Abstract group 34992.ns: $\He_3^2:(S_3\times D_4)$

• Label: 34992.ns
• Order: \( 2^{4} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $54$
• Trans deg.: $54$
• Rank: $3$

Group information

Description:	$\He_3^2:(S_3\times D_4)$
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4.F_9:D_6$, of order \(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Composition factors:	$C_2$ x 4, $C_3$ x 7
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	999	2186	1944	23058	6804	34992
Conjugacy classes	1	7	31	2	41	5	87
Divisions	1	7	31	2	41	4	86
Autjugacy classes	1	5	17	2	22	3	50

Dimension	1	2	4	8	16	36	72
Irr. complex chars.	8	14	23	22	8	8	4	87
Irr. rational chars.	8	14	21	23	8	8	4	86

Minimal presentations

Permutation degree:	$54$
Transitive degree:	$54$
Rank:	$3$
Inequivalent generating triples:	$139345920$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{6}=c^{12}=d^{3}=e^{3}=f^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $54$ $\langle(1,5,15)(2,9,23)(3,12,28)(4,14,31)(6,17,22)(7,19,36)(8,21,39)(10,25,35) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:S_3^2)$ $\,\rtimes\,$ $D_6$	$C_3^4$ $\,\rtimes\,$ $(D_6:S_3^2)$	$(\He_3^2:D_4)$ $\,\rtimes\,$ $S_3$	$\He_3^2$ $\,\rtimes\,$ $(S_3\times D_4)$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^3:S_3)$ . $S_3^3$	$C_3^4$ . $(S_3^3:C_2)$ (2)	$C_3^3$ . $(S_3^2:S_3^2)$	$(C_3^4:C_6)$ . $(S_3\times D_6)$ (2)	all 10
Aut. group:	$\Aut(C_3^3:S_3^2)$	$\Aut(C_3^4:S_3^2)$	$\Aut(\He_3^2:C_4)$	$\Aut(C_3^4:S_3^2)$	all 5

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 664769 subgroups in 4875 conjugacy classes, 58 normal (26 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $\He_3^2:(S_3\times D_4)$
Commutator:	$G' \simeq$ $C_3^4.C_3^3.C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_3\wr C_4:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3.C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2\times D_4$
Radical:	$R \simeq$ $\He_3^2:(S_3\times D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $C_3\wr C_4:D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3.C_3^4$

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 34992: $\He_3^2:(S_3\times D_4)$

Order 17496: $C_3^4.S_3^3$ x 2, $C_3^4:C_3.C_6.C_6.C_2$ x 2, $C_3^5.C_3:S_3.C_2^2$, $C_3^5.S_3\wr C_2$, $C_3^4.(C_3\times D_6:S_3)$

Order 11664: $\He_3^2:(C_2\times D_4)$ x 2, $C_3^3.S_3^3.C_2$

Order 8748: $C_3^4:C_3.C_6^2$ x 2, $C_3^5.(C_6\times S_3)$ x 2, $C_3^4.C_3^3.C_2^2$ x 2, $C_3^5:S_3^2$ x 2, $C_3^4.C_3^2.D_6$, $C_3^4.C_3:S_3.C_6$, $C_3^4:C_3.C_6.C_6$

Order 5832: $\He_3^2:D_4$ x 4, $\He_3^2:D_4$ x 4, $C_3^3.S_3^3$ x 2, $C_3^4.(C_3\times D_{12})$ x 2, $C_3^3.S_3^3$ x 2, $\He_3^2:C_2^3$ x 2, $C_3^3.S_3^3$ x 2, $\He_3^2:(C_2\times C_4)$ x 2, $\He_3^2:C_2^3$ x 2, $C_3^4.C_3:S_3.C_2^2$, $C_3^4.S_3\wr C_2$, $C_3^3.C_3^3.D_4$, $C_3^4.C_3:S_3.C_2^2$

Order 4374: $C_3^3.C_3^4.C_2$ x 2, $C_3^4.C_3^3.C_2$ x 2, $C_3^5.C_3.C_6$, $C_3^4.C_3^3.C_2$, $C_3^5.C_3.S_3$

