Abstract group 23328.hk: $\He_3^2:(C_2^2\times D_4)$

• Label: 23328.hk
• Order: \( 2^{5} \cdot 3^{6} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$\He_3^2:(C_2^2\times D_4)$
Order:	\(23328\)\(\medspace = 2^{5} \cdot 3^{6} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4.C_3^2.C_2^6.C_2^2$, of order \(186624\)\(\medspace = 2^{8} \cdot 3^{6} \)
Composition factors:	$C_2$ x 5, $C_3$ x 6
Derived length:	$3$

This group is nonabelian, solvable, and rational. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	1063	728	1944	15704	3888	23328
Conjugacy classes	1	15	17	4	75	4	116
Divisions	1	15	17	4	75	4	116
Autjugacy classes	1	8	14	1	38	1	63

Dimension	1	2	4	8	12	16	24	36	48
Irr. complex chars.	16	20	28	24	8	2	8	8	2	116
Irr. rational chars.	16	20	28	24	8	2	8	8	2	116

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	$452039253120$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{6}=d^{6}=e^{3}=f^{3}=g^{6}=[a,g]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2,6,12)(3,9)(4,8)(5,7)(10,15,18,14)(11,13,16,17), (1,3,10,13,4,12)(2,7,15,18,8,16) \!\cdots\! \rangle$
Transitive group:	36T12496	36T12499	36T12503	36T12505	more information
Direct product:	$C_2$ $\, \times\, $ $(\He_3^2:(C_2\times D_4))$
Semidirect product:	$(C_3^4:D_6^2)$ $\,\rtimes\,$ $C_2$	$C_3^4$ $\,\rtimes\,$ $(D_6^2:C_2)$	$C_3^4$ $\,\rtimes\,$ $(C_6^2:D_4)$	$(C_3^4:C_6:D_4)$ $\,\rtimes\,$ $S_3$	all 37
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:S_3)$ . $D_6^2$	$C_3^3$ . $(S_3^3:C_2^2)$	$(C_3^4.C_6^2)$ . $C_2^3$ (2)	$(\He_3^2:C_2)$ . $C_2^4$	all 11
Aut. group:	$\Aut(C_3^4:\SOPlus(4,2))$

Elements of the group are displayed as permutations of degree 20.

Homology

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

Subgroups

There are 506056 subgroups in 6108 conjugacy classes, 137 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\He_3^2:(C_2\times D_4)$
Commutator:	$G' \simeq$ $\He_3^2:C_2$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_2\times S_3^2:S_3^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times \He_3^2$	$G/\operatorname{Fit} \simeq$ $C_2\times D_4$
Radical:	$R \simeq$ $\He_3^2:(C_2^2\times D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $S_3^2:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $\He_3^2$

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 23328: $\He_3^2:(C_2^2\times D_4)$

Order 11664: $\He_3^2:(C_2\times D_4)$ x 8, $C_2\times \He_3^2:D_4$ x 2, $C_2\times \He_3^2:D_4$ x 2, $C_3^4.D_6^2$, $\He_3^2:(C_2^2\times C_4)$, $C_3^4:D_6^2$

Order 7776: $C_3^4:(D_4\times D_6)$ x 2

Order 5832: $\He_3^2:D_4$ x 8, $\He_3^2:D_4$ x 8, $\He_3^2:C_2^3$ x 6, $\He_3^2:C_2^3$ x 6, $\He_3^2:(C_2\times C_4)$ x 4, $\He_3^2:C_2^3$ x 3, $C_2\times \He_3^2:C_4$ x 2, $C_3^3.C_3^3.C_2^3$, $C_3^4.C_3^2.C_2^3$, $C_3^4.C_3^2.C_2^3$, $C_3^4.C_6^2.C_2$, $C_3^4.C_3^2.C_2^3$, $\He_3^2:C_2^3$

Order 3888: $C_3^4:(S_3\times D_4)$ x 16, $C_3^3:D_6^2$ x 5, $C_3^4:C_6:D_4$ x 4, $C_2\times C_3^4:D_{12}$ x 2, $C_3^4:(C_4\times D_6)$ x 2, $C_3^4:(C_6\times D_4)$ x 2, $C_3^3.D_6^2$, $C_3^3.C_6^2.C_2^2$

