2:S_3^2$), $C_6^2$ ($C_3^2:S_3^2$)

Order 27: $C_3^3$ ($C_3:D_6^2$), $C_3^3$ ($C_6^2:D_6$), $C_3\times C_9$ ($C_6^2:D_6$)

Order 18: $C_3\times C_6$ ($S_3\times C_3^2:D_6$) x 3, $C_3\times C_6$ ($C_2\times C_3^2:S_3^2$) x 3

Order 12: $C_2\times C_6$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($C_6^2:S_3^2$), $C_3^2$ ($C_6^2:S_3^2$)

Order 6: $C_6$ ($C_2\times C_3^3:S_3^2$) x 3

Order 4: $C_2^2$ ($C_3^4.S_3^2$)

Order 3: $C_3$ ($C_2^2\times C_3^4:C_3:C_2^2$)

Order 2: $C_2$ ($C_2\times C_3^4.S_3^2$) x 3

Order 1: $C_1$ ($C_3^4.D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11664: $C_3^4.D_6^2$ ($C_1$)

Order 5832: $C_2^2\times (C_3^3.C_3^3):C_2$ ($C_2$), $C_2^2\times (C_3^3.C_3^3):C_2$ ($C_2$), $C_2^2\times (C_3^4.C_3^2):C_2$ ($C_2$), $C_2\times C_3^4.S_3^2$ ($C_2$)

Order 2916: $C_2\times (C_3^4.C_3^2):C_2$ ($C_2^2$), $C_2^2\times C_3^3.C_3^3$ ($C_2^2$), $C_2\times (C_3^3.C_3^3):C_2$ ($C_2^2$), $C_2\times (C_3^3.C_3^3):C_2$ ($C_2^2$), $C_3^4.S_3^2$ ($C_2^2$)

Order 1944: $(C_3\times C_{18}):C_6^2$ ($S_3$), $C_2\times C_3^4:D_6$ ($S_3$), $(C_3\times C_{18}):C_6^2$ ($S_3$), $(C_3\times C_{18}):C_6^2$ ($S_3$), $(C_3\times C_{18}):C_6^2$ ($S_3$)

Order 1458: $(C_3\times C_{18}):\He_3$ ($C_2^3$), $C_3^4:C_3:S_3$ ($C_2^3$), $C_3^4.(C_3\times S_3)$ ($C_2^3$), $C_3^4.(C_3\times S_3)$ ($C_2^3$)

Order 972: $C_3^3.C_6^2$ ($D_6$), $(C_3\times C_9):C_6^2$ ($D_6$), $\He_3:C_6^2$ ($D_6$), $C_3^3.C_6^2$ ($D_6$), $C_3^3.C_6^2$ ($D_6$), $C_3^3:C_6^2$ ($D_6$), $C_3^4:D_6$ ($D_6$), $(C_3\times C_9):C_6^2$ ($D_6$), $(C_3\times C_9):C_6^2$ ($D_6$), $C_3^4.D_6$ ($D_6$)

Order 729: $C_3^3.C_3^3$ ($C_2^4$)

Order 648: $C_{18}:(C_6\times S_3)$ ($C_3:S_3$)

Order 486: $C_{18}.C_3^3$ ($C_2\times D_6$), $C_9.(S_3\times C_3^2)$ ($C_2\times D_6$), $(C_3\times C_{18}):C_3^2$ ($C_2\times D_6$), $(C_3\times C_{18}):C_3^2$ ($C_2\times D_6$), $C_3^4.C_6$ ($C_2\times D_6$), $C_3^4:C_6$ ($C_2\times D_6$), $C_3^4:S_3$ ($C_2\times D_6$), $C_3^3.(C_3\times S_3)$ ($C_2\times D_6$), $C_3^3.(C_3\times S_3)$ ($C_2\times D_6$), $C_3^4.S_3$ ($C_2\times D_6$)

Order 324: $C_3^2:C_6^2$ ($S_3^2$) x 3, $C_3^2\times C_6^2$ ($S_3^2$), $C_9:C_6^2$ ($C_6:S_3$), $C_3^3.D_6$ ($C_6:S_3$)

Order 243: $C_9.C_3^3$ ($C_2^2\times D_6$), $\He_3:C_3^2$ ($C_2^2\times D_6$), $C_3^2.C_3^3$ ($C_2^2\times D_6$), $C_3^4.C_3$ ($C_2^2\times D_6$), $C_3^4:C_3$ ($C_2^2\times D_6$)

Order 216: $C_6:D_{18}$ ($C_3^2:S_3$)

Order 162: $C_6\times \He_3$ ($S_3\times D_6$) x 3, $C_3^3\times C_6$ ($S_3\times D_6$), $C_{18}:C_3^2$ ($C_6:D_6$), $C_3^3.S_3$ ($C_6:D_6$)

Order 108: $C_3\times C_6^2$ ($C_3:S_3^2$), $C_3\times C_6^2$ ($C_3^2:D_6$), $C_6\times C_{18}$ ($C_3^2:D_6$), $C_3:D_{18}$ ($C_3^2:D_6$)

Order 81: $C_3\times \He_3$ ($D_6^2$) x 3, $C_3^4$ ($D_6^2$), $C_9:C_3^2$ ($C_6^2:C_2^2$)

Order 54: $C_3\times C_{18}$ ($C_6^2:S_3$), $C_3:D_9$ ($C_6^2:S_3$), $C_3^2\times C_6$ ($C_6:S_3^2$), $C_3^2\times C_6$ ($C_6.S_3^2$)

Order 36: $C_6^2$ ($C_3^2:S_3^2$), $C_6^2$ ($C_3^2:S_3^2$)

Order 27: $C_3^3$ ($C_3:D_6^2$), $C_3^3$ ($C_6^2:D_6$), $C_3\times C_9$ ($C_6^2:D_6$)

Order 18: $C_3\times C_6$ ($S_3\times C_3^2:D_6$), $C_3\times C_6$ ($C_2\times C_3^2:S_3^2$)

Order 12: $C_2\times C_6$ ($C_3^3:S_3^2$)

Order 9: $C_3^2$ ($C_6^2:S_3^2$), $C_3^2$ ($C_6^2:S_3^2$)

Order 6: $C_6$ ($C_2\times C_3^3:S_3^2$)

Order 4: $C_2^2$ ($C_3^4.S_3^2$)

Order 3: $C_3$ ($C_2^2\times C_3^4:C_3:C_2^2$)

Order 2: $C_2$ ($C_2\times C_3^4.S_3^2$)

Order 1: $C_1$ ($C_3^4.D_6^2$)

Series

Derived series

$C_3^4.D_6^2$

$\rhd$

$C_3^4.D_6^2$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^4.D_6^2$

$\rhd$

$C_3^4.D_6^2$

$\rhd$

$C_2\times C_3^4.S_3^2$

$\rhd$

$C_2\times C_3^4.S_3^2$

$\rhd$

$C_3^4.S_3^2$

$\rhd$

$C_3^4.S_3^2$

$\rhd$

$C_3^4.(C_3\times S_3)$

$\rhd$

$C_3^4.(C_3\times S_3)$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_9.C_3^3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3\times \He_3$

$\rhd$

$C_9:C_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^4.D_6^2$

$\rhd$

$C_3^4.D_6^2$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3.C_3^3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 1 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 $168 \times 168$ 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 $132 \times 132$ rational character table (warning: may be slow to load).