Order 3888: $C_3^4:(S_3\times D_4)$ x 3, $C_3\wr C_4:D_6$

Order 2916: $C_3^4:C_3.C_6.C_2$ x 6, $C_3^4:S_3^2$ x 6, $C_3^4:S_3^2$ x 4, $C_3^4:S_3^2$ x 4, $\He_3^2:C_4$ x 4, $C_3^4:C_3.D_6$ x 3, $C_3^4:C_3.D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3^5.D_6$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^4:C_3.D_6$ x 2, $C_3^5.C_6.C_2$ x 2, $C_3^4:C_3.C_{12}$ x 2, $C_3^5.D_6$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4:C_3.D_6$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4:C_3.D_6$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4.C_3:S_3.C_2$ x 2, $(C_3^2\times \He_3).D_6$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^4:C_3.C_6.C_2$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^3.C_3^3.C_4$, $C_3^4:C_3:C_2.C_6$

Order 2187: $C_3^3.C_3^4$

Order 1944: $C_3^2.S_3^3$ x 12, $C_3^3:\SOPlus(4,2)$ x 6, $C_3^2:S_3^3$ x 4, $C_3^4:(C_4\times S_3)$ x 4, $C_3^4:D_{12}$ x 3, $C_3^3:\SOPlus(4,2)$ x 3, $C_3^2:S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^2:S_3^3$ x 2, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3\wr C_4:S_3$

Order 1458: $\He_3^2:C_2$ x 11, $C_3^4:(C_3\times S_3)$ x 8, $\He_3\wr C_2$ x 8, $C_3^4:(C_3\times S_3)$ x 8, $(C_3^2\times \He_3):S_3$ x 8, $C_3^4:C_3:S_3$ x 6, $\He_3^2:C_2$ x 6, $C_3^4:(C_3\times C_6)$ x 4, $\He_3^2:C_2$ x 4, $\He_3^2:C_2$ x 4, $C_3^4:(C_3\times S_3)$ x 4, $C_3^4:(C_3\times C_6)$ x 4, $C_3^4:(C_3\times S_3)$ x 4, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2

Order 1296: $S_3^2:S_3^2$ x 2, $C_3^4:(C_2\times D_4)$, $C_3^3:(S_3\times D_4)$

Order 972: $C_3^3:S_3^2$ x 54, $C_3^3:S_3^2$ x 22, $C_3^4:D_6$ x 18, $C_3^4:D_6$ x 17, $C_3^3:S_3^2$ x 12, $C_3^3:S_3^2$ x 12, $C_3^3:S_3^2$ x 12, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^4:C_3:C_4$ x 4, $C_3^4:C_{12}$ x 4, $C_3^3:S_3^2$ 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, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$ x 2, $C_3^4:C_{12}$, $C_3^4:C_{12}$

Order 729: $\He_3^2$ x 11, $C_3^4:C_3^2$ x 8, $C_3^4:C_3^2$ x 6, $C_3^5:C_3$ x 2

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

Order 486: $C_3^4:S_3$ x 161, $C_3^4:S_3$ x 54, $C_3^4:C_6$ x 54, $C_3^4:S_3$ x 38, $C_3^4:C_6$ x 26, $C_3^4:C_6$ x 18, $C_3^4:C_6$ x 17, $C_3^4:S_3$ x 17, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$

Order 432: $S_3^3:C_2$ x 3, $D_6:S_3^2$

Order 324: $C_3^2:S_3^2$ x 48, $C_3^3:D_6$ x 46, $C_3\wr C_2^2$ x 36, $C_3^2:S_3^2$ x 26, $C_3^2:S_3^2$ x 24, $C_3^2:S_3^2$ x 20, $C_3^2:S_3^2$ x 20, $C_3^2:S_3^2$ x 14, $C_3^2:S_3^2$ x 10, $C_3^2:S_3^2$ x 6, $C_3\wr C_4$ x 6, $C_3^2\times S_3^2$ x 4, $C_3^4:C_4$ x 2, $C_3^3:D_6$ x 2, $C_3^3:D_6$ x 2, $C_3^2:C_6^2$ x 2, $C_3^4:C_4$, $C_3^3:C_{12}$

Order 243: $C_3^4:C_3$ x 161, $C_3^2\times \He_3$ x 32, $C_3^5$

Order 216: $S_3^3$ x 22, $C_6.S_3^2$ x 8, $S_3^2:S_3$ x 6, $C_6:S_3^2$ x 4, $C_3^2:C_4\times S_3$ x 4, $C_3^2:D_{12}$ x 3, $S_3^2:C_6$ x 3, $C_6:S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^3:D_4$

