Abstract group 4094064.a: $C_3^6:\SL(3,3)$

• Label: 4094064.a
• Order: \( 2^{4} \cdot 3^{9} \cdot 13 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 13 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 1 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{9} \cdot 13 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $39$
• Trans deg.: $39$
• Rank: $2$

Group information

Description:	$C_3^6:\SL(3,3)$
Order:	\(4094064\)\(\medspace = 2^{4} \cdot 3^{9} \cdot 13 \)
Exponent:	\(936\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 13 \)
Automorphism group:	$C_3^6.\GL(3,3)$, of order \(8188128\)\(\medspace = 2^{5} \cdot 3^{9} \cdot 13 \)
Composition factors:	$C_3$ x 6, $\SL(3,3)$
Derived length:	$0$

This group is nonabelian and perfect (hence nonsolvable).

Group statistics

Order	1	2	3	4	6	8	9	12	13	18	24
Elements	1	1053	76544	56862	311688	341172	454896	454896	1259712	454896	682344	4094064
Conjugacy classes	1	1	11	1	17	2	10	5	4	6	4	62
Divisions	1	1	8	1	10	1	5	3	1	3	1	35
Autjugacy classes	1	1	8	1	10	2	5	3	4	3	2	40

Dimension	1	12	13	16	26	27	39	52	64	78	104	156	208	234	312	416	468	624	702	936	1404
Irr. complex chars.	1	1	5	4	9	1	5	2	0	4	4	5	2	4	8	0	2	1	4	0	0	62
Irr. rational chars.	1	1	1	0	3	1	1	2	1	4	2	4	2	0	1	1	2	5	0	1	2	35

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$39$
Rank:	$2$
Inequivalent generating pairs:	$1764672$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	13	26	26
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $39$ $\langle(4,10,29)(5,12,28)(6,11,30)(7,32,26,36,39,20)(8,33,25,35,38,19)(9,31,27,34,37,21) \!\cdots\! \rangle$
Transitive group:	39T198				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^6$ $\,\rtimes\,$ $\SL(3,3)$				more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 39.

Homology

Abelianization:	$C_1 $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 9234772 subgroups in 1623 conjugacy classes, 3 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^6:\SL(3,3)$
Commutator:	$G' \simeq$ $C_3^6:\SL(3,3)$	$G/G' \simeq$ $C_1$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^6:\SL(3,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $\SL(3,3)$
Radical:	$R \simeq$ $C_3^6$	$G/R \simeq$ $\SL(3,3)$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $\SL(3,3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6:\He_3$
13-Sylow subgroup:	$P_{ 13 } \simeq$ $C_{13}$

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 4094064: $C_3^6:\SL(3,3)$

Order 314928: $C_3^6:(C_3^2:\GL(2,3))$, $C_3^6:C_3^2:\GL(2,3)$

Order 157464: $C_3^6:\PU(3,2)$, $C_3^6:\PU(3,2)$

Order 104976: $C_3^3.C_3^4:\GL(2,3)$, $C_3^5:F_9:C_6$, $C_3^6:F_9:C_2$

Order 78732: $C_3^6:C_3^2:D_6$

Order 52488: $C_3^5.C_3:S_3.C_6.C_2$, $C_3^5.\PU(3,2)$, $C_3^5:\PU(3,2)$, $C_3^6:\PSU(3,2)$, $C_3^6:F_9$, $C_3^6:\SOPlus(4,2)$, $C_3^6:F_9$, $C_3^6:\PSU(3,2)$

Order 39366: $C_3^6:C_3^2:S_3$, $C_3^6:C_3^2:S_3$, $C_3^6:C_3^2:C_6$

Order 34992: $C_3^5:F_9:C_2$, $C_3^5:F_9:C_2$, $C_3^6:\GL(2,3)$

Order 28431: $C_3^6.C_{13}:C_3$

Order 26244: $C_3^5.C_3^3.C_2^2$, $C_3^5.C_3^3.C_4$, $C_3^4.C_3\wr C_4$, $C_3^6:S_3^2$, $C_3^5:C_3^2:D_6$

