Abstract group 104976.qg: $C_3^4.C_3^3:\GL(2,3)$

• Label: 104976.qg
• Order: \( 2^{4} \cdot 3^{8} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $81$
• Trans deg.: $81$
• Rank: $3$

Group information

Description:	$C_3^4.C_3^3:\GL(2,3)$
Order:	\(104976\)\(\medspace = 2^{4} \cdot 3^{8} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_3:(\He_3:S_3).C_6^2.D_6$, of order \(209952\)\(\medspace = 2^{5} \cdot 3^{8} \)
Composition factors:	$C_2$ x 4, $C_3$ x 8
Derived length:	$6$

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

Group statistics

Order	1	2	3	4	6	8	9	12	24
Elements	1	1053	8018	486	44874	8748	11664	12636	17496	104976
Conjugacy classes	1	2	34	1	23	2	3	6	4	76
Divisions	1	2	32	1	20	1	3	5	1	66
Autjugacy classes	1	2	17	1	12	2	2	3	2	42

Dimension	1	2	3	4	6	8	12	16	18	24	27	48	54	81	108	162	216
Irr. complex chars.	2	6	2	4	10	2	9	4	3	6	4	12	6	4	2	0	0	76
Irr. rational chars.	2	4	2	5	10	2	9	4	3	6	0	12	2	0	1	2	2	66

Minimal presentations

Permutation degree:	$81$
Transitive degree:	$81$
Rank:	$3$
Inequivalent generating triples:	$2645395200$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	27	54	54
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid d^{12}=e^{3}=f^{3}=g^{3}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $81$ $\langle(1,3,7)(2,5,11)(4,14,27)(6,15,28)(8,16,29)(9,19,41)(10,22,39)(12,23,21) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4.C_3^3)$ $\,\rtimes\,$ $\GL(2,3)$	$C_3^4$ $\,\rtimes\,$ $(\He_3:\GL(2,3))$	$(C_3^4:C_3^2)$ $\,\rtimes\,$ $(C_3:\GL(2,3))$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^4.C_3^3.C_2)$ . $S_4$	$C_3^4$ . $(C_3^3:\GL(2,3))$	$C_3$ . $(C_3^3.C_3^3.Q_8.S_3)$	$C_3^2$ . $(C_3^4:C_3:\GL(2,3))$	all 7
Aut. group:	$\Aut(C_9.C_3^3)$	$\Aut(C_6\times C_8:D_4)$	$\Aut(C_{18}.C_3^3)$

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

Homology

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

Subgroups

There are 1299836 subgroups in 3915 conjugacy classes, 20 normal (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^3.C_3^3.Q_8.S_3$
Commutator:	$G' \simeq$ $C_3^4.C_3:S_3.A_4.C_3$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_3^4$	$G/\Phi \simeq$ $C_3^3:\GL(2,3)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4.C_3^3$	$G/\operatorname{Fit} \simeq$ $\GL(2,3)$
Radical:	$R \simeq$ $C_3^4.C_3^3:\GL(2,3)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3$	$G/\operatorname{soc} \simeq$ $C_3^3.C_3^3.Q_8.S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4.C_3^3.C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 104976: $C_3^4.C_3^3:\GL(2,3)$

Order 52488: $C_3^4.C_3:S_3.A_4.C_3$

Order 34992: $C_3^4.C_3:S_3.S_4$ x 2, $C_3^4.C_3:S_3.S_4$, $C_3^4.C_3^3.\SD_{16}$

Order 26244: $C_3^4.C_3^3.D_6$

Order 17496: $C_3^4.C_3:S_3.A_4$ x 2, $C_3^4.C_3:S_3.A_4$, $C_3^4.C_3:F_9$, $C_3^4.C_3^3.D_4$, $C_3^4.C_3:S_3.C_6.C_2$