Order 162: $C_3^3:S_3$ x 174, $C_3^3:C_6$ x 126, $C_3^2\wr C_2$ x 90, $C_3^3:C_6$ x 64, $C_3^3:S_3$ x 50, $S_3\times \He_3$ x 22, $C_3^3:C_6$ x 14, $C_3^3:S_3$ x 12, $S_3\times C_3^3$ x 10, $C_6\times \He_3$ x 2

Order 144: $S_3^2:C_2^2$ x 2, $D_6:D_6$ x 2, $C_{12}:D_6$

Order 108: $C_3:S_3^2$ x 93, $C_3\times S_3^2$ x 82, $C_3^2:D_6$ x 74, $C_3:S_3^2$ x 42, $C_3^2:D_6$ x 16, $C_3^2:D_6$ x 14, $C_3^2:D_6$ x 12, $C_3^2\times D_6$ x 4, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 81: $C_3\times \He_3$ x 228, $C_3^4$ x 34

Order 72: $S_3\times D_6$ x 26, $\SOPlus(4,2)$ x 9, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $C_6.D_6$ x 3, $C_6:D_6$ x 2, $C_2\times C_3^2:C_4$ x 2, $C_3\times D_{12}$ x 2, $D_6:S_3$ x 2, $C_{12}:S_3$

Order 54: $C_3^2:C_6$ x 205, $C_3^2:S_3$ x 182, $S_3\times C_3^2$ x 143, $C_3^2:C_6$ x 120, $C_3^2:S_3$ x 57, $C_2\times \He_3$ x 14, $C_3^2\times C_6$ x 7

Order 48: $S_3\times D_4$ x 3

Order 36: $S_3^2$ x 134, $C_6\times S_3$ x 60, $C_6:S_3$ x 35, $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 5, $C_6^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$

Order 27: $\He_3$ x 182, $C_3^3$ x 150

Order 24: $C_2\times D_6$ x 12, $C_3:D_4$ x 6, $C_4\times S_3$ x 4, $D_{12}$ x 3, $C_3\times D_4$ x 3

Order 18: $C_3\times S_3$ x 246, $C_3:S_3$ x 117, $C_3\times C_6$ x 45

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 71, $C_2\times C_6$ x 12, $C_{12}$ x 4, $C_3:C_4$ x 4

Order 9: $C_3^2$ x 160

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

Order 6: $S_3$ x 72, $C_6$ x 41

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

Order 3: $C_3$ x 31

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 34992: $\He_3^2:(S_3\times D_4)$

Order 17496: $C_3^5.C_3:S_3.C_2^2$, $C_3^5.S_3\wr C_2$, $C_3^4.S_3^3$, $C_3^4:C_3.C_6.C_6.C_2$, $C_3^4.(C_3\times D_6:S_3)$

Order 11664: $C_3^3.S_3^3.C_2$, $\He_3^2:(C_2\times D_4)$

Order 8748: $C_3^4.C_3^2.D_6$, $C_3^4:C_3.C_6^2$, $C_3^4.C_3:S_3.C_6$, $C_3^4:C_3.C_6.C_6$, $C_3^5.(C_6\times S_3)$, $C_3^4.C_3^3.C_2^2$, $C_3^5:S_3^2$

Order 5832: $\He_3^2:D_4$ x 2, $\He_3^2:D_4$ x 2, $C_3^3.S_3^3$, $C_3^4.(C_3\times D_{12})$, $C_3^3.S_3^3$, $C_3^4.C_3:S_3.C_2^2$, $C_3^4.S_3\wr C_2$, $C_3^3.C_3^3.D_4$, $C_3^4.C_3:S_3.C_2^2$, $\He_3^2:C_2^3$, $C_3^3.S_3^3$, $\He_3^2:(C_2\times C_4)$, $\He_3^2:C_2^3$

Order 4374: $C_3^5.C_3.C_6$, $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^5.C_3.S_3$, $C_3^4.C_3^3.C_2$

