Abstract group 11664.fe: $(C_3^3\times C_6^2):D_6$

• Label: 11664.fe
• Order: \( 2^{4} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $22$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$(C_3^3\times C_6^2):D_6$
Order:	\(11664\)\(\medspace = 2^{4} \cdot 3^{6} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^3.C_3^3.C_2^6$, of order \(46656\)\(\medspace = 2^{6} \cdot 3^{6} \)
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	4	6	9	12	18	36
Elements	1	183	404	72	6216	324	2196	1620	648	11664
Conjugacy classes	1	7	39	2	185	4	27	12	2	279
Divisions	1	7	23	2	102	2	15	6	1	159
Autjugacy classes	1	6	15	2	49	2	9	5	1	90

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	24	30	48	18	112	0	47	0	279
Irr. rational chars.	8	18	0	16	32	6	59	20	159

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$8491392$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{6}=b^{6}=c^{3}=d^{3}=e^{6}=f^{6}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(3,4)(5,7)(6,8)(10,12,11,13)(14,16,20)(17,22,18), (1,2,3)(4,5,6)(7,8,9) \!\cdots\! \rangle$
Transitive group:	36T9109				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^5:D_4)$ $\,\rtimes\,$ $S_3$	$(C_3^5:S_3)$ $\,\rtimes\,$ $D_4$	$C_3^5$ $\,\rtimes\,$ $(S_3\times D_4)$	$(C_3^4:D_6)$ $\,\rtimes\,$ $D_6$	all 51
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $(D_6:D_6)$	$C_3^4$ . $(C_{12}:D_6)$	$C_6^2$ . $(C_3^3:D_6)$	$C_3^3$ . $(C_6^2:D_6)$	all 39

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

Homology

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

Subgroups

There are 70852 subgroups in 4295 conjugacy classes, 98 normal (94 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_2\times C_3^4:D_6$
Commutator:	$G' \simeq$ $C_3^4:C_6$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^2\times C_6$	$G/\Phi \simeq$ $C_6\times S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_3^4.C_3^2$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $(C_3^3\times C_6^2):D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times C_3^2:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^5:C_3$

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 11664: $(C_3^3\times C_6^2):D_6$

Order 5832: $C_2^2\times (C_3^4.C_3^2):C_2$, $C_2\times C_3^5:D_6$, $C_3^5:(C_4\times S_3)$, $(C_3^4\times C_6).D_6$, $C_3^5:D_{12}$, $(C_3^3\times C_6^2):S_3$, $(C_3^3\times C_6^2):C_6$

Order 3888: $C_3^2\times (C_2\times C_6):S_3^2$, $(C_3\times C_6^2):S_3^2$, $(C_3\times C_6^2):S_3^2$

Order 2916: $C_2\times (C_3^4.C_3^2):C_2$ x 4, $C_3^5:D_6$ x 2, $C_2^2\times C_3^4.C_3^2$, $(C_3^4\times C_6).S_3$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^5:C_{12}$

Order 1944: $C_3^5:D_4$ x 2, $C_3^4:D_{12}$ x 2, $C_6\times C_3^3:D_6$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $C_3^2:C_6^3$, $C_3^5:C_2^3$, $C_3^3:(C_6\times C_{12})$, $C_2\times C_3^4:D_6$, $C_2\times C_3^3:S_3^2$, $C_3\wr S_3\times D_6$, $(C_3^2\times C_6^2):S_3$, $(C_3^2\times C_6^2):C_6$, $C_6^2.(S_3\times C_3^2)$, $(C_3^2\times C_6^2):C_6$, $(C_3\times \He_3):D_{12}$, $C_3^4:D_{12}$, $C_3^4:(C_4\times S_3)$, $(D_6\times C_3^3):S_3$, $C_3^4:D_{12}$, $C_3^4:(C_4\times S_3)$

Order 1458: $C_3^5:S_3$ x 3, $C_3^5:C_6$ x 2, $C_3^5:S_3$, $C_3^5:C_6$

Order 1296: $C_6^2:S_3^2$ x 4, $C_6^2:C_6^2$ x 2, $C_6^2:C_6^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_3\wr S_3\times D_4$