Order 13122: $C_3^4.C_3^3.C_6$, $C_3^4.\He_3:S_3$, $C_3^4.\He_3.C_6$

Order 11664: $\PSU(3,2).\He_3:S_3$, $C_3^4.C_3:S_3.D_4$

Order 8748: $C_3^4.C_3^2.D_6$ x 2, $C_3^4.C_3:S_3.C_6$, $C_3^5.S_3^2$, $C_3^4.C_3^2.D_6$, $C_3^5:S_3^2$

Order 6561: $C_3^4.C_3^3.C_3$

Order 5832: $\ASL(2,3)\times \He_3$, $C_3^4.C_3^2.Q_8$, $C_3^4.S_3\wr C_2$, $C_3^4.F_9$

Order 4374: $C_3^4.\He_3.C_2$ x 4, $C_3^3.\He_3.C_6$ x 4, $C_3^5.C_3.C_6$ x 2, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.\He_3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$

Order 3888: $C_3^2.Q_8:\He_3.C_2$ x 8, $C_3^4:\GL(2,3)$ x 4, $C_3^4.Q_8:S_3$, $C_3\times \SU(3,2).S_3$

Order 2916: $C_3^4:C_3.D_6$ x 3, $C_3^4.S_3^2$ x 3, $C_3^4:S_3^2$ x 3, $C_3^4:S_3^2$ x 3, $C_3^4:C_3.D_6$ x 2, $(C_3^2\times \He_3).D_6$ x 2, $C_3^4.S_3^2$ x 2, $C_3^4.C_3^2.C_4$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$

Order 2187: $C_3^4.C_3^3$ x 4, $C_3^4.C_3^3$ x 2, $C_3^3.C_3^4$, $C_3^4.C_3^3$, $C_3^4.C_3^3$

Order 1944: $C_3^4:\SL(2,3)$ x 8, $C_3^4:\SL(2,3)$ x 4, $\He_3\times \PSU(3,2)$, $C_3^4:D_{12}$, $\He_3:F_9$, $C_3\times \Unitary(3,2)$

Order 1458: $\He_3^2:C_2$ x 12, $(C_3^2\times \He_3):S_3$ x 11, $C_3^4:(C_3\times S_3)$ x 8, $C_3^4.(C_3\times S_3)$ x 6, $C_3^4:C_3:S_3$ x 6, $C_3^4:(C_3\times S_3)$ x 6, $\He_3\wr C_2$ x 6, $C_3^3:C_9:S_3$ x 4, $\He_3^2:C_2$ x 3, $\He_3^2:C_2$ 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, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $(C_3^2\times \He_3):C_6$, $C_3^4:D_9$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1296: $C_3^3:\GL(2,3)$ x 9, $C_3^4:\SD_{16}$ x 4, $C_3^3:\GL(2,3)$ x 4, $\SU(3,2):C_6$, $\He_3:\GL(2,3)$, $\He_3:\GL(2,3)$

Order 972: $C_3^3:S_3^2$ x 48, $C_3^3:S_3^2$ x 15, $C_3^3:S_3^2$ x 4, $C_3^4:D_6$ x 3, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_{12}$

Order 729: $\He_3^2$ x 12, $C_3^4:C_3^2$ x 12, $C_3^4.C_3^2$ x 8, $C_3^4:C_3^2$ x 6, $C_3^3.C_3^3$ x 3, $C_3^4:C_9$ x 2, $C_3^5:C_3$ x 2, $C_3^5:C_3$ x 2, $C_3^3.C_3^3$ x 2

Order 648: $C_3^3:\SL(2,3)$ x 17, $C_3^4:Q_8$ x 4, $C_3^3:D_{12}$ x 4, $C_3^3:C_{24}$ x 4, $C_3\times \He_3:D_4$ x 2, $\Unitary(3,2)$ x 2, $C_3\times \SU(3,2)$, $\He_3:C_{24}$, $\He_3\times \SL(2,3)$