Order 3888: $C_3^4:(S_3\times D_4)$ x 2, $C_3\wr C_4:D_6$

Order 2916: $C_3^4:C_3.C_6.C_2$ x 3, $C_3^4:S_3^2$ x 3, $C_3^4:C_3.D_6$ x 2, $C_3^4:C_3.D_6$ x 2, $C_3^4:C_3.C_{12}$ x 2, $C_3^4:S_3^2$ x 2, $C_3^4:S_3^2$ x 2, $\He_3^2:C_4$ x 2, $C_3^4:C_3.D_6$, $C_3^4.C_6^2$, $C_3^5.D_6$, $C_3^3.C_3^3.C_4$, $C_3^4.(C_6\times S_3)$, $C_3^3.C_3^3.C_2^2$, $C_3^5.C_6.C_2$, $C_3^5.D_6$, $C_3^4.S_3^2$, $C_3^4:C_3.D_6$, $C_3^4.S_3^2$, $C_3^4:C_3.D_6$, $C_3^4.S_3^2$, $C_3^4.C_3:S_3.C_2$, $(C_3^2\times \He_3).D_6$, $C_3^4:C_3:C_2.C_6$, $C_3^3.C_3^3.C_2^2$, $C_3^4:C_3.C_6.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^3.C_3^4$

Order 1944: $C_3^2.S_3^3$ x 6, $C_3^3:\SOPlus(4,2)$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3^2:S_3^3$ x 2, $C_3^4:D_{12}$ x 2, $C_3^3:\SOPlus(4,2)$ x 2, $C_3^2:S_3^3$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3\wr C_4:S_3$, $C_3^2:S_3^3$

Order 1458: $\He_3^2:C_2$ x 6, $C_3^4:C_3:S_3$ x 4, $C_3^4:(C_3\times C_6)$ x 4, $C_3^4:(C_3\times S_3)$ x 4, $\He_3^2:C_2$ x 4, $\He_3\wr C_2$ x 4, $C_3^4:(C_3\times S_3)$ x 4, $(C_3^2\times \He_3):S_3$ x 4, $\He_3^2:C_2$ x 2, $\He_3^2:C_2$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^4:(C_2\times D_4)$, $C_3^3:(S_3\times D_4)$, $S_3^2:S_3^2$

Order 972: $C_3^3:S_3^2$ x 27, $C_3^3:S_3^2$ x 11, $C_3^4:D_6$ x 10, $C_3^4:D_6$ x 9, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 3, $C_3^4:C_3:C_4$ x 3, $C_3^4:C_{12}$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$

Order 729: $\He_3^2$ x 6, $C_3^4:C_3^2$ x 4, $C_3^4:C_3^2$ x 4, $C_3^5:C_3$

Order 648: $C_3.S_3^3$ x 6, $C_3:S_3^3$ x 4, $C_3^4:D_4$ x 3, $C_3:S_3^3$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3\wr D_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3:S_3^3$, $C_3^3:D_{12}$, $C_3^4:(C_2\times C_4)$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^2:S_3^2$

Order 486: $C_3^4:S_3$ x 82, $C_3^4:S_3$ x 27, $C_3^4:C_6$ x 27, $C_3^4:S_3$ x 19, $C_3^4:C_6$ x 13, $C_3^4:C_6$ x 10, $C_3^4:S_3$ x 10, $C_3^4:C_6$ x 9, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 2, $D_6:S_3^2$

Order 324: $C_3^2:S_3^2$ x 24, $C_3^3:D_6$ x 23, $C_3\wr C_2^2$ x 18, $C_3^2:S_3^2$ x 14, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 10, $C_3^2:S_3^2$ x 10, $C_3^2:S_3^2$ x 7, $C_3^2:S_3^2$ x 5, $C_3\wr C_4$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2\times S_3^2$ x 2, $C_3^4:C_4$ x 2, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 243: $C_3^4:C_3$ x 82, $C_3^2\times \He_3$ x 16, $C_3^5$

Order 216: $S_3^3$ x 11, $C_6.S_3^2$ x 4, $S_3^2:S_3$ x 3, $C_3^2:C_4\times S_3$ x 3, $C_6:S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $S_3^2:C_6$ x 2, $C_6:S_3^2$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^3:D_4$