Order 972: $C_3^4:D_6$ x 10, $C_3^3:C_6^2$ x 6, $C_3^3:C_6^2$ x 4, $C_3^4:D_6$ x 4, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $(C_3\times \He_3):C_{12}$ x 2, $C_3^3\times C_6^2$, $C_3^3.C_6^2$, $C_3^3:C_6^2$, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $(C_3\times \He_3):C_{12}$, $C_3^4:C_{12}$, $C_3^4.C_{12}$, $C_3^4:C_{12}$

Order 729: $C_3^5:C_3$

Order 648: $C_3^4:D_4$ x 14, $C_3^4:C_2^3$ x 9, $C_3^3:D_{12}$ x 8, $C_2^2\times C_3\wr S_3$ x 6, $C_3^4:D_4$ x 5, $C_3^4:C_2^3$ x 4, $C_3\times C_6.S_3^2$ x 4, $C_3^3:D_{12}$ x 4, $C_3\times C_6^2:S_3$ x 4, $C_3:C_6^3$ x 3, $C_3^4:C_2^3$ x 3, $(C_3\times C_6^2):S_3$ x 3, $C_3^4:D_4$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3\times C_6^2:C_6$ x 2, $C_3^3:D_{12}$ x 2, $C_3^2:(C_6\times C_{12})$, $S_3\times C_3^2:D_6$, $C_6\times C_3^2:D_6$, $(C_3\times C_6^2):S_3$, $(C_6\times C_{18}):C_6$, $C_3:D_4\times \He_3$, $C_3^3:D_{12}$, $(C_3^2\times D_6):S_3$, $C_3^3:(C_4\times S_3)$, $C_3\times C_6.S_3^2$, $C_3\wr C_3\times D_4$, $C_6^2:C_{18}$, $C_3^3:D_{12}$, $C_4\times C_3\wr S_3$

Order 486: $C_3^4:C_6$ x 10, $C_3^4:S_3$ x 7, $C_3^4:C_6$ x 5, $C_3^4:S_3$ x 3, $C_3^4\times C_6$ x 2, $C_3^4.C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:C_6$

Order 432: $C_6^2:D_6$ x 9, $C_6^2:D_6$ x 4, $C_{12}:C_6^2$ x 2, $D_6:C_6^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$

Order 324: $C_3^2:C_6^2$ x 58, $C_3^3:D_6$ x 32, $C_3^2\times C_6^2$ x 21, $C_3^3:D_6$ x 20, $D_6\times C_3^3$ x 19, $C_3^3:C_{12}$ x 14, $C_3^2.C_6^2$ x 11, $C_3\wr C_2^2$ x 8, $C_3^2\times S_3^2$ x 6, $C_3^3:C_{12}$ x 5, $C_3^2:C_6^2$ x 4, $C_3^3:C_{12}$ x 2, $C_6^2:C_9$ x 2, $C_9:C_6^2$ x 2, $C_3^2:C_6^2$ x 2, $\He_3:C_{12}$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$ x 2, $C_3^3.C_{12}$, $C_3^3:C_{12}$, $C_3^2:C_{36}$, $C_2\times C_3^2:C_{18}$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $(C_3\times \He_3):C_4$

Order 243: $C_3^4:C_3$ x 4, $C_3^5$, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $C_6^2:C_6$ x 50, $S_3\times C_6^2$ x 36, $C_3^2:D_{12}$ x 17, $C_6^2:C_6$ x 15, $C_6\times S_3^2$ x 13, $C_6^2:C_6$ x 9, $C_3^3:D_4$ x 9, $C_6.S_3^2$ x 9, $C_6^2:S_3$ x 5, $C_6.S_3^2$ x 4, $C_6^2:S_3$ x 3, $C_6^2:C_6$ x 3, $C_3^2:D_{12}$ x 2, $D_4\times C_3^3$ x 2, $C_3^2\times D_{12}$ x 2, $C_6.C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_{36}:C_6$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $D_6:C_{18}$, $C_6.S_3^2$, $C_6^3$, $C_6:S_3^2$, $C_6.C_6^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 162: $C_3^3\times C_6$ x 51, $C_3^2\wr C_2$ x 45, $C_3^3:C_6$ x 29, $C_3\wr S_3$ x 21, $S_3\times C_3^3$ x 15, $C_3^3:S_3$ x 14, $C_6\times \He_3$ x 10, $C_{18}:C_3^2$ x 5, $C_3^2:C_{18}$ x 5, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$