Order 486: $C_3^4:S_3$ x 203, $C_3^4:S_3$ x 51, $C_3^4:C_6$ x 51, $C_3^4:S_3$ x 41, $C_3^4:C_6$ x 15, $C_3^4:C_6$ x 15, $C_3^3:D_9$ x 9, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 4, $(C_3\times \He_3):C_6$ x 3, $C_3^4.S_3$ x 3, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $C_3^2:\GL(2,3)$ x 9, $C_3^2:\GL(2,3)$ x 8, $C_3^3:\SD_{16}$ x 4, $C_3^2:\GL(2,3)$ x 4, $F_9:C_6$, $\He_3:\SD_{16}$, $\He_3:\SD_{16}$

Order 324: $C_3^3:D_6$ x 42, $C_3\wr C_2^2$ x 16, $C_3^2:S_3^2$ x 15, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 5, $C_3^3:C_{12}$ x 4, $C_3^3:D_6$ x 3, $\He_3:C_{12}$ x 2, $C_3^2:S_3^2$, $C_3^3:D_6$

Order 243: $C_3^4:C_3$ x 209, $C_3^2\times \He_3$ x 38, $C_3^3:C_9$ x 9, $C_3^4:C_3$ x 6, $C_3^3.C_3^2$ x 6, $C_3^4.C_3$ x 3, $C_3^5$

Order 216: $\PU(3,2)$ x 10, $C_3^2:\SL(2,3)$ x 8, $C_3^3:Q_8$ x 5, $C_3^2:D_{12}$ x 4, $C_3:F_9$ x 4, $C_3^2\times \SL(2,3)$ x 4, $\He_3:D_4$ x 2, $\SU(3,2)$, $\He_3:C_8$, $Q_8\times \He_3$, $C_3^2:D_{12}$, $\He_3:C_8$, $S_3^2:C_6$, $C_3\times F_9$

Order 162: $C_3^3:S_3$ x 213, $C_3^3:C_6$ x 120, $C_3^2\wr C_2$ x 68, $C_3^3:S_3$ x 49, $C_3^3:C_6$ x 15, $S_3\times \He_3$ x 14, $C_3^3:C_6$ x 7, $C_3^3:S_3$ x 6, $C_3\wr S_3$ x 6, $S_3\times C_3^3$ x 5, $D_9:C_3^2$ x 3, $C_3^2:D_9$ x 3, $C_3^3:C_6$ x 3, $C_6\times \He_3$

Order 144: $C_3\times \GL(2,3)$ x 9, $C_{12}.D_6$ x 4, $C_3:\GL(2,3)$ x 4, $F_9:C_2$

Order 108: $C_3^2:D_6$ x 42, $C_3\times S_3^2$ x 29, $C_3:S_3^2$ x 16, $C_3^2:D_6$ x 10, $C_3:S_3^2$ x 5, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 4, $\He_3:C_4$ x 2, $C_4\times \He_3$

Order 81: $C_3\times \He_3$ x 292, $C_3^4$ x 40, $C_3\wr C_3$ x 12, $C_3^2:C_9$ x 6, $C_9:C_3^2$ x 3

Order 72: $C_3\times \SL(2,3)$ x 17, $Q_8\times C_3^2$ x 4, $C_3\times D_{12}$ x 4, $C_3:C_{24}$ x 4, $\PSU(3,2)$, $\SOPlus(4,2)$, $F_9$

Order 54: $C_3^2:S_3$ x 219, $C_3^2:C_6$ x 163, $C_3^2:C_6$ x 93, $S_3\times C_3^2$ x 86, $C_2\times \He_3$ x 10, $C_9:C_6$ x 9, $C_3^2:S_3$ x 5, $C_3^2\times C_6$ x 4, $C_3\times D_9$ x 3

Order 48: $\GL(2,3)$ x 9, $Q_8:S_3$ x 4, $C_3\times \SD_{16}$

Order 36: $S_3^2$ x 29, $C_6\times S_3$ x 17, $C_3\times C_{12}$ x 4, $C_6:S_3$ x 4, $C_3^2:C_4$

Order 27: $\He_3$ x 238, $C_3^3$ x 164, $C_9:C_3$ x 9, $C_3\times C_9$ x 3

Order 24: $\SL(2,3)$ x 10, $C_3\times Q_8$ x 5, $D_{12}$ x 4, $C_3:C_8$ x 4, $C_{24}$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 169, $C_3:S_3$ x 54, $C_3\times C_6$ x 25, $D_9$ x 3