Order 19683: $C_3^6:\He_3$

Order 17496: $C_3^5.S_3\wr C_2$, $C_3^5:\PSU(3,2)$, $C_3^5:F_9$, $C_3^5:\SOPlus(4,2)$, $C_3^5:F_9$, $C_3^5:\SOPlus(4,2)$, $C_3^5:\PSU(3,2)$, $C_3^6:\SL(2,3)$, $C_3\wr S_4$

Order 13122: $C_3^5.C_3^3.C_2$, $C_3^4.C_3^3.C_6$, $C_3^5:C_3^2:C_6$, $C_3^4.C_3^2\wr C_2$, $C_3^5:C_3^2:S_3$, $C_3^6:(C_3\times S_3)$, $C_3^5:C_3^2:S_3$, $C_3^5:C_3^2:S_3$, $C_3^4.C_3^3:C_6$, $C_3^6:(C_3\times S_3)$

Order 11664: $C_3^5:\GL(2,3)$ x 2, $C_3^3:C_3^2:\GL(2,3)$, $C_3^4:F_9:C_2$, $C_3^6:\SD_{16}$

Order 9477: $C_3^6:C_{13}$

Order 8748: $C_3^5:S_3^2$ x 2, $C_3^5.C_3^2.C_4$, $C_3^5.S_3^2$, $C_3^4:C_3.C_6^2$, $C_3^5:S_3^2$, $C_3^5:C_3^2:C_4$, $C_3^5:S_3^2$, $C_3^3.C_3\wr C_4$, $C_3^5:S_3^2$, $C_3^3.C_3\wr C_4$, $C_3\wr A_4$

Order 6561: $C_3^5.C_3^3$, $C_3^5.\He_3$, $C_3^5.\He_3$, $C_3^5.\He_3$, $C_3^5:\He_3$, $C_3^6:C_3^2$, $C_3^6:C_3^2$

Order 5832: $C_3^5:\SL(2,3)$ x 3, $C_3^4.C_6^2.C_2$, $C_3^4:\SOPlus(4,2)$, $C_3^4:F_9$, $C_3^3:\PU(3,2)$, $C_3^4:\PSU(3,2)$, $C_3^4:\SOPlus(4,2)$, $C_3^6:Q_8$, $C_3^5:C_{24}$, $C_3^5:S_4$

Order 5616: $\SL(3,3)$

Order 4374: $C_3^5:(C_3\times S_3)$ x 4, $C_3^5:(C_3\times S_3)$ x 3, $C_3^5:(C_3\times S_3)$ x 2, $C_3^5:(C_3\times S_3)$ x 2, $C_3^5.(C_3\times S_3)$ x 2, $C_3^3.C_3^2\wr C_2$ x 2, $C_3^5:(C_3\times S_3)$ x 2, $C_3^5.C_3.S_3$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^6:C_6$, $C_3^3.C_3^2\wr C_2$, $C_3^3.C_3\wr S_3$, $C_3^3.C_3^2\wr C_2$, $C_3^5.(C_3\times S_3)$, $C_3^3:C_3^2:C_{18}$, $C_3^2\wr S_3$

Order 3888: $C_3^5:\SD_{16}$ x 2, $C_3^3:F_9:C_2$, $C_3^5:\SD_{16}$, $C_3^4:\GL(2,3)$

Order 2916: $C_3^5:D_6$ x 4, $C_3^4:S_3^2$ x 2, $C_3^6.C_2^2$, $C_3^4:C_3.C_6.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^5:D_6$, $C_3^5:C_{12}$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^4:C_3^2:C_4$, $C_3^3:C_3^2:C_{12}$, $C_3^4:C_3^2:C_4$, $C_3^5:A_4$, $C_3^5.A_4$

Order 2187: $C_3^5:C_3^2$ x 5, $C_3^5:C_3^2$ x 3, $C_3^4.\He_3$ x 3, $C_3^5:C_3^2$ x 2, $C_3^4.C_3^3$ x 2, $C_3^4.\He_3$ x 2, $C_3^4.\He_3$ x 2, $C_3^3.C_3^4$, $C_3^6.C_3$, $C_3^4.C_3^3$, $C_3^4.C_3^3$, $C_3^4.C_3^3$, $C_3^2\wr C_3$