Order 2916: $C_3^4:S_3^2$ x 10, $C_3^4.S_3^2$ x 4, $\He_3.\He_3.C_2^2$ x 4, $C_3^4:S_3^2$ x 4, $C_3^4:S_3^2$ x 4, $\He_3^2:C_4$ x 4, $C_3^4.C_6^2$ x 3, $C_3^4.C_6^2$ x 2, $(C_3^2\times \He_3).D_6$ x 2, $\He_3^2:C_2^2$ x 2, $C_3^4.C_6^2$, $C_3^4.C_3:S_3.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^4.S_3^2$

Order 2592: $C_2\times S_3^2:S_3^2$, $C_3^4:(C_2^2\times D_4)$

Order 1944: $C_3^2.S_3^3$ x 32, $C_3^3:\SOPlus(4,2)$ x 16, $C_2\times C_3^3:S_3^2$ x 11, $C_3^2:S_3^3$ x 8, $C_3^2:S_3^3$ x 8, $C_3^4:D_{12}$ x 8, $C_3^3:\SOPlus(4,2)$ x 8, $C_3^4:(C_4\times S_3)$ x 8, $C_2\times C_3^4:D_6$ x 6, $C_2\times C_3^3:S_3^2$ x 5, $C_2\times C_3^3:S_3^2$ x 5, $C_6\times C_3^2:S_3^2$ x 4, $C_2\times C_3^3:S_3^2$ x 3, $C_2\times C_3^4:D_6$ x 3, $C_3^4:C_6:C_4$ x 2, $C_2\times C_3^4:C_{12}$ x 2, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_3^3:C_6\times D_6$, $C_6\times C_3^2:S_3^2$

Order 1458: $\He_3^2:C_2$ x 4, $\He_3\wr C_2$ x 4, $\He_3^2:C_2$ x 2, $\He_3^2:C_2$ x 2, $\He_3^2:C_2$ x 2, $C_2\times \He_3^2$

Order 1296: $C_3^4:(C_2\times D_4)$ x 8, $S_3^2:S_3^2$ x 8, $C_3^2:D_6^2$ x 6, $C_2\times C_3^4:D_4$ x 4, $C_6:S_3^3$ x 3, $C_2\times C_3^3:D_{12}$ x 2, $C_2\times C_3\wr D_4$ x 2, $C_6:S_3^3$, $C_6:S_3^3$, $C_3^4:(C_2^2\times C_4)$, $C_3^3:(C_4\times D_6)$, $C_6^2:S_3^2$

Order 972: $C_3^3:S_3^2$ x 44, $C_3^4:D_6$ x 24, $C_3^3:S_3^2$ 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 12, $C_3^4:D_6$ x 12, $C_3^4:D_6$ x 11, $C_3^3.C_6^2$ x 11, $C_3^4:D_6$ x 9, $C_3^3:C_6^2$ x 5, $C_3^4:D_6$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3^4:C_3:C_4$ x 4, $C_3^4:C_{12}$ x 4, $C_3^3:C_6^2$ x 3, $C_2\times C_3^4:C_6$ x 3, $C_3^3:C_6^2$ x 3, $C_3^4:D_6$ x 2, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^3:C_6^2$

Order 864: $S_3^3:C_2^2$ x 2

Order 729: $\He_3^2$

Order 648: $C_3.S_3^3$ x 48, $C_2\times C_3^2:S_3^2$ x 27, $C_3:S_3^3$ x 18, $C_3^4:D_4$ x 16, $C_6\times C_3^2:D_6$ x 15, $S_3\times C_3^2:D_6$ x 12, $C_3^4:C_2^3$ x 10, $S_3\times C_3^2:D_6$ x 9, $C_3:S_3^3$ x 8, $C_3^3:D_{12}$ x 8, $C_3\wr D_4$ x 8, $C_2\times C_3^2:S_3^2$ x 8, $C_3^4:C_2^3$ x 7, $C_3:S_3^3$ x 6, $C_3^4:C_2^3$ x 5, $C_3^4:C_2^3$ x 4, $C_3^4:(C_2\times C_4)$ x 4, $C_3^3:(C_4\times S_3)$ x 4, $C_2\times C_3^4:C_4$ x 2, $C_2\times C_3\wr C_4$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_2\times C_3^3:D_6$, $C_2\times C_3^3:D_6$, $C_3\times C_6^2:C_6$