Order 16: $\SD_{16}$

Order 12: $D_6$ x 17, $C_{12}$ x 5, $C_2\times C_6$

Order 9: $C_3^2$ x 174, $C_9$ x 3

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

Order 6: $S_3$ x 43, $C_6$ x 20

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

Order 3: $C_3$ x 32

Order 2: $C_2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 104976: $C_3^4.C_3^3:\GL(2,3)$

Order 52488: $C_3^4.C_3:S_3.A_4.C_3$

Order 34992: $C_3^4.C_3:S_3.S_4$, $C_3^4.C_3^3.\SD_{16}$, $C_3^4.C_3:S_3.S_4$

Order 26244: $C_3^4.C_3^3.D_6$

Order 17496: $C_3^4.C_3:S_3.A_4$, $C_3^4.C_3:F_9$, $C_3^4.C_3^3.D_4$, $C_3^4.C_3:S_3.A_4$, $C_3^4.C_3:S_3.C_6.C_2$

Order 13122: $C_3^4.C_3^3.C_6$, $C_3^4.\He_3:S_3$, $C_3^4.\He_3.C_6$

Order 11664: $\PSU(3,2).\He_3:S_3$, $C_3^4.C_3:S_3.D_4$

Order 8748: $C_3^4.C_3:S_3.C_6$, $C_3^4.C_3^2.D_6$, $C_3^5.S_3^2$, $C_3^4.C_3^2.D_6$, $C_3^5:S_3^2$

Order 6561: $C_3^4.C_3^3.C_3$

Order 5832: $\ASL(2,3)\times \He_3$, $C_3^4.C_3^2.Q_8$, $C_3^4.S_3\wr C_2$, $C_3^4.F_9$

Order 4374: $C_3^5.C_3.C_6$ x 2, $C_3^4.\He_3.C_2$ x 2, $C_3^3.\He_3.C_6$ x 2, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.\He_3.C_2$, $C_3^3.C_3^4.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.\He_3.C_2$, $C_3^4.C_3^3.C_2$

Order 3888: $C_3^2.Q_8:\He_3.C_2$ x 3, $C_3^4:\GL(2,3)$ x 2, $C_3^4.Q_8:S_3$, $C_3\times \SU(3,2).S_3$

Order 2916: $(C_3^2\times \He_3).D_6$ x 2, $C_3^4:S_3^2$ x 2, $C_3^4.C_3^2.C_4$, $C_3^4:C_3.D_6$, $C_3^4:C_3.D_6$, $C_3^4.S_3^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.S_3^2$, $C_3^4:S_3^2$, $C_3^5:D_6$

Order 2187: $C_3^4.C_3^3$ x 2, $C_3^4.C_3^3$ x 2, $C_3^3.C_3^4$, $C_3^4.C_3^3$, $C_3^4.C_3^3$

Order 1944: $C_3^4:\SL(2,3)$ x 3, $C_3^4:\SL(2,3)$ x 2, $\He_3\times \PSU(3,2)$, $C_3^4:D_{12}$, $\He_3:F_9$, $C_3\times \Unitary(3,2)$

Order 1458: $\He_3^2:C_2$ x 6, $C_3^4:(C_3\times S_3)$ x 6, $(C_3^2\times \He_3):S_3$ x 6, $C_3^4.(C_3\times S_3)$ x 3, $C_3^3:C_9:S_3$ x 2, $C_3^4:C_3:S_3$ x 2, $C_3^4:(C_3\times S_3)$ x 2, $\He_3^2:C_2$ x 2, $\He_3^2:C_2$ x 2, $\He_3\wr C_2$ x 2, $C_3^5:S_3$ x 2, $C_3^4.(C_3\times S_3)$ x 2, $C_3^5:S_3$ x 2, $(C_3^2\times \He_3):C_6$, $C_3^4:D_9$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$