Order 1944: $C_3^5:D_4$ x 4, $C_3^5:Q_8$ x 2, $C_3^3:F_9$ x 2, $C_3^5:Q_8$, $C_3^5:D_4$, $C_3^4:C_{24}$, $C_3\wr S_4$, $C_3^4:\SL(2,3)$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$, $C_3^3.F_9$

Order 1458: $C_3^4:(C_3\times S_3)$ x 9, $C_3^5:S_3$ x 7, $C_3^5:C_6$ x 7, $(C_3^2\times \He_3):C_6$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $C_3^5:S_3$ x 4, $C_3\wr S_3:C_3^2$ x 3, $C_3^4:(C_3\times S_3)$ x 3, $C_3^4.(C_3\times S_3)$ x 3, $C_3^4.(C_3\times S_3)$ x 3, $C_3^4.(C_3\times S_3)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:C_3:S_3$ x 2, $(C_3^2\times \He_3):S_3$, $(C_3\times \He_3):C_{18}$, $C_3^5:C_6$, $(C_3^2\times \He_3):S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.C_6$, $C_3^4:C_3:S_3$

Order 1296: $C_3^4:\SD_{16}$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^3:\GL(2,3)$

Order 1053: $C_3^3:C_{13}:C_3$ x 2

Order 972: $C_3^3:C_6^2$ x 6, $C_3^4:C_{12}$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^4:C_{12}$ x 2, $C_3^4:D_6$ x 2, $C_3^3\times S_3^2$, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_3^2:C_4$, $(C_3^2\times \He_3):C_4$, $C_3^4.A_4$, $C_3^4:D_6$

Order 729: $C_3^3.C_3^3$ x 12, $C_3^5:C_3$ x 10, $C_3^4:C_3^2$ x 9, $C_3^3.C_3^3$ x 7, $C_3^3.C_3^3$ x 6, $C_3^3.C_3^3$ x 6, $C_3^4:C_3^2$ x 4, $C_3^2.C_3^4$ x 3, $C_3^3.C_3^3$ x 3, $C_3^4.C_3^2$ x 2, $C_3^4.C_3^2$ x 2, $C_3^5:C_3$ x 2, $C_3^5:C_3$ x 2, $C_3^4.C_3^2$ x 2, $C_3^4.C_3^2$ x 2, $C_3^5.C_3$ x 2, $C_3^4:C_3^2$ x 2, $C_3^4:C_9$, $C_3^6$, $C_3^5.C_3$, $C_3^3:\He_3$

Order 648: $C_3\wr D_4$ x 5, $C_3^4:Q_8$ x 2, $C_3^4:D_4$ x 2, $C_3^2:F_9$ x 2, $C_3^3:\SL(2,3)$ x 2, $C_3^3:\SL(2,3)$ x 2, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^3:C_{24}$, $C_3^3:S_4$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^3:C_{24}$, $\Unitary(3,2)$, $C_3^3:S_4$

Order 486: $C_3^4:S_3$ x 19, $C_3\wr C_3:C_6$ x 15, $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 9, $C_3^4:C_6$ x 8, $C_3^4:S_3$ x 6, $C_3\wr S_3:C_3$ x 6, $C_3^4.C_6$ x 6, $C_3^4:S_3$ x 5, $C_3^4:S_3$ x 5, $C_3^4:C_6$ 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 3, $C_3^3:(C_3\times S_3)$ x 3, $C_3^4:S_3$ x 3, $(C_3^2\times C_9):S_3$ x 3, $C_3^3:C_{18}$, $S_3\times C_3^4$, $(C_3\times \He_3):S_3$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $C_3^2:\GL(2,3)$ x 3, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$

Order 351: $C_3^3:C_{13}$ x 2

Order 324: $C_3^2\times S_3^2$ x 11, $C_3\wr C_4$ x 9, $C_3^3:D_6$ x 9, $C_3\wr C_2^2$ x 8, $C_3^3:D_6$ x 3, $C_3^4:C_4$ x 2, $C_3^2:S_3^2$ x 2, $(C_3\times \He_3):C_4$ x 2, $C_3^3:A_4$, $D_6\times C_3^3$, $C_3^3:C_{12}$, $C_3^3:A_4$, $C_3^3:C_{12}$, $C_3^3:D_6$

