Abstract group 34992.ln: $C_3^3.S_3^2:S_3^2$

• Label: 34992.ln
• Order: \( 2^{4} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^3.S_3^2:S_3^2$
Order:	\(34992\)\(\medspace = 2^{4} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^3.C_3^4.(C_3\times D_4^2)$, of order \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \)
Composition factors:	$C_2$ x 4, $C_3$ x 7
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	9	12	18
Elements	1	1683	1214	972	15570	972	7776	6804	34992
Conjugacy classes	1	7	31	2	48	12	10	15	126
Divisions	1	7	31	2	48	4	6	5	104
Autjugacy classes	1	6	14	1	23	3	5	3	56

Dimension	1	2	4	6	8	12	16	24	36	48	72	144
Irr. complex chars.	8	10	14	8	12	32	1	38	0	3	0	0	126
Irr. rational chars.	8	10	14	4	12	18	1	28	4	0	4	1	104

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$26127360$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid c^{3}=d^{6}=e^{3}=f^{3}=g^{3}=h^{9}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,4,5,13,11,18,14,3,10,17,2,7,15,8,16,6,9,12)(19,20,22,23,21,24), (1,3,9,17,10,7,14,4,11,13,16,18,15,8,2,6,5,12) \!\cdots\! \rangle$
Transitive group:	36T14146				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4.S_3^2)$ $\,\rtimes\,$ $D_6$	$(C_3^4.S_3^3)$ $\,\rtimes\,$ $C_2$	$(C_3^3.S_3^2)$ $\,\rtimes\,$ $S_3^2$	$(C_3^5.S_3^2)$ $\,\rtimes\,$ $C_2^2$	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_3^4$ . $(S_3^3:C_2)$	$C_3^3$ . $(S_3^2:S_3^2)$	$C_3^5$ . $(S_3^2:C_2^2)$	$C_3^3$ . $(S_3\times \He_3:D_4)$	all 12

Elements of the group are displayed as permutations of degree 24.

Homology

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

Subgroups

There are 1084052 subgroups in 5010 conjugacy classes, 49 normal (23 characteristic).

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

Special subgroups

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

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 34992: $C_3^3.S_3^2:S_3^2$

Order 17496: $C_3^5.\SOPlus(4,2)$ x 2, $C_3^5.\SOPlus(4,2)$ x 2, $C_3^4.S_3^3$, $C_3^3.C_3^3:(C_4\times S_3)$, $C_3^4.S_3^3$

Order 11664: $C_3^3.S_3^3:C_2$ x 2

Order 8748: $C_3^3.C_3\wr C_4$ x 2, $C_3^5.S_3^2$ x 2, $C_3^5.S_3^2$, $C_3^4.C_3^2.D_6$, $C_3^5.(C_6\times S_3)$, $C_3^3.C_3^3.D_6$, $C_3^4.C_3^2.D_6$, $C_3^3.C_3^4.C_2^2$, $C_3^5.(C_6\times S_3)$

Order 5832: $C_3^3.S_3^3$ x 3, $C_3^3.(C_3^2:C_4\times S_3)$ x 2, $C_3^3.S_3^3$ x 2, $C_3^4.\SOPlus(4,2)$ x 2, $C_3^4.\SOPlus(4,2)$ x 2, $C_3^4.\SOPlus(4,2)$ x 2, $C_3^4.\SOPlus(4,2)$ x 2, $C_3^5.C_6.C_2^2$, $C_3^3.S_3^3$

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

Order 3888: $C_3^3.S_3^2:C_2^2$, $C_3^3:(S_3\times D_{12})$

Order 2916: $C_3.C_3\wr C_3.D_6$ x 4, $C_3^4.(C_6\times S_3)$ x 4, $C_3^4.S_3^2$ x 4, $C_3.C_3\wr C_3.D_6$ x 4, $C_3^3.C_3^3.C_2^2$ x 3, $C_3.C_3\wr C_3.D_6$ x 3, $C_3^4.C_3:S_3.C_2$ x 2, $C_3.C_3\wr C_3.D_6$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^3.C_3^3:C_4$ x 2, $C_3^3.C_3^2:C_{12}$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4:D_{18}$ x 2, $C_3^4.(C_6\times S_3)$, $C_3.C_3^5.C_2^2$, $C_3^4.S_3^2$, $C_3^5.C_6.C_2$, $(C_3^2\times \He_3).D_6$, $C_3.C_3^5.C_2^2$, $C_3^4.S_3^2$, $C_3^4.C_6^2$, $C_3.C_3^5.C_2^2$, $C_3^4:D_{18}$, $C_3^3:(S_3\times C_{18})$