Order 162: $C_3^3:S_3$ x 87, $C_3^3:C_6$ x 63, $C_3^2\wr C_2$ x 49, $C_3^3:C_6$ x 32, $C_3^3:S_3$ x 25, $S_3\times \He_3$ x 11, $C_3^3:S_3$ x 8, $C_3^3:C_6$ x 7, $S_3\times C_3^3$ x 5, $C_6\times \He_3$

Order 144: $S_3^2:C_2^2$ x 2, $D_6:D_6$, $C_{12}:D_6$

Order 108: $C_3:S_3^2$ x 48, $C_3\times S_3^2$ x 41, $C_3^2:D_6$ x 37, $C_3:S_3^2$ x 21, $C_3^2:D_6$ x 8, $C_3^2:D_6$ x 7, $C_3^2:D_6$ x 6, $C_3^3:C_4$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 81: $C_3\times \He_3$ x 114, $C_3^4$ x 19

Order 72: $S_3\times D_6$ x 13, $\SOPlus(4,2)$ x 7, $C_2\times C_3^2:C_4$ x 2, $C_6\wr C_2$ x 2, $C_3:D_{12}$ x 2, $D_6:S_3$ x 2, $C_6.D_6$ x 2, $C_6:D_6$, $C_{12}:S_3$, $C_3\times D_{12}$

Order 54: $C_3^2:C_6$ x 104, $C_3^2:S_3$ x 91, $S_3\times C_3^2$ x 73, $C_3^2:C_6$ x 60, $C_3^2:S_3$ x 31, $C_2\times \He_3$ x 7, $C_3^2\times C_6$ x 4

Order 48: $S_3\times D_4$ x 2

Order 36: $S_3^2$ x 67, $C_6\times S_3$ x 30, $C_6:S_3$ x 20, $C_3^2:C_4$ x 5, $C_3:C_{12}$ x 4, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$

Order 27: $\He_3$ x 91, $C_3^3$ x 77

Order 24: $C_2\times D_6$ x 6, $C_3:D_4$ x 3, $C_4\times S_3$ x 3, $D_{12}$ x 2, $C_3\times D_4$ x 2

Order 18: $C_3\times S_3$ x 123, $C_3:S_3$ x 64, $C_3\times C_6$ x 25

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 37, $C_2\times C_6$ x 6, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 83

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

Order 6: $S_3$ x 39, $C_6$ x 22

Order 4: $C_2^2$ x 5, $C_4$ x 2

Order 3: $C_3$ x 17

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 34992: $\He_3^2:(S_3\times D_4)$ ($C_1$)

Order 17496: $C_3^4.S_3^3$ ($C_2$) x 2, $C_3^4:C_3.C_6.C_6.C_2$ ($C_2$) x 2, $C_3^5.C_3:S_3.C_2^2$ ($C_2$), $C_3^5.S_3\wr C_2$ ($C_2$), $C_3^4.(C_3\times D_6:S_3)$ ($C_2$)

Order 8748: $C_3^4:C_3.C_6^2$ ($C_2^2$) x 2, $C_3^4.C_3^3.C_2^2$ ($C_2^2$) x 2, $C_3^4.C_3^2.D_6$ ($C_2^2$), $C_3^4.C_3:S_3.C_6$ ($C_2^2$), $C_3^4:C_3.C_6.C_6$ ($C_2^2$)

Order 5832: $\He_3^2:D_4$ ($S_3$) x 2, $C_3^4.S_3\wr C_2$ ($S_3$)

Order 4374: $C_3^5.C_3.C_6$ ($D_4$), $C_3^4.C_3^3.C_2$ ($C_2^3$), $C_3^5.C_3.S_3$ ($D_4$)

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

Order 2187: $C_3^3.C_3^4$ ($C_2\times D_4$)

Order 1458: $\He_3^2:C_2$ ($C_2\times D_6$) x 2, $C_3^4:(C_3\times C_6)$ ($C_2\times D_6$)

Order 972: $C_3^3:S_3^2$ ($S_3^2$) x 2, $C_3^4:C_3:C_4$ ($S_3^2$)

Order 729: $\He_3^2$ ($S_3\times D_4$) x 2, $C_3^4:C_3^2$ ($S_3\times D_4$)

Order 486: $C_3^4:C_6$ ($S_3\times D_6$) x 3, $C_3^4:S_3$ ($\SOPlus(4,2)$)