Order 144: $C_{12}:D_6$ x 9, $C_6^2:C_2^2$ x 4, $D_6:D_6$ x 3, $C_4:C_6^2$, $D_6:D_6$

Order 108: $C_3^2\times D_6$ x 228, $C_3\times C_6^2$ x 150, $C_3^2:D_6$ x 59, $C_3^2:C_{12}$ x 46, $C_3\times S_3^2$ x 26, $C_3^2:D_6$ x 24, $C_3^2:C_{12}$ x 15, $C_2^2\times \He_3$ x 9, $C_{18}:C_6$ x 6, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 2, $C_3^2\times C_{12}$ x 2, $C_6\times C_{18}$ x 2, $C_3^2:D_6$ x 2, $\He_3:C_4$ x 2, $C_3:C_{36}$, $S_3\times C_{18}$, $C_9:C_{12}$, $C_4\times \He_3$

Order 81: $C_3^4$ x 21, $C_3\wr C_3$ x 10, $C_3\times \He_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$ x 2

Order 72: $C_6\wr C_2$ x 64, $C_6\times D_6$ x 51, $D_4\times C_3^2$ x 17, $C_2\times C_6^2$ x 10, $C_3\times D_{12}$ x 9, $S_3\times C_{12}$ x 9, $C_3:D_{12}$ x 5, $S_3\times D_6$ x 4, $C_6:C_{12}$ x 4, $C_6^2:C_2$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_6:D_6$, $C_6\times C_{12}$, $C_6.D_6$, $D_4\times C_9$

Order 54: $C_3^2\times C_6$ x 391, $S_3\times C_3^2$ x 167, $C_3^2:C_6$ x 46, $C_2\times \He_3$ x 21, $C_3^2:S_3$ x 16, $C_9:C_6$ x 15, $C_3\times C_{18}$ x 5, $C_3^2:C_6$ x 3, $S_3\times C_9$

Order 48: $C_6\times D_4$ x 4, $S_3\times D_4$ x 3, $C_6:D_4$

Order 36: $C_6\times S_3$ x 277, $C_6^2$ x 210, $C_3:C_{12}$ x 47, $C_3\times C_{12}$ x 15, $S_3^2$ x 8, $C_6:S_3$ x 7, $C_3^2:C_4$ x 3, $C_2\times C_{18}$ x 3, $C_{36}$

Order 27: $C_3^3$ x 144, $\He_3$ x 8, $C_9:C_3$ x 5, $C_3\times C_9$ x 2

Order 24: $C_3\times D_4$ x 23, $C_2^2\times C_6$ x 13, $C_3:D_4$ x 11, $C_2\times D_6$ x 9, $C_2\times C_{12}$ x 4, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_6:C_4$

Order 18: $C_3\times C_6$ x 446, $C_3\times S_3$ x 187, $C_3:S_3$ x 6, $C_{18}$ x 6

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 90, $D_6$ x 38, $C_{12}$ x 15, $C_3:C_4$ x 6

Order 9: $C_3^2$ x 149, $C_9$ x 2

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

Order 6: $C_6$ x 102, $S_3$ x 23

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

Order 3: $C_3$ x 23

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 11664: $(C_3^3\times C_6^2):D_6$

Order 5832: $C_2^2\times (C_3^4.C_3^2):C_2$, $C_2\times C_3^5:D_6$, $C_3^5:(C_4\times S_3)$, $(C_3^4\times C_6).D_6$, $C_3^5:D_{12}$, $(C_3^3\times C_6^2):S_3$, $(C_3^3\times C_6^2):C_6$

Order 3888: $C_3^2\times (C_2\times C_6):S_3^2$, $(C_3\times C_6^2):S_3^2$, $(C_3\times C_6^2):S_3^2$