Order 2187: $C_3^2\times C_3.C_3\wr C_3$

Order 1944: $C_3^3.\SOPlus(4,2)$ x 4, $C_3^2:S_3^3$ x 4, $C_3^2.S_3^3$ x 3, $C_3^4:D_{12}$ x 2, $C_3^4:D_{12}$ x 2, $C_3^2.S_3^3$ x 2, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^3.S_3^2$, $(C_3\times \He_3):C_6:C_4$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^3:(S_3\times C_{12})$, $C_9:S_3^3$

Order 1458: $(C_3^2\times \He_3):S_3$ x 10, $(C_3^2\times \He_3):C_6$ x 8, $(C_3^2\times \He_3):C_6$ x 8, $C_3^4.(C_3\times S_3)$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $(C_3^2\times \He_3):S_3$ x 3, $S_3\times C_3^3.C_3^2$ x 3, $C_3^4:D_9$ x 2, $C_3^4:C_{18}$ x 2, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^4:D_9$, $C_3^4:D_9$, $C_3^4:C_{18}$

Order 1296: $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$

Order 972: $C_3^3:S_3^2$ x 26, $C_3^3:S_3^2$ x 8, $C_3^3.S_3^2$ x 7, $C_3^3.S_3^2$ x 7, $C_3^3:S_3^2$ 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, $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3^3:D_{18}$ x 4, $C_3^3:D_{18}$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:D_{18}$ x 3, $C_3^3:D_{18}$ x 3, $C_3^4:D_6$ x 3, $C_3^3:D_{18}$ x 3, $C_3^3.C_3^2:C_4$ 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, $C_3^4:C_{12}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $S_3\times C_3^2:C_{18}$ x 2, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_9\times C_3:S_3^2$, $(C_6\times \He_3):S_3$, $C_3^3.(C_6\times S_3)$, $(C_6\times \He_3):C_6$

Order 729: $C_3^3.C_3^3$ x 10, $C_3^3\times \He_3$, $C_3^4:C_9$

Order 648: $C_3.S_3^3$ x 12, $C_3:S_3^3$ x 9, $C_3^3:D_{12}$ x 8, $C_3:S_3^3$ x 6, $D_9\times S_3^2$ x 3, $C_3^3:D_{12}$ x 2, $C_3^3:D_{12}$ x 2, $C_3\wr D_4$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3^4:C_2^3$, $C_3:S_3^3$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^2:S_3^2$, $C_2\times C_3^2:D_{18}$

Order 486: $C_3^4:S_3$ x 27, $C_3^4:C_6$ x 26, $C_3^4:C_6$ x 26, $(C_3\times \He_3):S_3$ x 16, $C_3^4:C_6$ x 16, $C_3^3.(C_3\times S_3)$ x 10, $(C_3\times \He_3):C_6$ x 10, $C_3^3:D_9$ x 6, $C_3^3:D_9$ x 6, $C_3^3:D_9$ x 6, $C_3^4:C_6$ x 5, $C_3^3:C_{18}$ x 4, $C_3^4:C_6$ x 4, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 4, $C_3^3:C_{18}$ x 4, $C_3^3:C_{18}$ x 3, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $C_3^3:C_{18}$ x 3, $C_3^3:D_9$ x 3, $C_3^3:D_9$ x 2, $(C_3\times \He_3):C_6$, $C_3^4:S_3$, $S_3\times C_3^4$, $C_3^3:D_9$

Order 432: $S_3^3:C_2$ x 2, $C_2\times C_3^2:D_{12}$, $D_6:S_3^2$

Order 324: $C_3^2:S_3^2$ x 96, $C_3^2:S_3^2$ x 33, $C_3^2:S_3^2$ x 32, $C_3^2:S_3^2$ x 29, $C_3^2:S_3^2$ x 24, $C_3^2:S_3^2$ x 19, $C_3^2:S_3^2$ x 12, $C_3^2:S_3^2$ x 12, $C_3^3:D_6$ x 12, $C_3\wr C_2^2$ x 11, $C_3^2\times S_3^2$ x 9, $C_9:S_3^2$ x 9, $C_3^2:D_{18}$ x 8, $C_3\wr C_4$ x 6, $C_3^2:D_{18}$ x 6, $C_3^2:D_{18}$ x 5, $C_9:S_3^2$ x 3, $C_9\times S_3^2$ x 3, $C_3^2:C_6^2$ x 2, $C_3^3:C_{12}$ x 2, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $C_2\times C_3^2:C_{18}$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 243: $C_3^2\times \He_3$ x 27, $C_3^3.C_3^2$ x 15, $C_3^3:C_9$ x 9, $C_3^5$ x 2, $C_3^3\times C_9$