Order 486: $C_3^4:S_3$ x 22, $C_3^4:C_6$ x 22, $C_3^4:S_3$ x 18, $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 9, $C_3^4:S_3$ x 8, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 6, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2

Order 432: $S_3^3:C_2$ x 16, $C_2\times S_3^3$ x 11, $C_6^2:D_6$ x 5, $S_3^2:D_6$ x 4, $C_2\times C_3^2:D_{12}$ x 2, $C_6\times \SOPlus(4,2)$ x 2, $C_3^2:C_4\times D_6$ x 2, $C_3:D_6^2$, $C_3:D_6^2$

Order 324: $C_3^2:S_3^2$ x 76, $C_3^3:D_6$ x 60, $C_3^2:S_3^2$ x 48, $C_3^2:C_6^2$ x 40, $C_3\wr C_2^2$ x 38, $C_3^2:S_3^2$ x 36, $C_3^2:S_3^2$ x 32, $C_3^3:D_6$ x 28, $C_3^3:D_6$ x 26, $C_3^2:S_3^2$ x 24, $C_3^2:C_6^2$ x 20, $C_3^2:S_3^2$ x 20, $C_3^3:D_6$ x 19, $C_3^2:S_3^2$ x 15, $D_6\times \He_3$ x 9, $C_3^3:D_6$ x 5, $C_3^2:S_3^2$ x 4, $C_3^4:C_4$ x 4, $C_3\wr C_4$ x 4, $D_6\times C_3^3$ x 3, $C_3^2\times S_3^2$ x 3, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$

Order 288: $C_6^2:D_4$ x 2, $D_6^2:C_2$

Order 243: $C_3^4:C_3$ x 9, $C_3^2\times \He_3$ x 3

Order 216: $S_3^3$ x 80, $C_6.S_3^2$ x 73, $C_6:S_3^2$ x 42, $C_6\times S_3^2$ x 38, $C_6:S_3^2$ x 27, $S_3^2:S_3$ x 16, $C_6^2:C_6$ x 10, $C_3^2:D_{12}$ x 8, $S_3^2:C_6$ x 8, $C_3^2:C_4\times S_3$ x 8, $C_6^2:S_3$ x 8, $C_6^2:C_6$ x 5, $C_2\times C_3^3:C_4$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $C_6^2:S_3$, $S_3\times C_6^2$

Order 162: $C_3^3:C_6$ x 72, $C_3^3:S_3$ x 56, $C_3^3:C_6$ x 44, $C_3^2\wr C_2$ x 40, $C_6\times \He_3$ x 38, $C_3^3:S_3$ x 30, $S_3\times \He_3$ x 18, $C_3^3:C_6$ x 10, $C_3^3\times C_6$ x 6, $S_3\times C_3^3$ x 6, $C_3^3:S_3$ x 4

Order 144: $S_3^2:C_2^2$ x 26, $D_6^2$ x 16, $D_6:D_6$ x 8, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$ x 2, $C_6:D_{12}$ x 2, $C_6^2:C_2^2$, $C_2^2.S_3^2$

Order 108: $C_3\times S_3^2$ x 148, $C_3:S_3^2$ x 136, $C_3^2:D_6$ x 132, $C_3^2:D_6$ x 90, $C_3^2:D_6$ x 86, $C_3:S_3^2$ x 72, $C_3^2:D_6$ x 72, $C_3^2\times D_6$ x 57, $C_3^2:D_6$ x 18, $C_2^2\times \He_3$ x 8, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 4, $C_3\times C_6^2$ x 3

Order 96: $D_4\times D_6$ x 2

Order 81: $C_3\times \He_3$ x 36, $C_3^4$ x 6

Order 72: $S_3\times D_6$ x 196, $\SOPlus(4,2)$ x 40, $C_6\times D_6$ x 35, $C_6:D_6$ x 25, $C_2\times C_3^2:C_4$ x 14, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_6.D_6$ x 4, $C_6:C_{12}$ x 2, $C_2\times C_6^2$