Order 243: $C_3^4:C_3$ ($D_6:D_6$) x 2, $C_3^5$ ($S_3^2:C_2^2$), $C_3^4:C_3$ ($C_{12}:D_6$)

Order 162: $C_3^3:S_3$ ($S_3^3$)

Order 81: $C_3^4$ ($S_3^3:C_2$) x 3, $C_3^4$ ($D_6:S_3^2$)

Order 27: $C_3^3$ ($S_3^2:S_3^2$) x 2, $C_3^3$ ($C_3^3:(S_3\times D_4)$)

Order 9: $C_3^2$ ($C_3\wr C_4:D_6$)

Order 1: $C_1$ ($\He_3^2:(S_3\times D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 34992: $\He_3^2:(S_3\times D_4)$ ($C_1$)

Order 17496: $C_3^5.C_3:S_3.C_2^2$ ($C_2$), $C_3^5.S_3\wr C_2$ ($C_2$), $C_3^4.S_3^3$ ($C_2$), $C_3^4:C_3.C_6.C_6.C_2$ ($C_2$), $C_3^4.(C_3\times D_6:S_3)$ ($C_2$)

Order 8748: $C_3^4.C_3^2.D_6$ ($C_2^2$), $C_3^4:C_3.C_6^2$ ($C_2^2$), $C_3^4.C_3:S_3.C_6$ ($C_2^2$), $C_3^4:C_3.C_6.C_6$ ($C_2^2$), $C_3^4.C_3^3.C_2^2$ ($C_2^2$)

Order 5832: $C_3^4.S_3\wr C_2$ ($S_3$), $\He_3^2:D_4$ ($S_3$)

Order 4374: $C_3^5.C_3.C_6$ ($D_4$), $C_3^4.C_3^3.C_2$ ($C_2^3$), $C_3^5.C_3.S_3$ ($D_4$)

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

Order 2187: $C_3^3.C_3^4$ ($C_2\times D_4$)

Order 1458: $C_3^4:(C_3\times C_6)$ ($C_2\times D_6$), $\He_3^2:C_2$ ($C_2\times D_6$)

Order 972: $C_3^3:S_3^2$ ($S_3^2$), $C_3^4:C_3:C_4$ ($S_3^2$)

Order 729: $C_3^4:C_3^2$ ($S_3\times D_4$), $\He_3^2$ ($S_3\times D_4$)

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

Order 243: $C_3^5$ ($S_3^2:C_2^2$), $C_3^4:C_3$ ($D_6:D_6$), $C_3^4:C_3$ ($C_{12}:D_6$)

Order 162: $C_3^3:S_3$ ($S_3^3$)

Order 81: $C_3^4$ ($S_3^3:C_2$) x 2, $C_3^4$ ($D_6:S_3^2$)

Order 27: $C_3^3$ ($C_3^3:(S_3\times D_4)$), $C_3^3$ ($S_3^2:S_3^2$)

Order 9: $C_3^2$ ($C_3\wr C_4:D_6$)

Order 1: $C_1$ ($\He_3^2:(S_3\times D_4)$)

Series

Derived series

$\He_3^2:(S_3\times D_4)$

$\rhd$

$\He_3^2:(S_3\times D_4)$

$\rhd$

$C_3^4.C_3^3.C_2$

$\rhd$

$C_3^4.C_3^3.C_2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$\He_3^2:(S_3\times D_4)$

$\rhd$

$\He_3^2:(S_3\times D_4)$

$\rhd$

$C_3^5.C_3:S_3.C_2^2$

$\rhd$

$C_3^5.C_3:S_3.C_2^2$

$\rhd$

$C_3^4:C_3.C_6.C_6$

$\rhd$

$C_3^4:C_3.C_6.C_6$

$\rhd$

$\He_3^2:C_4$

$\rhd$

$\He_3^2:C_4$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2:C_2$

$\rhd$

$\He_3^2$

$\rhd$

$\He_3^2$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$\He_3^2:(S_3\times D_4)$

$\rhd$

$\He_3^2:(S_3\times D_4)$

$\rhd$

$C_3^4.C_3^3.C_2$

$\rhd$

$C_3^4.C_3^3.C_2$

$\rhd$

$C_3^3.C_3^4$

$\rhd$

$C_3^3.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_1$

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 $87 \times 87$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $86 \times 86$ rational character table.