Order 216: $S_3^3$ x 24, $C_6:S_3^2$ x 10, $C_3^2:D_{12}$ x 8, $S_3^2:S_3$ x 4, $C_6:S_3^2$ x 3, $C_6.S_3^2$ x 3, $S_3\times D_{18}$ x 3, $S_3^2:C_6$ x 2, $C_3^2:C_4\times S_3$ x 2, $C_3^2:D_{12}$ x 2, $C_3^2:D_{12}$ x 2, $C_2\times C_3^2:C_{12}$, $C_6.S_3^2$

Order 162: $C_3^3:S_3$ x 134, $C_3^3:C_6$ x 94, $C_3^3:C_6$ x 94, $C_3^3:C_6$ x 62, $C_3^2\wr C_2$ x 57, $S_3\times C_3^3$ x 37, $C_3^3:S_3$ x 36, $C_3^3:S_3$ x 12, $S_3\times \He_3$ x 12, $C_3^2:D_9$ x 10, $C_3^2:D_9$ x 9, $C_3^2:C_{18}$ x 9, $C_3^2:D_9$ x 8, $C_3^2:C_{18}$ x 8, $C_3^2:D_9$ x 6, $C_3^2:C_{18}$ x 5, $C_3^2\times D_9$ x 5, $C_3^3\times C_6$, $C_6\times \He_3$, $C_3^2:C_{18}$

Order 144: $S_3\times D_{12}$ x 2, $S_3^2:C_2^2$, $D_6:D_6$

Order 108: $C_3:S_3^2$ x 190, $C_3\times S_3^2$ x 71, $C_3:S_3^2$ x 47, $C_3^2:D_6$ x 36, $C_3^2:D_6$ x 26, $S_3\times D_9$ x 23, $C_3^2:D_6$ x 15, $C_3^2\times D_6$ x 10, $C_3^2:C_{12}$ x 6, $C_3^2:D_6$ x 6, $C_3^2:D_6$ x 3, $C_3:D_{18}$ x 3, $S_3\times C_{18}$ x 3, $C_3\times D_{18}$ x 3, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2

Order 81: $C_3\times \He_3$ x 133, $C_3^4$ x 49, $C_3^2:C_9$ x 10, $C_3^2\times C_9$ x 10

Order 72: $S_3\times D_6$ x 31, $C_3:D_{12}$ x 8, $\SOPlus(4,2)$ x 4, $C_6:D_6$ x 2, $C_3:D_{12}$ x 2, $C_6\wr C_2$ x 2, $C_3\times D_{12}$ x 2, $S_3\times C_{12}$ x 2, $C_2\times C_3^2:C_4$, $C_6.D_6$, $C_2\times D_{18}$

Order 54: $C_3^2:C_6$ x 235, $C_3^2:S_3$ x 228, $S_3\times C_3^2$ x 178, $C_3^2:S_3$ x 78, $C_3^2:C_6$ x 60, $C_3^2\times C_6$ x 27, $C_3:D_9$ x 22, $S_3\times C_9$ x 17, $C_3\times D_9$ x 17, $C_3\times C_{18}$ x 3, $C_2\times \He_3$ x 3

Order 48: $S_3\times D_4$ x 2, $C_2\times D_{12}$

Order 36: $S_3^2$ x 159, $C_6:S_3$ x 99, $C_6\times S_3$ x 60, $D_{18}$ x 8, $C_3:C_{12}$ x 6, $C_3^2:C_4$ x 2, $C_3\times C_{12}$ x 2, $C_6^2$ x 2, $C_2\times C_{18}$

Order 27: $C_3^3$ x 235, $\He_3$ x 75, $C_3\times C_9$ x 19

Order 24: $C_2\times D_6$ x 12, $D_{12}$ x 8, $C_3:D_4$ x 4, $C_4\times S_3$ x 2, $C_3\times D_4$ x 2, $C_2\times C_{12}$