Order 54: $C_3^2:C_6$ x 132, $S_3\times C_3^2$ x 106, $C_3^2:C_6$ x 100, $C_3^2:S_3$ x 80, $C_2\times \He_3$ x 56, $C_3^2\times C_6$ x 43, $C_3^2:S_3$ x 28

Order 48: $S_3\times D_4$ x 16, $C_2^2\times D_6$ x 9, $C_6:D_4$ x 4, $C_6\times D_4$ x 2, $C_2\times D_{12}$ x 2, $C_4\times D_6$ x 2

Order 36: $S_3^2$ x 300, $C_6\times S_3$ x 260, $C_6:S_3$ x 127, $C_6^2$ x 28, $C_3^2:C_4$ x 12, $C_3:C_{12}$ x 4

Order 32: $C_2^2\times D_4$

Order 27: $\He_3$ x 40, $C_3^3$ x 37

Order 24: $C_2\times D_6$ x 116, $C_3:D_4$ x 16, $C_2^2\times C_6$ x 9, $D_{12}$ x 8, $C_4\times S_3$ x 8, $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2

Order 18: $C_3\times S_3$ x 244, $C_3\times C_6$ x 113, $C_3:S_3$ x 104

Order 16: $C_2\times D_4$ x 12, $C_2^4$ x 2, $C_2^2\times C_4$

Order 12: $D_6$ x 250, $C_2\times C_6$ x 61, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 63

Order 8: $C_2^3$ x 21, $D_4$ x 16, $C_2\times C_4$ x 6

Order 6: $S_3$ x 92, $C_6$ x 75

Order 4: $C_2^2$ x 39, $C_4$ x 4

Order 3: $C_3$ x 17

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 23328: $\He_3^2:(C_2^2\times D_4)$

Order 11664: $C_3^4.D_6^2$, $C_2\times \He_3^2:D_4$, $\He_3^2:(C_2^2\times C_4)$, $C_3^4:D_6^2$, $C_2\times \He_3^2:D_4$, $\He_3^2:(C_2\times D_4)$

Order 7776: $C_3^4:(D_4\times D_6)$

Order 5832: $\He_3^2:C_2^3$ x 2, $\He_3^2:C_2^3$ x 2, $\He_3^2:C_2^3$ x 2, $C_3^3.C_3^3.C_2^3$, $C_3^4.C_3^2.C_2^3$, $C_3^4.C_3^2.C_2^3$, $C_3^4.C_6^2.C_2$, $C_3^4.C_3^2.C_2^3$, $\He_3^2:C_2^3$, $C_2\times \He_3^2:C_4$, $\He_3^2:D_4$, $\He_3^2:(C_2\times C_4)$, $\He_3^2:D_4$

Order 3888: $C_3^3:D_6^2$ x 3, $C_3^4:C_6:D_4$ x 2, $C_3^3.D_6^2$, $C_3^3.C_6^2.C_2^2$, $C_2\times C_3^4:D_{12}$, $C_3^4:(C_4\times D_6)$, $C_3^4:(C_6\times D_4)$, $C_3^4:(S_3\times D_4)$

Order 2916: $C_3^4:S_3^2$ x 3, $C_3^4.S_3^2$ x 2, $C_3^4.C_6^2$ x 2, $C_3^4.C_6^2$ x 2, $C_3^4.C_6^2$, $C_3^4.C_3:S_3.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^4.S_3^2$, $(C_3^2\times \He_3).D_6$, $\He_3.\He_3.C_2^2$, $C_3^4:S_3^2$, $\He_3^2:C_2^2$, $C_3^4:S_3^2$, $\He_3^2:C_4$