Order 243: $C_3^4:C_3$ x 32, $C_3^3:C_3^2$ x 24, $C_3^2.C_3^3$ x 12, $C_3^4:C_3$ x 11, $\He_3:C_3^2$ x 9, $C_3^4.C_3$ x 9, $C_3^2\times \He_3$ x 6, $C_9.\He_3$ x 6, $C_3^3:C_9$ x 6, $C_3^5$ x 5, $C_9.C_3^3$ x 5, $C_3^3:C_3^2$ x 3, $C_3^2.C_3^3$ x 3, $C_9.\He_3$ x 3, $C_9:C_3^3$, $C_3^3:C_9$

Order 216: $S_3^2:C_6$ x 5, $\PU(3,2)$ x 5, $C_3^3:D_4$ x 3, $C_3^3:Q_8$ x 2, $C_3^3:Q_8$ x 2, $S_3^2:S_3$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2, $C_3^2:S_4$, $\SU(3,2)$, $\He_3:C_8$, $C_6^2:C_6$, $C_3^3:Q_8$, $C_3^3:Q_8$, $C_3^3:C_8$, $C_3^2:C_{24}$

Order 162: $C_3\wr S_3$ x 29, $C_3^3:S_3$ x 26, $C_3^2\wr C_2$ x 25, $C_3^3:C_6$ x 22, $S_3\times C_3^3$ x 19, $C_3^3.C_6$ x 8, $S_3\times \He_3$ x 6, $C_3^3:C_6$ x 6, $C_3^2:C_{18}$ x 4, $\He_3.C_6$ x 4, $\He_3.C_6$ x 4, $\He_3.C_6$ x 3, $C_3^3:S_3$ x 2, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^2:C_{18}$

Order 144: $F_9:C_2$ x 3, $C_{12}.D_6$, $C_3\times \GL(2,3)$, $C_3^2:\SD_{16}$

Order 108: $C_3\times S_3^2$ x 17, $C_3^2\times D_6$ x 10, $C_3^2:C_{12}$ x 5, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 2, $C_3^2:C_{12}$ x 2, $\He_3:C_4$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2:A_4$, $C_3^2.A_4$

Order 81: $C_3\wr C_3$ x 51, $C_3\times \He_3$ x 35, $C_3^4$ x 21, $C_9:C_3^2$ x 16, $C_3.\He_3$ x 12, $\He_3:C_3$ x 9, $C_3^2:C_9$ x 9, $C_9.C_3^2$ x 7, $C_3.\He_3$ x 3, $C_3^2\times C_9$

Order 72: $C_6\wr C_2$ x 4, $\PSU(3,2)$ x 3, $\SOPlus(4,2)$ x 3, $F_9$ x 3, $C_3\times S_4$, $D_4\times C_3^2$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3^2:Q_8$, $D_6:S_3$, $C_3^2:C_8$, $C_3:C_{24}$

Order 54: $S_3\times C_3^2$ x 66, $C_3^2:C_6$ x 18, $C_3^2:S_3$ x 12, $C_3^2:C_6$ x 11, $C_3^2\times C_6$ x 9, $S_3\times C_9$ x 6, $C_9:C_6$ x 5, $C_2\times \He_3$ x 3, $C_3\times C_{18}$

Order 48: $\GL(2,3)$, $C_3\times \SD_{16}$, $C_3:\SD_{16}$

Order 39: $C_{13}:C_3$

Order 36: $C_6\times S_3$ x 13, $C_3:C_{12}$ x 7, $C_6^2$ x 5, $S_3^2$ x 4, $C_3^2:C_4$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_2^2:C_9$, $C_3\times A_4$

Order 27: $C_3^3$ x 56, $\He_3$ x 34, $C_9:C_3$ x 25, $C_3\times C_9$ x 13

Order 24: $C_3\times D_4$ x 3, $\SL(2,3)$ x 2, $C_3:D_4$, $C_3:Q_8$, $C_{24}$, $S_4$, $C_3\times Q_8$, $C_3:C_8$

Order 18: $C_3\times S_3$ x 43, $C_3\times C_6$ x 21, $C_3:S_3$ x 4, $C_{18}$ x 3