Order 18: $C_3:S_3$ x 289, $C_3\times S_3$ x 210, $C_3\times C_6$ x 92, $D_9$ x 9, $C_{18}$ x 5

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 81, $C_2\times C_6$ x 11, $C_{12}$ x 6, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 204, $C_9$ x 4

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

Order 6: $S_3$ x 79, $C_6$ x 48

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: $C_3^3.S_3^2:S_3^2$

Order 17496: $C_3^4.S_3^3$, $C_3^3.C_3^3:(C_4\times S_3)$, $C_3^4.S_3^3$, $C_3^5.\SOPlus(4,2)$, $C_3^5.\SOPlus(4,2)$

Order 11664: $C_3^3.S_3^3:C_2$

Order 8748: $C_3^5.S_3^2$, $C_3^4.C_3^2.D_6$, $C_3^5.(C_6\times S_3)$, $C_3^3.C_3^3.D_6$, $C_3^4.C_3^2.D_6$, $C_3^3.C_3^4.C_2^2$, $C_3^5.(C_6\times S_3)$, $C_3^3.C_3\wr C_4$, $C_3^5.S_3^2$

Order 5832: $C_3^3.S_3^3$ x 2, $C_3^5.C_6.C_2^2$, $C_3^3.(C_3^2:C_4\times S_3)$, $C_3^3.S_3^3$, $C_3^4.\SOPlus(4,2)$, $C_3^4.\SOPlus(4,2)$, $C_3^4.\SOPlus(4,2)$, $C_3^4.\SOPlus(4,2)$, $C_3^3.S_3^3$

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

Order 3888: $C_3^3.S_3^2:C_2^2$, $C_3^3:(S_3\times D_{12})$

Order 2916: $C_3.C_3\wr C_3.D_6$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^4.S_3^2$ x 2, $C_3.C_3\wr C_3.D_6$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3.C_3\wr C_3.D_6$ x 2, $C_3^4.(C_6\times S_3)$, $C_3^4.C_3:S_3.C_2$, $C_3.C_3^5.C_2^2$, $C_3^4.S_3^2$, $C_3^5.C_6.C_2$, $C_3.C_3\wr C_3.D_6$, $(C_3^2\times \He_3).D_6$, $C_3.C_3^5.C_2^2$, $C_3^4.S_3^2$, $C_3^4.C_6^2$, $C_3.C_3^5.C_2^2$, $C_3^4.(C_6\times S_3)$, $C_3^3.C_3^3:C_4$, $C_3^3.C_3^2:C_{12}$, $C_3^4.S_3^2$, $C_3^4.S_3^2$, $C_3^4:D_{18}$, $C_3^4:D_{18}$, $C_3^3:(S_3\times C_{18})$

Order 2187: $C_3^2\times C_3.C_3\wr C_3$

Order 1944: $C_3^3.\SOPlus(4,2)$ x 2, $C_3^2:S_3^3$ x 2, $C_2\times C_3^3.S_3^2$, $C_2\times C_3^3.S_3^2$, $(C_3\times \He_3):C_6:C_4$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^3:(S_3\times C_{12})$, $C_9:S_3^3$, $C_3^2.S_3^3$, $C_3^2.S_3^3$

Order 1458: $(C_3^2\times \He_3):S_3$ x 6, $(C_3^2\times \He_3):C_6$ x 4, $(C_3^2\times \He_3):C_6$ x 4, $(C_3^2\times \He_3):S_3$ x 2, $S_3\times C_3^3.C_3^2$ x 2, $C_3^4.(C_3\times S_3)$ x 2, $C_3^4.(C_3\times S_3)$ x 2, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^4:D_9$, $C_3^4:D_9$, $C_3^4:D_9$, $C_3^4:C_{18}$, $C_3^4:C_{18}$

Order 1296: $S_3\times C_3^2:D_{12}$, $S_3^2:S_3^2$

Order 972: $C_3^3:S_3^2$ x 8, $C_3^3.S_3^2$ x 5, $C_3^3:S_3^2$ x 4, $C_3^3.S_3^2$ x 4, $C_3^3:S_3^2$ 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, $C_3^3:C_6^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:D_{18}$ 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, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3:D_{18}$ x 2, $C_3^3.C_3^2:C_4$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_9\times C_3:S_3^2$, $C_3^4:D_6$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $C_3^3:D_{18}$, $S_3\times C_3^2:C_{18}$, $(C_6\times \He_3):S_3$, $C_3^3.(C_6\times S_3)$, $(C_6\times \He_3):C_6$