Order 2592: $C_2\times S_3^2:S_3^2$, $C_3^4:(C_2^2\times D_4)$

Order 1944: $C_2\times C_3^3:S_3^2$ x 6, $C_3^2.S_3^3$ x 6, $C_6\times C_3^2:S_3^2$ x 4, $C_2\times C_3^3:S_3^2$ x 3, $C_2\times C_3^3:S_3^2$ x 3, $C_2\times C_3^4:D_6$ x 3, $C_2\times C_3^3:S_3^2$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_2\times C_3^4:D_6$ x 2, $C_3^3:\SOPlus(4,2)$ x 2, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^3:S_3^2$, $C_3^3:C_6\times D_6$, $C_6\times C_3^2:S_3^2$, $C_3^4:C_6:C_4$, $C_2\times C_3^4:C_{12}$, $C_3^4:D_{12}$, $C_3^3:\SOPlus(4,2)$, $C_3^4:(C_4\times S_3)$

Order 1458: $\He_3^2:C_2$ x 3, $C_2\times \He_3^2$, $\He_3^2:C_2$, $\He_3^2:C_2$, $\He_3^2:C_2$, $\He_3\wr C_2$

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

Order 972: $C_3^3:S_3^2$ x 6, $C_3^4:D_6$ x 6, $C_3^4:D_6$ x 6, $C_3^3.C_6^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:C_6^2$ x 5, $C_3^3:S_3^2$ x 5, $C_3^3:S_3^2$ x 5, $C_3^4:D_6$ x 4, $C_3^3:S_3^2$ x 4, $C_3^4:D_6$ x 4, $C_3^4:D_6$ x 4, $C_3^3:C_6^2$ x 3, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_2\times C_3^4:C_6$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:D_6$, $C_3^3.C_6^2$, $C_3^3:C_6^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_3:C_4$, $C_3^4:C_{12}$

Order 864: $S_3^3:C_2^2$

Order 729: $\He_3^2$

Order 648: $C_2\times C_3^2:S_3^2$ x 17, $C_6\times C_3^2:D_6$ x 10, $C_3^4:C_2^3$ x 9, $S_3\times C_3^2:D_6$ x 8, $C_3.S_3^3$ x 8, $C_3^4:C_2^3$ x 6, $C_3:S_3^3$ x 6, $C_2\times C_3^2:S_3^2$ x 6, $S_3\times C_3^2:D_6$ x 6, $C_3^4:C_2^3$ x 5, $C_3^4:C_2^3$ x 4, $C_3:S_3^3$ x 2, $C_3:S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3^4:C_2^3$, $C_3^4:C_2^3$, $C_2\times C_3^4:C_4$, $C_2\times C_3\wr C_4$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^3:D_6$, $C_2\times C_3^3:D_6$, $C_3\times C_6^2:C_6$

Order 486: $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 6, $C_3^4:S_3$ x 6, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 6, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $C_2\times S_3^3$ x 9, $C_6^2:D_6$ x 3, $S_3^2:D_6$ x 2, $C_3:D_6^2$, $C_3:D_6^2$, $C_2\times C_3^2:D_{12}$, $C_6\times \SOPlus(4,2)$, $C_3^2:C_4\times D_6$, $S_3^3:C_2$

Order 324: $C_3^2:C_6^2$ x 27, $C_3^2:S_3^2$ x 22, $C_3^3:D_6$ x 20, $C_3^3:D_6$ x 19, $C_3^2:C_6^2$ x 17, $C_3^3:D_6$ x 13, $C_3^2:S_3^2$ x 12, $C_3^2:S_3^2$ x 12, $C_3^2:S_3^2$ x 11, $C_3\wr C_2^2$ x 10, $C_3^3:D_6$ x 10, $C_3^2:S_3^2$ x 7, $C_3^2:S_3^2$ x 6, $D_6\times \He_3$ x 6, $C_3^2:S_3^2$ x 6, $C_3^3:D_6$ x 5, $D_6\times C_3^3$ x 3, $C_3^3:D_6$ x 2, $C_3^2\times S_3^2$ x 2, $C_3^2:S_3^2$, $C_3^4:C_4$, $C_3\wr C_4$, $C_3^2:C_6^2$