Order 2916: $C_2\times (C_3^4.C_3^2):C_2$ x 3, $C_2^2\times C_3^4.C_3^2$, $(C_3^4\times C_6).S_3$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^5:C_{12}$, $C_3^5:D_6$

Order 1944: $C_3^5:D_4$ x 2, $C_3^4:D_{12}$ x 2, $C_6\times C_3^3:D_6$ x 2, $(C_3^2\times C_6^2):S_3$ x 2, $C_3^2:C_6^3$, $C_3^5:C_2^3$, $C_3^3:(C_6\times C_{12})$, $C_2\times C_3^4:D_6$, $C_2\times C_3^3:S_3^2$, $C_3\wr S_3\times D_6$, $(C_3^2\times C_6^2):S_3$, $(C_3^2\times C_6^2):C_6$, $C_6^2.(S_3\times C_3^2)$, $(C_3^2\times C_6^2):C_6$, $(C_3\times \He_3):D_{12}$, $C_3^4:D_{12}$, $C_3^4:(C_4\times S_3)$, $(D_6\times C_3^3):S_3$, $C_3^4:D_{12}$, $C_3^4:(C_4\times S_3)$

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

Order 1296: $C_6^2:S_3^2$ x 2, $C_6^2:C_6^2$ x 2, $C_6^2:C_6^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_3\wr S_3\times D_4$

Order 972: $C_3^4:D_6$ x 6, $C_3^3:C_6^2$ x 5, $C_3^3:C_6^2$ x 4, $C_3^4:D_6$ x 3, $C_3^4:C_{12}$ x 2, $C_3^4:D_6$ x 2, $(C_3\times \He_3):C_{12}$ x 2, $C_3^3\times C_6^2$, $C_3^3:C_6^2$, $C_3^3.C_6^2$, $C_3^3:C_6^2$, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $C_3^3.C_6^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $(C_3\times \He_3):C_{12}$, $C_3^4:C_{12}$, $C_3^4.C_{12}$, $C_3^4:C_{12}$

Order 729: $C_3^5:C_3$

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

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

Order 432: $C_6^2:D_6$ x 5, $C_{12}:C_6^2$ x 2, $C_6^2:D_6$ x 2, $D_6:C_6^2$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$

Order 324: $C_3^2:C_6^2$ x 23, $C_3^3:D_6$ x 15, $D_6\times C_3^3$ x 14, $C_3^2\times C_6^2$ x 13, $C_3^3:D_6$ x 12, $C_3^2.C_6^2$ x 9, $C_3^3:C_{12}$ x 8, $C_3^3:C_{12}$ x 5, $C_3^2:C_6^2$ x 4, $C_3^2\times S_3^2$ x 3, $C_3^3:C_{12}$ x 2, $C_6^2:C_9$ x 2, $C_3\wr C_2^2$ x 2, $C_9:C_6^2$ x 2, $C_3^2:C_6^2$ x 2, $\He_3:C_{12}$ x 2, $C_3^3.C_{12}$, $C_3^3:C_{12}$, $C_3^2:C_{36}$, $C_2\times C_3^2:C_{18}$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^2.C_6^2$, $D_6\times \He_3$, $C_3^2:S_3^2$, $C_3^3:D_6$, $(C_3\times \He_3):C_4$

Order 243: $C_3^4:C_3$ x 4, $C_3^5$, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $C_6^2:C_6$ x 30, $S_3\times C_6^2$ x 22, $C_6^2:C_6$ x 9, $C_3^2:D_{12}$ x 9, $C_6\times S_3^2$ x 7, $C_6^2:C_6$ x 5, $C_3^3:D_4$ x 5, $C_6.S_3^2$ x 5, $C_6^2:S_3$ x 5, $C_6^2:S_3$ x 3, $C_6^2:C_6$ x 3, $C_3^2:D_{12}$ x 2, $D_4\times C_3^3$ x 2, $C_3^2\times D_{12}$ x 2, $C_6.C_6^2$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_{36}:C_6$, $D_4\times \He_3$, $C_3^2:D_{12}$, $C_4\times C_3^2:S_3$, $D_6:C_{18}$, $C_6.S_3^2$, $C_6^3$, $C_6:S_3^2$, $C_6.C_6^2$, $C_6.S_3^2$, $C_6.S_3^2$