Order 16: $\SD_{16}$

Order 13: $C_{13}$

Order 12: $C_2\times C_6$ x 5, $C_{12}$ x 3, $D_6$ x 2, $C_3:C_4$ x 2, $A_4$

Order 9: $C_3^2$ x 40, $C_9$ x 5

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 10, $S_3$ x 5

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

Order 3: $C_3$ x 8

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4094064: $C_3^6:\SL(3,3)$

Order 314928: $C_3^6:(C_3^2:\GL(2,3))$, $C_3^6:C_3^2:\GL(2,3)$

Order 157464: $C_3^6:\PU(3,2)$, $C_3^6:\PU(3,2)$

Order 104976: $C_3^3.C_3^4:\GL(2,3)$, $C_3^5:F_9:C_6$, $C_3^6:F_9:C_2$

Order 78732: $C_3^6:C_3^2:D_6$

Order 52488: $C_3^5.C_3:S_3.C_6.C_2$, $C_3^5.\PU(3,2)$, $C_3^5:\PU(3,2)$, $C_3^6:\PSU(3,2)$, $C_3^6:F_9$, $C_3^6:\SOPlus(4,2)$, $C_3^6:F_9$, $C_3^6:\PSU(3,2)$

Order 39366: $C_3^6:C_3^2:S_3$, $C_3^6:C_3^2:S_3$, $C_3^6:C_3^2:C_6$

Order 34992: $C_3^5:F_9:C_2$, $C_3^5:F_9:C_2$, $C_3^6:\GL(2,3)$

Order 28431: $C_3^6.C_{13}:C_3$

Order 26244: $C_3^5.C_3^3.C_2^2$, $C_3^5.C_3^3.C_4$, $C_3^4.C_3\wr C_4$, $C_3^6:S_3^2$, $C_3^5:C_3^2:D_6$

Order 19683: $C_3^6:\He_3$

Order 17496: $C_3^5.S_3\wr C_2$, $C_3^5:\PSU(3,2)$, $C_3^5:F_9$, $C_3^5:\SOPlus(4,2)$, $C_3^5:F_9$, $C_3^5:\SOPlus(4,2)$, $C_3^5:\PSU(3,2)$, $C_3^6:\SL(2,3)$, $C_3\wr S_4$

Order 13122: $C_3^5.C_3^3.C_2$, $C_3^4.C_3^3.C_6$, $C_3^5:C_3^2:C_6$, $C_3^4.C_3^2\wr C_2$, $C_3^5:C_3^2:S_3$, $C_3^6:(C_3\times S_3)$, $C_3^5:C_3^2:S_3$, $C_3^5:C_3^2:S_3$, $C_3^4.C_3^3:C_6$, $C_3^6:(C_3\times S_3)$

Order 11664: $C_3^5:\GL(2,3)$ x 2, $C_3^3:C_3^2:\GL(2,3)$, $C_3^4:F_9:C_2$, $C_3^6:\SD_{16}$

Order 9477: $C_3^6:C_{13}$

Order 8748: $C_3^5:S_3^2$ x 2, $C_3^5.C_3^2.C_4$, $C_3^5.S_3^2$, $C_3^4:C_3.C_6^2$, $C_3^5:S_3^2$, $C_3^5:C_3^2:C_4$, $C_3^5:S_3^2$, $C_3^3.C_3\wr C_4$, $C_3^5:S_3^2$, $C_3^3.C_3\wr C_4$, $C_3\wr A_4$

Order 6561: $C_3^5.C_3^3$, $C_3^5.\He_3$, $C_3^5.\He_3$, $C_3^5.\He_3$, $C_3^5:\He_3$, $C_3^6:C_3^2$, $C_3^6:C_3^2$

Order 5832: $C_3^5:\SL(2,3)$ x 3, $C_3^4.C_6^2.C_2$, $C_3^4:\SOPlus(4,2)$, $C_3^4:F_9$, $C_3^3:\PU(3,2)$, $C_3^4:\PSU(3,2)$, $C_3^4:\SOPlus(4,2)$, $C_3^6:Q_8$, $C_3^5:C_{24}$, $C_3^5:S_4$