Order 288: $C_6^2:D_4$ x 2, $D_6^2:C_2$

Order 243: $C_3^4:C_3$ x 6, $C_3^2\times \He_3$ x 3

Order 216: $C_6:S_3^2$ x 30, $C_6\times S_3^2$ x 29, $C_6.S_3^2$ x 27, $C_6:S_3^2$ x 19, $S_3^3$ x 18, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 5, $C_6^2:S_3$ x 5, $S_3^2:S_3$ x 2, $C_6^2:S_3$, $S_3\times C_6^2$, $C_2\times C_3^3:C_4$, $C_2\times C_3^2:C_{12}$, $C_3^2:D_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$

Order 162: $C_6\times \He_3$ x 29, $C_3^3:C_6$ x 27, $C_3^3:C_6$ x 24, $C_3^3:S_3$ x 20, $C_3^2\wr C_2$ x 18, $C_3^3:S_3$ x 12, $C_3^3\times C_6$ x 6, $C_3^3:C_6$ x 6, $S_3\times \He_3$ x 6, $C_3^3:S_3$ x 3, $S_3\times C_3^3$ x 3

Order 144: $D_6^2$ x 13, $S_3^2:C_2^2$ x 7, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $D_6:D_6$, $C_6:D_{12}$, $C_2^2.S_3^2$

Order 108: $C_3^2:D_6$ x 63, $C_3:S_3^2$ x 45, $C_3^2\times D_6$ x 44, $C_3^2:D_6$ x 41, $C_3\times S_3^2$ x 34, $C_3^2:D_6$ x 32, $C_3^2:D_6$ x 30, $C_3:S_3^2$ x 25, $C_3^2:D_6$ x 14, $C_2^2\times \He_3$ x 5, $C_3\times C_6^2$ x 3, $C_3^3:C_4$, $C_3^2:C_{12}$

Order 96: $D_4\times D_6$

Order 81: $C_3\times \He_3$ x 28, $C_3^4$ x 6

Order 72: $S_3\times D_6$ x 86, $C_6\times D_6$ x 26, $C_6:D_6$ x 18, $C_2\times C_3^2:C_4$ x 5, $\SOPlus(4,2)$ x 5, $C_2\times C_6^2$, $C_6\wr C_2$, $C_6:C_{12}$, $C_3:D_{12}$, $C_6.D_6$

Order 54: $C_3^2:C_6$ x 63, $S_3\times C_3^2$ x 44, $C_3^2:C_6$ x 38, $C_3^2\times C_6$ x 37, $C_2\times \He_3$ x 33, $C_3^2:S_3$ x 27, $C_3^2:S_3$ x 18

Order 48: $C_2^2\times D_6$ x 7, $C_6:D_4$ x 2, $C_6\times D_4$, $S_3\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$

Order 36: $C_6\times S_3$ x 122, $S_3^2$ x 90, $C_6:S_3$ x 66, $C_6^2$ x 22, $C_3^2:C_4$ x 3, $C_3:C_{12}$

Order 32: $C_2^2\times D_4$

Order 27: $C_3^3$ x 34, $\He_3$ x 28

Order 24: $C_2\times D_6$ x 53, $C_2^2\times C_6$ x 7, $C_3:D_4$ x 2, $C_2\times C_{12}$, $C_6:C_4$, $D_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 97, $C_3\times C_6$ x 71, $C_3:S_3$ x 53

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

Order 12: $D_6$ x 91, $C_2\times C_6$ x 32, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 50

Order 8: $C_2^3$ x 12, $D_4$ x 2, $C_2\times C_4$ x 2

Order 6: $S_3$ x 41, $C_6$ x 38

Order 4: $C_2^2$ x 18, $C_4$

Order 3: $C_3$ x 14

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 23328: $\He_3^2:(C_2^2\times D_4)$ ($C_1$)

Order 11664: $\He_3^2:(C_2\times D_4)$ ($C_2$) x 8, $C_2\times \He_3^2:D_4$ ($C_2$) x 2, $C_2\times \He_3^2:D_4$ ($C_2$) x 2, $C_3^4.D_6^2$ ($C_2$), $\He_3^2:(C_2^2\times C_4)$ ($C_2$), $C_3^4:D_6^2$ ($C_2$)