Order 162: $C_3^3\times C_6$ x 26, $C_3^2\wr C_2$ x 18, $C_3^3:C_6$ x 17, $S_3\times C_3^3$ x 11, $C_3\wr S_3$ x 10, $C_6\times \He_3$ x 8, $C_3^3:S_3$ x 8, $C_{18}:C_3^2$ x 4, $C_3^2:C_{18}$ x 4, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$

Order 144: $C_{12}:D_6$ x 5, $D_6:D_6$ x 3, $C_6^2:C_2^2$ x 2, $C_4:C_6^2$, $D_6:D_6$

Order 108: $C_3^2\times D_6$ x 86, $C_3\times C_6^2$ x 73, $C_3^2:C_{12}$ x 26, $C_3^2:D_6$ x 24, $C_3^2:D_6$ x 15, $C_3^2:C_{12}$ x 9, $C_2^2\times \He_3$ x 9, $C_3\times S_3^2$ x 7, $C_{18}:C_6$ x 6, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 3, $C_3^2\times C_{12}$ x 2, $C_6\times C_{18}$ x 2, $\He_3:C_4$ x 2, $C_3:C_{36}$, $C_3:S_3^2$, $S_3\times C_{18}$, $C_3^2:D_6$, $C_9:C_{12}$, $C_4\times \He_3$

Order 81: $C_3^4$ x 13, $C_3\wr C_3$ x 8, $C_3\times \He_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$ x 2

Order 72: $C_6\wr C_2$ x 36, $C_6\times D_6$ x 30, $D_4\times C_3^2$ x 11, $C_2\times C_6^2$ x 6, $C_3\times D_{12}$ x 5, $S_3\times C_{12}$ x 5, $C_3:D_{12}$ x 5, $S_3\times D_6$ x 4, $C_6^2:C_2$ x 3, $D_6:S_3$ x 3, $C_6.D_6$ x 3, $C_6:C_{12}$ x 2, $C_6:D_6$, $C_6\times C_{12}$, $C_6.D_6$, $D_4\times C_9$

Order 54: $C_3^2\times C_6$ x 140, $S_3\times C_3^2$ x 64, $C_3^2:C_6$ x 19, $C_2\times \He_3$ x 17, $C_9:C_6$ x 11, $C_3^2:S_3$ x 10, $C_3\times C_{18}$ x 4, $C_3^2:C_6$ x 3, $S_3\times C_9$

Order 48: $S_3\times D_4$ x 3, $C_6\times D_4$ x 2, $C_6:D_4$

Order 36: $C_6\times S_3$ x 103, $C_6^2$ x 97, $C_3:C_{12}$ x 27, $C_3\times C_{12}$ x 9, $C_6:S_3$ x 6, $S_3^2$ x 4, $C_3^2:C_4$ x 3, $C_2\times C_{18}$ x 3, $C_{36}$

Order 27: $C_3^3$ x 68, $\He_3$ x 8, $C_9:C_3$ x 5, $C_3\times C_9$ x 2

Order 24: $C_3\times D_4$ x 13, $C_3:D_4$ x 11, $C_2\times D_6$ x 9, $C_2^2\times C_6$ x 7, $D_{12}$ x 3, $C_4\times S_3$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$

Order 18: $C_3\times C_6$ x 163, $C_3\times S_3$ x 73, $C_3:S_3$ x 5, $C_{18}$ x 5

Order 16: $C_2\times D_4$

Order 12: $C_2\times C_6$ x 41, $D_6$ x 25, $C_{12}$ x 9, $C_3:C_4$ x 6

Order 9: $C_3^2$ x 72, $C_9$ x 2

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

Order 6: $C_6$ x 49, $S_3$ x 16

Order 4: $C_2^2$ x 7, $C_4$ x 2

Order 3: $C_3$ x 15

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 11664: $(C_3^3\times C_6^2):D_6$ ($C_1$)