Order 729: $C_3^3.C_3^3$ x 6, $C_3^3\times \He_3$, $C_3^4:C_9$

Order 648: $C_3:S_3^3$ x 5, $C_3:S_3^3$ x 4, $C_3^3:D_{12}$ x 4, $C_3.S_3^3$ x 2, $D_9\times S_3^2$ x 2, $C_3^4:C_2^3$, $C_3:S_3^3$, $C_3^3:D_{12}$, $C_3^3:D_{12}$, $C_3\wr D_4$, $C_3^3:(C_4\times S_3)$, $S_3\times C_3^2:C_{12}$, $C_2\times C_3^2:S_3^2$, $C_2\times C_3^2:D_{18}$

Order 486: $(C_3\times \He_3):S_3$ x 9, $C_3^4:S_3$ x 9, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 8, $(C_3\times \He_3):C_6$ x 6, $C_3^4:C_6$ x 5, $C_3^3:D_9$ x 4, $C_3^3.(C_3\times S_3)$ x 4, $C_3^3:D_9$ x 4, $C_3^4:C_6$ x 3, $C_3^4:C_6$ x 3, $C_3^3:D_9$ x 3, $C_3^3:C_{18}$ x 2, $C_3^3:C_{18}$ x 2, $C_3^4:S_3$ x 2, $C_3^3:C_{18}$ x 2, $C_3^3:D_9$ x 2, $C_3^4:C_6$ x 2, $C_3^3:C_{18}$ x 2, $(C_3\times \He_3):C_6$, $C_3^4:S_3$, $S_3\times C_3^4$, $C_3^3:D_9$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^3:D_9$

Order 432: $C_2\times C_3^2:D_{12}$, $S_3^3:C_2$, $D_6:S_3^2$

Order 324: $C_3^2:S_3^2$ x 20, $C_3^2:S_3^2$ x 18, $C_3^2:S_3^2$ x 16, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 13, $C_3\wr C_2^2$ x 7, $C_3^2\times S_3^2$ x 5, $C_9:S_3^2$ x 5, $C_3^2:D_{18}$ x 4, $C_3^2:S_3^2$ x 4, $C_3\wr C_4$ x 3, $C_3^2:D_{18}$ x 3, $C_3^2:D_{18}$ x 3, $C_3^2:C_6^2$ x 2, $C_3^2:S_3^2$ x 2, $C_9:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$ x 2, $C_9\times S_3^2$ x 2, $C_3^2:D_{18}$, $C_3^2:D_{18}$, $C_2\times C_3^2:C_{18}$, $C_3^3:D_6$, $C_3^3:C_{12}$, $C_3^3:D_6$, $C_3^3:D_6$, $C_3^2:C_6^2$

Order 243: $C_3^2\times \He_3$ x 9, $C_3^3.C_3^2$ x 8, $C_3^3:C_9$ x 6, $C_3^5$ x 2, $C_3^3\times C_9$

Order 216: $S_3^3$ x 9, $C_6:S_3^2$ x 6, $C_3^2:D_{12}$ x 4, $C_6:S_3^2$ x 3, $S_3^2:S_3$ x 2, $S_3\times D_{18}$ x 2, $C_2\times C_3^2:C_{12}$, $S_3^2:C_6$, $C_3^2:C_4\times S_3$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$

Order 162: $C_3^2\wr C_2$ x 34, $C_3^3:C_6$ x 29, $C_3^3:S_3$ x 24, $C_3^3:C_6$ x 18, $C_3^3:C_6$ x 18, $C_3^3:S_3$ x 16, $S_3\times C_3^3$ x 16, $C_3^2:D_9$ x 7, $C_3^2:D_9$ x 6, $C_3^2:D_9$ x 5, $C_3^2:C_{18}$ x 5, $C_3^2:C_{18}$ x 4, $C_3^2:C_{18}$ x 3, $C_3^2\times D_9$ x 3, $C_3^2:D_9$ x 3, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2, $C_3^3\times C_6$, $C_6\times \He_3$, $C_3^2:C_{18}$