Order 5616: $\SL(3,3)$

Order 4374: $C_3^5:(C_3\times S_3)$ x 4, $C_3^5:(C_3\times S_3)$ x 3, $C_3^5:(C_3\times S_3)$ x 2, $C_3^5:(C_3\times S_3)$ x 2, $C_3^3.C_3^2\wr C_2$ x 2, $C_3^5:(C_3\times S_3)$ x 2, $C_3^5.C_3.S_3$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^6:C_6$, $C_3^3.C_3^2\wr C_2$, $C_3^3.C_3\wr S_3$, $C_3^3.C_3^2\wr C_2$, $C_3^5.(C_3\times S_3)$, $C_3^5.(C_3\times S_3)$, $C_3^3:C_3^2:C_{18}$, $C_3^2\wr S_3$

Order 3888: $C_3^5:\SD_{16}$ x 2, $C_3^3:F_9:C_2$, $C_3^5:\SD_{16}$, $C_3^4:\GL(2,3)$

Order 2916: $C_3^5:D_6$ x 4, $C_3^4:S_3^2$ x 2, $C_3^6.C_2^2$, $C_3^4:C_3.C_6.C_2$, $C_3^4.(C_6\times S_3)$, $C_3^5:D_6$, $C_3^5:C_{12}$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^4:S_3^2$, $C_3^4:C_3^2:C_4$, $C_3^3:C_3^2:C_{12}$, $C_3^4:C_3^2:C_4$, $C_3^5:A_4$, $C_3^5.A_4$

Order 2187: $C_3^5:C_3^2$ x 5, $C_3^5:C_3^2$ x 3, $C_3^5:C_3^2$ x 2, $C_3^4.\He_3$ x 2, $C_3^4.\He_3$ x 2, $C_3^4.\He_3$ x 2, $C_3^3.C_3^4$, $C_3^6.C_3$, $C_3^4.C_3^3$, $C_3^4.C_3^3$, $C_3^4.C_3^3$, $C_3^4.C_3^3$, $C_3^2\wr C_3$

Order 1944: $C_3^5:D_4$ x 4, $C_3^5:Q_8$ x 2, $C_3^3:F_9$ x 2, $C_3^5:Q_8$, $C_3^5:D_4$, $C_3^4:C_{24}$, $C_3\wr S_4$, $C_3^4:\SL(2,3)$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$, $C_3^3.F_9$

Order 1458: $C_3^4:(C_3\times S_3)$ x 8, $C_3^5:S_3$ x 7, $C_3^5:C_6$ x 7, $(C_3^2\times \He_3):C_6$ x 4, $C_3^4.(C_3\times S_3)$ x 4, $C_3^5:S_3$ x 4, $C_3\wr S_3:C_3^2$ x 3, $C_3^4:(C_3\times S_3)$ x 3, $C_3^4.(C_3\times S_3)$ x 3, $C_3^4.(C_3\times S_3)$ x 3, $C_3^4:(C_3\times S_3)$ 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, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $C_3^4:C_3:S_3$ x 2, $(C_3^2\times \He_3):S_3$, $C_3^4.(C_3\times S_3)$, $(C_3\times \He_3):C_{18}$, $C_3^5:C_6$, $(C_3^2\times \He_3):S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.C_6$, $C_3^4:C_3:S_3$

Order 1296: $C_3^4:\SD_{16}$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^3:\GL(2,3)$

Order 1053: $C_3^3:C_{13}:C_3$ x 2

Order 972: $C_3^3:C_6^2$ x 6, $C_3^4:C_{12}$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^4:C_{12}$ x 2, $C_3^4:D_6$ x 2, $C_3^3\times S_3^2$, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_3^2:C_4$, $(C_3^2\times \He_3):C_4$, $C_3^4.A_4$, $C_3^4:D_6$

Order 729: $C_3^3.C_3^3$ x 11, $C_3^5:C_3$ x 10, $C_3^4:C_3^2$ x 8, $C_3^3.C_3^3$ x 7, $C_3^3.C_3^3$ x 6, $C_3^3.C_3^3$ x 6, $C_3^4:C_3^2$ x 4, $C_3^2.C_3^4$ x 3, $C_3^3.C_3^3$ x 3, $C_3^4.C_3^2$ x 2, $C_3^4.C_3^2$ x 2, $C_3^5:C_3$ x 2, $C_3^5:C_3$ x 2, $C_3^5.C_3$ x 2, $C_3^4:C_3^2$ x 2, $C_3^4:C_9$, $C_3^6$, $C_3^5.C_3$, $C_3^4.C_3^2$, $C_3^4.C_3^2$, $C_3^3:\He_3$