Order 5832: $C_2^2\times (C_3^4.C_3^2):C_2$ ($C_2$), $C_2\times C_3^5:D_6$ ($C_2$), $C_3^5:(C_4\times S_3)$ ($C_2$), $(C_3^4\times C_6).D_6$ ($C_2$), $C_3^5:D_{12}$ ($C_2$), $(C_3^3\times C_6^2):S_3$ ($C_2$), $(C_3^3\times C_6^2):C_6$ ($C_2$)

Order 3888: $(C_3\times C_6^2):S_3^2$ ($C_3$)

Order 2916: $C_2\times (C_3^4.C_3^2):C_2$ ($C_2^2$) x 2, $C_2^2\times C_3^4.C_3^2$ ($C_2^2$), $(C_3^4\times C_6).S_3$ ($C_2^2$), $C_3^5:D_6$ ($C_2^2$), $C_3^4.C_6^2$ ($C_2^2$), $C_3^5:C_{12}$ ($C_2^2$)

Order 1944: $C_3^5:D_4$ ($S_3$), $C_2\times C_3^4:D_6$ ($C_6$), $C_2\times C_3^3:S_3^2$ ($C_6$), $(C_3^2\times C_6^2):S_3$ ($S_3$), $(C_3^2\times C_6^2):S_3$ ($C_6$), $(C_3^2\times C_6^2):C_6$ ($C_6$), $(C_3\times \He_3):D_{12}$ ($C_6$), $C_3^4:D_{12}$ ($C_6$), $C_3^4:(C_4\times S_3)$ ($C_6$)

Order 1458: $C_3^5:S_3$ ($D_4$) x 2, $C_3^5:C_6$ ($C_2^3$)

Order 972: $C_3^4:D_6$ ($C_2\times C_6$) x 2, $C_3^3\times C_6^2$ ($D_6$), $C_3^3:C_6^2$ ($D_6$), $C_3^4:C_{12}$ ($D_6$), $C_3^3:C_6^2$ ($D_6$), $C_3^3:C_6^2$ ($C_2\times C_6$), $C_3^4:D_6$ ($D_6$), $C_3^4:D_6$ ($C_2\times C_6$), $C_3^3.C_6^2$ ($C_2\times C_6$), $(C_3\times \He_3):C_{12}$ ($D_6$), $(C_3\times \He_3):C_{12}$ ($C_2\times C_6$), $C_3^4:C_{12}$ ($C_2\times C_6$)

Order 729: $C_3^5:C_3$ ($C_2\times D_4$)

Order 648: $C_3^4:D_4$ ($C_3\times S_3$), $(C_3\times C_6^2):S_3$ ($C_3\times S_3$)

Order 486: $C_3^4:S_3$ ($C_3\times D_4$) x 2, $C_3^4\times C_6$ ($C_2\times D_6$), $C_3^4:C_6$ ($C_2\times D_6$), $C_3^4:C_6$ ($C_2^2\times C_6$)

Order 324: $C_3^2\times C_6^2$ ($C_6\times S_3$), $C_3^2\times C_6^2$ ($S_3^2$), $C_3^2:C_6^2$ ($C_6\times S_3$), $C_3^3:C_{12}$ ($C_6\times S_3$), $C_3^2:C_6^2$ ($C_6\times S_3$), $C_3^3:D_6$ ($C_6\times S_3$), $(C_3\times \He_3):C_4$ ($C_6\times S_3$)

Order 243: $C_3^5$ ($S_3\times D_4$), $C_3^4:C_3$ ($S_3\times D_4$), $C_3^4:C_3$ ($C_6\times D_4$)

Order 216: $C_6^2:C_6$ ($C_3^2:C_6$)

Order 162: $C_3^3\times C_6$ ($C_6\times D_6$), $C_3^3\times C_6$ ($S_3\times D_6$), $C_6\times \He_3$ ($C_6\times D_6$)

Order 108: $C_3\times C_6^2$ ($C_3\times S_3^2$), $C_3\times C_6^2$ ($C_3^2:D_6$), $C_3\times C_6^2$ ($C_3^2:D_6$), $C_3^2:D_6$ ($C_3^2:D_6$), $C_3^2:C_{12}$ ($C_3^2:D_6$)

Order 81: $C_3^4$ ($C_{12}:D_6$), $C_3^4$ ($D_6:D_6$), $C_3\times \He_3$ ($C_{12}:D_6$)