Order 144: $S_3^2:C_2^2$, $D_6:D_6$, $S_3\times D_{12}$

Order 108: $C_3:S_3^2$ x 65, $C_3\times S_3^2$ x 25, $C_3:S_3^2$ x 24, $S_3\times D_9$ x 11, $C_3^2:D_6$ x 10, $C_3^2:D_6$ x 10, $C_3^2:D_6$ x 8, $C_3^2\times D_6$ x 6, $C_3^2:C_{12}$ x 3, $C_3:D_{18}$ x 2, $C_3^2:D_6$ x 2, $S_3\times C_{18}$ x 2, $C_3\times D_{18}$ x 2, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 81: $C_3^4$ x 24, $C_3\times \He_3$ x 23, $C_3^2:C_9$ x 7, $C_3^2\times C_9$ x 7

Order 72: $S_3\times D_6$ x 14, $C_3:D_{12}$ x 4, $C_6:D_6$ x 2, $\SOPlus(4,2)$ x 2, $C_2\times C_3^2:C_4$, $C_3:D_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_6.D_6$, $C_2\times D_{18}$

Order 54: $C_3^2:C_6$ x 89, $C_3^2:S_3$ x 63, $S_3\times C_3^2$ x 59, $C_3:D_9$ x 13, $C_3^2:S_3$ x 12, $C_3^2:C_6$ x 12, $C_3^2\times C_6$ x 11, $S_3\times C_9$ x 8, $C_3\times D_9$ x 8, $C_3\times C_{18}$ x 2, $C_2\times \He_3$

Order 48: $S_3\times D_4$, $C_2\times D_{12}$

Order 36: $S_3^2$ x 59, $C_6:S_3$ x 35, $C_6\times S_3$ x 25, $D_{18}$ x 5, $C_3:C_{12}$ x 3, $C_6^2$ x 2, $C_3^2:C_4$, $C_3\times C_{12}$, $C_2\times C_{18}$

Order 27: $C_3^3$ x 73, $\He_3$ x 11, $C_3\times C_9$ x 11

Order 24: $C_2\times D_6$ x 8, $D_{12}$ x 4, $C_3:D_4$ x 2, $C_2\times C_{12}$, $C_4\times S_3$, $C_3\times D_4$

Order 18: $C_3:S_3$ x 81, $C_3\times S_3$ x 77, $C_3\times C_6$ x 29, $D_9$ x 6, $C_{18}$ x 3

Order 16: $C_2\times D_4$

Order 12: $D_6$ x 42, $C_2\times C_6$ x 7, $C_{12}$ x 3, $C_3:C_4$

Order 9: $C_3^2$ x 60, $C_9$ x 3

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

Order 6: $S_3$ x 35, $C_6$ x 23

Order 4: $C_2^2$ x 8, $C_4$

Order 3: $C_3$ x 14

Order 2: $C_2$ x 6

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 34992: $C_3^3.S_3^2:S_3^2$ ($C_1$)

Order 17496: $C_3^5.\SOPlus(4,2)$ ($C_2$) x 2, $C_3^5.\SOPlus(4,2)$ ($C_2$) x 2, $C_3^4.S_3^3$ ($C_2$), $C_3^3.C_3^3:(C_4\times S_3)$ ($C_2$), $C_3^4.S_3^3$ ($C_2$)

Order 8748: $C_3^3.C_3\wr C_4$ ($C_2^2$) x 2, $C_3^5.S_3^2$ ($C_2^2$) x 2, $C_3^3.C_3^3.D_6$ ($C_2^2$), $C_3^4.C_3^2.D_6$ ($C_2^2$), $C_3^3.C_3^4.C_2^2$ ($C_2^2$)

Order 5832: $C_3^4.\SOPlus(4,2)$ ($S_3$) x 2

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

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

Order 2187: $C_3^2\times C_3.C_3\wr C_3$ ($C_2\times D_4$)

Order 1458: $(C_3^2\times \He_3):S_3$ ($C_2\times D_6$) x 2

Order 972: $C_3^3.S_3^2$ ($S_3^2$)

Order 729: $C_3^3.C_3^3$ ($S_3\times D_4$) x 2

Order 486: $(C_3\times \He_3):S_3$ ($S_3\times D_6$), $C_3^4:C_6$ ($\SOPlus(4,2)$)

Order 243: $C_3^5$ ($S_3^2:C_2^2$), $C_3^3.C_3^2$ ($D_6:D_6$)

Order 162: $C_3^2\wr C_2$ ($\He_3:D_4$)