Order 648: $C_3\wr D_4$ x 5, $C_3^4:Q_8$ x 2, $C_3^4:D_4$ x 2, $C_3^2:F_9$ x 2, $C_3^3:\SL(2,3)$ x 2, $C_3^3:\SL(2,3)$ x 2, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^3:C_{24}$, $C_3^3:S_4$, $C_3^4:Q_8$, $C_3^4:D_4$, $C_3^3:C_{24}$, $\Unitary(3,2)$, $C_3^3:S_4$

Order 486: $C_3^4:S_3$ x 17, $C_3\wr C_3:C_6$ x 11, $C_3^4:C_6$ x 10, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 8, $C_3^4:S_3$ x 6, $C_3^4:S_3$ x 5, $C_3^4:S_3$ x 5, $C_3^4.C_6$ x 5, $C_3^3.(C_3\times S_3)$ x 4, $C_3^3.(C_3\times S_3)$ x 4, $C_3^3.(C_3\times S_3)$ x 3, $C_3^3:(C_3\times S_3)$ x 3, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $C_3\wr S_3:C_3$ x 3, $(C_3^2\times C_9):S_3$ x 3, $C_3^3:C_{18}$, $S_3\times C_3^4$, $(C_3\times \He_3):S_3$, $C_3^4:S_3$, $C_3^4.C_6$, $C_3^4:C_6$, $C_3^4:C_6$

Order 432: $C_3^2:\GL(2,3)$ x 3, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $C_3^3:\SD_{16}$, $C_3^3:\SD_{16}$

Order 351: $C_3^3:C_{13}$ x 2

Order 324: $C_3^2\times S_3^2$ x 11, $C_3\wr C_4$ x 9, $C_3\wr C_2^2$ x 8, $C_3^3:D_6$ x 8, $C_3^3:D_6$ x 3, $C_3^4:C_4$ x 2, $C_3^2:S_3^2$ x 2, $(C_3\times \He_3):C_4$ x 2, $C_3^3:A_4$, $D_6\times C_3^3$, $C_3^3:C_{12}$, $C_3^3:A_4$, $C_3^3:C_{12}$, $C_3^3:D_6$

Order 243: $C_3^4:C_3$ x 29, $C_3^3:C_3^2$ x 19, $C_3^2.C_3^3$ x 11, $C_3^4:C_3$ x 10, $\He_3:C_3^2$ x 8, $C_3^4.C_3$ x 8, $C_3^2\times \He_3$ x 6, $C_3^5$ x 5, $C_9.C_3^3$ x 5, $C_3^3:C_9$ x 4, $C_3^3:C_3^2$ x 3, $C_9.\He_3$ x 3, $C_3^2.C_3^3$ x 3, $C_9.\He_3$ x 3, $C_9:C_3^3$, $C_3^3:C_9$

Order 216: $S_3^2:C_6$ x 5, $\PU(3,2)$ x 5, $C_3^3:D_4$ x 3, $C_3^3:Q_8$ x 2, $C_3^3:Q_8$ x 2, $S_3^2:S_3$ x 2, $C_3:F_9$ x 2, $C_3\times F_9$ x 2, $C_3^2:S_4$, $\SU(3,2)$, $\He_3:C_8$, $C_6^2:C_6$, $C_3^3:Q_8$, $C_3^3:Q_8$, $C_3^3:C_8$, $C_3^2:C_{24}$

Order 162: $C_3^2\wr C_2$ x 25, $C_3\wr S_3$ x 25, $C_3^3:S_3$ x 24, $S_3\times C_3^3$ x 19, $C_3^3:C_6$ x 17, $C_3^3.C_6$ x 7, $S_3\times \He_3$ x 6, $C_3^3:C_6$ x 6, $C_3^2:C_{18}$ x 4, $\He_3.C_6$ x 4, $\He_3.C_6$ x 4, $\He_3.C_6$ x 3, $C_3^3:S_3$ x 2, $C_3^3\times C_6$, $C_{18}:C_3^2$, $C_6\times \He_3$, $C_3^2:C_{18}$