Order 72: $C_6^2:C_2$ ($C_3\wr S_3$)

Order 54: $C_3^2\times C_6$ ($C_6\times S_3^2$), $C_3^2\times C_6$ ($C_6^2:C_6$), $C_3^2\times C_6$ ($C_6.S_3^2$)

Order 36: $C_3^2:C_4$ ($C_3^3:D_6$), $C_6^2$ ($C_3^3:D_6$), $C_6^2$ ($C_3^3:D_6$), $C_6^2$ ($C_3^2:S_3^2$), $C_6:S_3$ ($C_3^3:D_6$)

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

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

Order 12: $C_2\times C_6$ ($C_3^3:S_3^2$), $C_2\times C_6$ ($C_3^4:D_6$)

Order 9: $C_3^2$ ($C_6^2.S_3^2$), $C_3^2$ ($C_6^2:S_3^2$), $C_3^2$ ($C_3\wr S_3\times D_4$)

Order 6: $C_6$ ($C_3\wr S_3\times D_6$), $C_6$ ($C_2\times C_3^4:D_6$)

Order 4: $C_2^2$ ($C_3^5:D_6$)

Order 3: $C_3$ ($C_6^2.(C_3\times S_3^2)$), $C_3$ ($C_6^2.(C_3\times S_3^2)$)

Order 2: $C_2$ ($C_2\times C_3^5:D_6$)

Order 1: $C_1$ ($(C_3^3\times C_6^2):D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 11664: $(C_3^3\times C_6^2):D_6$ ($C_1$)

Order 5832: $C_2^2\times (C_3^4.C_3^2):C_2$ ($C_2$), $C_2\times C_3^5:D_6$ ($C_2$), $C_3^5:(C_4\times S_3)$ ($C_2$), $(C_3^4\times C_6).D_6$ ($C_2$), $C_3^5:D_{12}$ ($C_2$), $(C_3^3\times C_6^2):S_3$ ($C_2$), $(C_3^3\times C_6^2):C_6$ ($C_2$)

Order 3888: $(C_3\times C_6^2):S_3^2$ ($C_3$)

Order 2916: $C_2\times (C_3^4.C_3^2):C_2$ ($C_2^2$) x 2, $C_2^2\times C_3^4.C_3^2$ ($C_2^2$), $(C_3^4\times C_6).S_3$ ($C_2^2$), $C_3^5:D_6$ ($C_2^2$), $C_3^4.C_6^2$ ($C_2^2$), $C_3^5:C_{12}$ ($C_2^2$)

Order 1944: $C_3^5:D_4$ ($S_3$), $C_2\times C_3^4:D_6$ ($C_6$), $C_2\times C_3^3:S_3^2$ ($C_6$), $(C_3^2\times C_6^2):S_3$ ($S_3$), $(C_3^2\times C_6^2):S_3$ ($C_6$), $(C_3^2\times C_6^2):C_6$ ($C_6$), $(C_3\times \He_3):D_{12}$ ($C_6$), $C_3^4:D_{12}$ ($C_6$), $C_3^4:(C_4\times S_3)$ ($C_6$)

Order 1458: $C_3^5:C_6$ ($C_2^3$), $C_3^5:S_3$ ($D_4$)

Order 972: $C_3^4:D_6$ ($C_2\times C_6$) x 2, $C_3^3\times C_6^2$ ($D_6$), $C_3^3:C_6^2$ ($D_6$), $C_3^4:C_{12}$ ($D_6$), $C_3^3:C_6^2$ ($D_6$), $C_3^3:C_6^2$ ($C_2\times C_6$), $C_3^4:D_6$ ($D_6$), $C_3^4:D_6$ ($C_2\times C_6$), $C_3^3.C_6^2$ ($C_2\times C_6$), $(C_3\times \He_3):C_{12}$ ($D_6$), $(C_3\times \He_3):C_{12}$ ($C_2\times C_6$), $C_3^4:C_{12}$ ($C_2\times C_6$)

Order 729: $C_3^5:C_3$ ($C_2\times D_4$)

Order 648: $C_3^4:D_4$ ($C_3\times S_3$), $(C_3\times C_6^2):S_3$ ($C_3\times S_3$)