Order 5832: $\He_3^2:D_4$ ($C_2^2$) x 8, $\He_3^2:D_4$ ($C_2^2$) x 8, $\He_3^2:C_2^3$ ($C_2^2$) x 4, $\He_3^2:(C_2\times C_4)$ ($C_2^2$) x 4, $\He_3^2:C_2^3$ ($C_2^2$) x 4, $C_2\times \He_3^2:C_4$ ($C_2^2$) x 2, $\He_3^2:C_2^3$ ($C_2^2$) x 2, $C_3^4.C_3^2.C_2^3$ ($C_2^2$), $C_3^4.C_6^2.C_2$ ($C_2^2$), $C_3^4.C_3^2.C_2^3$ ($C_2^2$)

Order 3888: $C_3^4:C_6:D_4$ ($S_3$) x 2

Order 2916: $\He_3^2:C_4$ ($C_2^3$) x 4, $C_3^4:S_3^2$ ($C_2^3$) x 4, $C_3^4.S_3^2$ ($C_2^3$) x 2, $C_3^4.S_3^2$ ($D_4$) x 2, $C_3^4.C_6^2$ ($C_2^3$) x 2, $(C_3^2\times \He_3).D_6$ ($C_2^3$) x 2, $C_3^4.C_3:S_3.C_2$ ($D_4$), $C_3^4.C_6^2$ ($C_2^3$), $C_3^4.C_6^2$ ($D_4$)

Order 1944: $C_3^3:\SOPlus(4,2)$ ($D_6$) x 8, $C_2\times C_3^3:S_3^2$ ($D_6$) x 2, $C_2\times C_3^3:S_3^2$ ($D_6$) x 2, $C_3^4:C_6:C_4$ ($D_6$) x 2

Order 1458: $\He_3^2:C_2$ ($C_2\times D_4$) x 3, $\He_3^2:C_2$ ($C_2\times D_4$) x 2, $C_2\times \He_3^2$ ($C_2\times D_4$), $\He_3^2:C_2$ ($C_2^4$)

Order 972: $C_3^3:S_3^2$ ($C_2\times D_6$) x 4, $C_3^3:S_3^2$ ($C_2\times D_6$) x 4, $C_3^4:C_3:C_4$ ($C_2\times D_6$) x 4, $C_2\times C_3^4:C_6$ ($C_2\times D_6$) x 2

Order 729: $\He_3^2$ ($C_2^2\times D_4$)

Order 648: $C_3^4:C_2^3$ ($S_3^2$)

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

Order 324: $C_3^2:S_3^2$ ($S_3\times D_6$) x 2, $C_3^3:D_6$ ($S_3\times D_6$), $C_3^3:D_6$ ($\SOPlus(4,2)$)

Order 243: $C_3^4:C_3$ ($D_4\times D_6$) x 2

Order 162: $C_3^3:S_3$ ($S_3^2:C_2^2$) x 2, $C_3^3\times C_6$ ($S_3^2:C_2^2$), $C_3^3\times C_6$ ($D_6:D_6$), $C_3^3:S_3$ ($D_6^2$), $C_3^3:S_3$ ($D_6:D_6$)

Order 81: $C_3^4$ ($D_6^2:C_2$), $C_3^4$ ($C_6^2:D_4$)

Order 54: $C_3^2\times C_6$ ($S_3^3:C_2$) x 2

Order 27: $C_3^3$ ($S_3^3:C_2^2$) x 2

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

Order 9: $C_3^2$ ($C_2\times S_3^2:S_3^2$)

Order 2: $C_2$ ($\He_3^2:(C_2\times D_4)$)

Order 1: $C_1$ ($\He_3^2:(C_2^2\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 23328: $\He_3^2:(C_2^2\times D_4)$ ($C_1$)

Order 11664: $C_3^4.D_6^2$ ($C_2$), $C_2\times \He_3^2:D_4$ ($C_2$), $\He_3^2:(C_2^2\times C_4)$ ($C_2$), $C_3^4:D_6^2$ ($C_2$), $C_2\times \He_3^2:D_4$ ($C_2$), $\He_3^2:(C_2\times D_4)$ ($C_2$)