Order 144: $F_9:C_2$ x 3, $C_{12}.D_6$, $C_3\times \GL(2,3)$, $C_3^2:\SD_{16}$

Order 108: $C_3\times S_3^2$ x 17, $C_3^2\times D_6$ x 10, $C_3^2:C_{12}$ x 5, $C_3^3:C_4$ x 4, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 3, $C_3:S_3^2$ x 2, $C_3^2:C_{12}$ x 2, $\He_3:C_4$ x 2, $C_3\times C_6^2$, $C_3^2:D_6$, $C_3^2:A_4$, $C_3^2.A_4$

Order 81: $C_3\wr C_3$ x 44, $C_3\times \He_3$ x 32, $C_3^4$ x 21, $C_9:C_3^2$ x 14, $C_3.\He_3$ x 11, $C_3^2:C_9$ x 9, $\He_3:C_3$ x 8, $C_9.C_3^2$ x 7, $C_3.\He_3$ x 3, $C_3^2\times C_9$

Order 72: $C_6\wr C_2$ x 4, $\PSU(3,2)$ x 3, $\SOPlus(4,2)$ x 3, $F_9$ x 3, $C_3\times S_4$, $D_4\times C_3^2$, $C_3^2:Q_8$, $C_3\times \SL(2,3)$, $C_3^2:Q_8$, $D_6:S_3$, $C_3^2:C_8$, $C_3:C_{24}$

Order 54: $S_3\times C_3^2$ x 65, $C_3^2:C_6$ x 18, $C_3^2:S_3$ x 12, $C_3^2:C_6$ x 9, $C_3^2\times C_6$ x 9, $S_3\times C_9$ x 6, $C_9:C_6$ x 4, $C_2\times \He_3$ x 3, $C_3\times C_{18}$

Order 48: $\GL(2,3)$, $C_3\times \SD_{16}$, $C_3:\SD_{16}$

Order 39: $C_{13}:C_3$

Order 36: $C_6\times S_3$ x 13, $C_3:C_{12}$ x 7, $C_6^2$ x 5, $S_3^2$ x 4, $C_3^2:C_4$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_2^2:C_9$, $C_3\times A_4$

Order 27: $C_3^3$ x 55, $\He_3$ x 31, $C_9:C_3$ x 19, $C_3\times C_9$ x 12

Order 24: $C_3\times D_4$ x 3, $\SL(2,3)$ x 2, $C_3:D_4$, $C_3:Q_8$, $C_{24}$, $S_4$, $C_3\times Q_8$, $C_3:C_8$

Order 18: $C_3\times S_3$ x 42, $C_3\times C_6$ x 21, $C_3:S_3$ x 4, $C_{18}$ x 3

Order 16: $\SD_{16}$

Order 13: $C_{13}$

Order 12: $C_2\times C_6$ x 5, $C_{12}$ x 3, $D_6$ x 2, $C_3:C_4$ x 2, $A_4$

Order 9: $C_3^2$ x 39, $C_9$ x 5

Order 8: $Q_8$, $D_4$, $C_8$

Order 6: $C_6$ x 10, $S_3$ x 5

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

Order 3: $C_3$ x 8

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4094064: $C_3^6:\SL(3,3)$ ($C_1$)

Order 729: $C_3^6$ ($\SL(3,3)$)

Order 1: $C_1$ ($C_3^6:\SL(3,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4094064: $C_3^6:\SL(3,3)$ ($C_1$)

Order 729: $C_3^6$ ($\SL(3,3)$)

Order 1: $C_1$ ($C_3^6:\SL(3,3)$)

Series

Derived series

$C_3^6:\SL(3,3)$

$\rhd$

$C_3^6:\SL(3,3)$

Chief series

$C_3^6:\SL(3,3)$

$\rhd$

$C_3^6:\SL(3,3)$

$\rhd$

$C_3^6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^6:\SL(3,3)$

$\rhd$

$C_3^6:\SL(3,3)$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

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

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $62 \times 62$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $35 \times 35$ rational character table.