Order 486: $C_3^4\times C_6$ ($C_2\times D_6$), $C_3^4:C_6$ ($C_2\times D_6$), $C_3^4:C_6$ ($C_2^2\times C_6$), $C_3^4:S_3$ ($C_3\times D_4$)

Order 324: $C_3^2\times C_6^2$ ($C_6\times S_3$), $C_3^2\times C_6^2$ ($S_3^2$), $C_3^2:C_6^2$ ($C_6\times S_3$), $C_3^3:C_{12}$ ($C_6\times S_3$), $C_3^2:C_6^2$ ($C_6\times S_3$), $C_3^3:D_6$ ($C_6\times S_3$), $(C_3\times \He_3):C_4$ ($C_6\times S_3$)

Order 243: $C_3^5$ ($S_3\times D_4$), $C_3^4:C_3$ ($S_3\times D_4$), $C_3^4:C_3$ ($C_6\times D_4$)

Order 216: $C_6^2:C_6$ ($C_3^2:C_6$)

Order 162: $C_3^3\times C_6$ ($C_6\times D_6$), $C_3^3\times C_6$ ($S_3\times D_6$), $C_6\times \He_3$ ($C_6\times D_6$)

Order 108: $C_3\times C_6^2$ ($C_3\times S_3^2$), $C_3\times C_6^2$ ($C_3^2:D_6$), $C_3\times C_6^2$ ($C_3^2:D_6$), $C_3^2:D_6$ ($C_3^2:D_6$), $C_3^2:C_{12}$ ($C_3^2:D_6$)

Order 81: $C_3^4$ ($C_{12}:D_6$), $C_3^4$ ($D_6:D_6$), $C_3\times \He_3$ ($C_{12}:D_6$)

Order 72: $C_6^2:C_2$ ($C_3\wr S_3$)

Order 54: $C_3^2\times C_6$ ($C_6\times S_3^2$), $C_3^2\times C_6$ ($C_6^2:C_6$), $C_3^2\times C_6$ ($C_6.S_3^2$)

Order 36: $C_3^2:C_4$ ($C_3^3:D_6$), $C_6^2$ ($C_3^3:D_6$), $C_6^2$ ($C_3^3:D_6$), $C_6^2$ ($C_3^2:S_3^2$), $C_6:S_3$ ($C_3^3:D_6$)

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

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

Order 12: $C_2\times C_6$ ($C_3^3:S_3^2$), $C_2\times C_6$ ($C_3^4:D_6$)

Order 9: $C_3^2$ ($C_6^2.S_3^2$), $C_3^2$ ($C_6^2:S_3^2$), $C_3^2$ ($C_3\wr S_3\times D_4$)

Order 6: $C_6$ ($C_3\wr S_3\times D_6$), $C_6$ ($C_2\times C_3^4:D_6$)

Order 4: $C_2^2$ ($C_3^5:D_6$)

Order 3: $C_3$ ($C_6^2.(C_3\times S_3^2)$), $C_3$ ($C_6^2.(C_3\times S_3^2)$)

Order 2: $C_2$ ($C_2\times C_3^5:D_6$)

Order 1: $C_1$ ($(C_3^3\times C_6^2):D_6$)

Series

Derived series

$(C_3^3\times C_6^2):D_6$

$\rhd$

$(C_3^3\times C_6^2):D_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(C_3^3\times C_6^2):D_6$

$\rhd$

$(C_3^3\times C_6^2):D_6$

$\rhd$

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

$\rhd$

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

$\rhd$

$(C_3^2\times C_6^2):C_6$

$\rhd$

$(C_3^2\times C_6^2):C_6$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3^3:C_6^2$

$\rhd$

$C_3^2\times C_6^2$

$\rhd$

$C_3^2\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_3\times C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_6^2$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(C_3^3\times C_6^2):D_6$

$\rhd$

$(C_3^3\times C_6^2):D_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4:C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_6$

$\lhd$

$C_2\times C_6$

$\lhd$

$C_2\times C_6$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $279 \times 279$ 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 $159 \times 159$ rational character table (warning: may be slow to load).