Order 5832: $C_3^4.C_3^2.C_2^3$ ($C_2^2$), $C_3^4.C_6^2.C_2$ ($C_2^2$), $C_3^4.C_3^2.C_2^3$ ($C_2^2$), $\He_3^2:C_2^3$ ($C_2^2$), $C_2\times \He_3^2:C_4$ ($C_2^2$), $\He_3^2:C_2^3$ ($C_2^2$), $\He_3^2:D_4$ ($C_2^2$), $\He_3^2:(C_2\times C_4)$ ($C_2^2$), $\He_3^2:C_2^3$ ($C_2^2$), $\He_3^2:D_4$ ($C_2^2$)

Order 3888: $C_3^4:C_6:D_4$ ($S_3$)

Order 2916: $C_3^4.S_3^2$ ($C_2^3$), $C_3^4.S_3^2$ ($D_4$), $C_3^4.C_3:S_3.C_2$ ($D_4$), $C_3^4.C_6^2$ ($C_2^3$), $C_3^4.C_6^2$ ($D_4$), $C_3^4.C_6^2$ ($C_2^3$), $(C_3^2\times \He_3).D_6$ ($C_2^3$), $\He_3^2:C_4$ ($C_2^3$), $C_3^4:S_3^2$ ($C_2^3$)

Order 1944: $C_2\times C_3^3:S_3^2$ ($D_6$), $C_2\times C_3^3:S_3^2$ ($D_6$), $C_3^4:C_6:C_4$ ($D_6$), $C_3^3:\SOPlus(4,2)$ ($D_6$)

Order 1458: $\He_3^2:C_2$ ($C_2\times D_4$) x 2, $C_2\times \He_3^2$ ($C_2\times D_4$), $\He_3^2:C_2$ ($C_2^4$), $\He_3^2:C_2$ ($C_2\times D_4$)

Order 972: $C_2\times C_3^4:C_6$ ($C_2\times D_6$), $C_3^3:S_3^2$ ($C_2\times D_6$), $C_3^3:S_3^2$ ($C_2\times D_6$), $C_3^4:C_3:C_4$ ($C_2\times D_6$)

Order 729: $\He_3^2$ ($C_2^2\times D_4$)

Order 648: $C_3^4:C_2^3$ ($S_3^2$)

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

Order 324: $C_3^3:D_6$ ($S_3\times D_6$), $C_3^3:D_6$ ($\SOPlus(4,2)$), $C_3^2:S_3^2$ ($S_3\times D_6$)

Order 243: $C_3^4:C_3$ ($D_4\times D_6$)

Order 162: $C_3^3\times C_6$ ($S_3^2:C_2^2$), $C_3^3\times C_6$ ($D_6:D_6$), $C_3^3:S_3$ ($D_6^2$), $C_3^3:S_3$ ($S_3^2:C_2^2$), $C_3^3:S_3$ ($D_6:D_6$)

Order 81: $C_3^4$ ($D_6^2:C_2$), $C_3^4$ ($C_6^2:D_4$)

Order 54: $C_3^2\times C_6$ ($S_3^3:C_2$)

Order 27: $C_3^3$ ($S_3^3:C_2^2$)

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

Order 9: $C_3^2$ ($C_2\times S_3^2:S_3^2$)

Order 2: $C_2$ ($\He_3^2:(C_2\times D_4)$)

Order 1: $C_1$ ($\He_3^2:(C_2^2\times D_4)$)

Series

Derived series

$\He_3^2:(C_2^2\times D_4)$

$\rhd$

$\He_3^2:(C_2^2\times D_4)$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$\He_3^2:(C_2^2\times D_4)$

$\rhd$

$\He_3^2:(C_2^2\times D_4)$

$\rhd$

$\He_3^2:(C_2\times D_4)$

$\rhd$

$\He_3^2:(C_2\times D_4)$

$\rhd$

$\He_3^2:C_2^3$

$\rhd$

$\He_3^2:C_2^3$

$\rhd$

$C_3^4.C_6^2$

$\rhd$

$C_3^4.C_6^2$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$\He_3^2:(C_2^2\times D_4)$

$\rhd$

$\He_3^2:(C_2^2\times D_4)$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2$

$\rhd$

$\He_3^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $116 \times 116$ rational character table. Alternatively, you may search for characters of this group with desired properties.