Order 81: $C_3^4$ ($S_3^3:C_2$) x 2, $C_3^4$ ($C_2\times \He_3:D_4$)

Order 27: $C_3^3$ ($S_3\times \He_3:D_4$) x 2, $C_3^3$ ($S_3^2:S_3^2$)

Order 18: $C_3:S_3$ ($C_3^3.\SOPlus(4,2)$)

Order 9: $C_3^2$ ($C_3^2:D_6:S_3^2$), $C_3^2$ ($C_3^3.S_3^2:C_2^2$)

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

Order 1: $C_1$ ($C_3^3.S_3^2:S_3^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 34992: $C_3^3.S_3^2:S_3^2$ ($C_1$)

Order 17496: $C_3^4.S_3^3$ ($C_2$), $C_3^3.C_3^3:(C_4\times S_3)$ ($C_2$), $C_3^4.S_3^3$ ($C_2$), $C_3^5.\SOPlus(4,2)$ ($C_2$), $C_3^5.\SOPlus(4,2)$ ($C_2$)

Order 8748: $C_3^3.C_3^3.D_6$ ($C_2^2$), $C_3^4.C_3^2.D_6$ ($C_2^2$), $C_3^3.C_3^4.C_2^2$ ($C_2^2$), $C_3^3.C_3\wr C_4$ ($C_2^2$), $C_3^5.S_3^2$ ($C_2^2$)

Order 5832: $C_3^4.\SOPlus(4,2)$ ($S_3$)

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

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

Order 2187: $C_3^2\times C_3.C_3\wr C_3$ ($C_2\times D_4$)

Order 1458: $(C_3^2\times \He_3):S_3$ ($C_2\times D_6$)

Order 972: $C_3^3.S_3^2$ ($S_3^2$)

Order 729: $C_3^3.C_3^3$ ($S_3\times D_4$)

Order 486: $(C_3\times \He_3):S_3$ ($S_3\times D_6$), $C_3^4:C_6$ ($\SOPlus(4,2)$)

Order 243: $C_3^5$ ($S_3^2:C_2^2$), $C_3^3.C_3^2$ ($D_6:D_6$)

Order 162: $C_3^2\wr C_2$ ($\He_3:D_4$)

Order 81: $C_3^4$ ($S_3^3:C_2$), $C_3^4$ ($C_2\times \He_3:D_4$)

Order 27: $C_3^3$ ($S_3^2:S_3^2$), $C_3^3$ ($S_3\times \He_3:D_4$)

Order 18: $C_3:S_3$ ($C_3^3.\SOPlus(4,2)$)

Order 9: $C_3^2$ ($C_3^2:D_6:S_3^2$), $C_3^2$ ($C_3^3.S_3^2:C_2^2$)

Order 3: $C_3$ ($C_3^3.S_3^3:C_2$)

Order 1: $C_1$ ($C_3^3.S_3^2:S_3^2$)

Series

Derived series

$C_3^3.S_3^2:S_3^2$

$\rhd$

$C_3^3.S_3^2:S_3^2$

$\rhd$

$C_3^3.C_3^4.C_2$

$\rhd$

$C_3^3.C_3^4.C_2$

$\rhd$

$C_3^3.C_3^2$

$\rhd$

$C_3^3.C_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^3.S_3^2:S_3^2$

$\rhd$

$C_3^3.S_3^2:S_3^2$

$\rhd$

$C_3^4.S_3^3$

$\rhd$

$C_3^4.S_3^3$

$\rhd$

$C_3^4.C_3^2.D_6$

$\rhd$

$C_3^4.C_3^2.D_6$

$\rhd$

$C_3^3.C_3^4.C_2$

$\rhd$

$C_3^3.C_3^4.C_2$

$\rhd$

$C_3^2\times C_3.C_3\wr C_3$

$\rhd$

$C_3^2\times C_3.C_3\wr C_3$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3.C_3^3$

$\rhd$

$C_3^3.C_3^2$

$\rhd$

$C_3^3.C_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^3.S_3^2:S_3^2$

$\rhd$

$C_3^3.S_3^2:S_3^2$

$\rhd$

$C_3^3.C_3^4.C_2$

$\rhd$

$C_3^3.C_3^4.C_2$

$\rhd$

$C_3^2\times C_3.C_3\wr C_3$

$\rhd$

$C_3^2\times C_3.C_3\wr C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 3 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 $126 \times 126$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $104 \times 104$ rational character table.