Order 1296: $C_3^3:\GL(2,3)$ x 4, $C_3^4:\SD_{16}$ x 2, $C_3^3:\GL(2,3)$ x 2, $\SU(3,2):C_6$, $\He_3:\GL(2,3)$, $\He_3:\GL(2,3)$

Order 972: $C_3^3:S_3^2$ x 18, $C_3^3:S_3^2$ x 5, $C_3^3:S_3^2$ x 3, $C_3^4:D_6$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_{12}$

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

Order 648: $C_3^3:\SL(2,3)$ x 8, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:C_{24}$ x 2, $C_3\times \He_3:D_4$ x 2, $\Unitary(3,2)$ x 2, $C_3\times \SU(3,2)$, $\He_3:C_{24}$, $\He_3\times \SL(2,3)$

Order 486: $C_3^4:S_3$ x 59, $C_3^4:S_3$ x 20, $C_3^4:S_3$ x 20, $C_3^4:C_6$ x 19, $C_3^4:C_6$ x 5, $C_3^4:C_6$ x 5, $C_3^3:D_9$ x 5, $C_3^4:C_6$ x 3, $C_3^4:S_3$ x 3, $(C_3\times \He_3):C_6$ x 2, $C_3^4:C_6$ x 2, $C_3^4.S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$ x 2, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $C_3^2:\GL(2,3)$ x 4, $C_3^2:\GL(2,3)$ x 3, $C_3^3:\SD_{16}$ x 2, $C_3^2:\GL(2,3)$ x 2, $F_9:C_6$, $\He_3:\SD_{16}$, $\He_3:\SD_{16}$

Order 324: $C_3^3:D_6$ x 15, $C_3\wr C_2^2$ x 8, $C_3^2:S_3^2$ x 7, $C_3^2:S_3^2$ x 5, $C_3^2:S_3^2$ x 3, $C_3^3:D_6$ x 2, $C_3^3:C_{12}$ x 2, $\He_3:C_{12}$ x 2, $C_3^2:S_3^2$, $C_3^3:D_6$

Order 243: $C_3^4:C_3$ x 62, $C_3^2\times \He_3$ x 18, $C_3^3:C_9$ x 5, $C_3^4:C_3$ x 3, $C_3^3.C_3^2$ x 3, $C_3^4.C_3$ x 2, $C_3^5$

Order 216: $\PU(3,2)$ x 5, $C_3^2:\SL(2,3)$ x 3, $C_3^3:Q_8$ x 3, $\He_3:D_4$ x 2, $C_3^2:D_{12}$ x 2, $C_3:F_9$ x 2, $C_3^2\times \SL(2,3)$ x 2, $\SU(3,2)$, $\He_3:C_8$, $Q_8\times \He_3$, $C_3^2:D_{12}$, $\He_3:C_8$, $S_3^2:C_6$, $C_3\times F_9$

Order 162: $C_3^3:S_3$ x 67, $C_3^3:C_6$ x 39, $C_3^2\wr C_2$ x 34, $C_3^3:S_3$ x 24, $S_3\times \He_3$ x 8, $C_3^3:C_6$ x 5, $C_3^3:C_6$ x 5, $C_3\wr S_3$ x 4, $S_3\times C_3^3$ x 3, $C_3^3:S_3$ x 3, $D_9:C_3^2$ x 2, $C_3^2:D_9$ x 2, $C_3^3:C_6$ x 2, $C_6\times \He_3$

Order 144: $C_3\times \GL(2,3)$ x 4, $C_{12}.D_6$ x 2, $C_3:\GL(2,3)$ x 2, $F_9:C_2$

Order 108: $C_3\times S_3^2$ x 15, $C_3^2:D_6$ x 15, $C_3:S_3^2$ x 8, $C_3^2:D_6$ x 5, $C_3:S_3^2$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$ x 2, $\He_3:C_4$ x 2, $C_4\times \He_3$

Order 81: $C_3\times \He_3$ x 101, $C_3^4$ x 20, $C_3\wr C_3$ x 6, $C_3^2:C_9$ x 4, $C_9:C_3^2$ x 2

Order 72: $C_3\times \SL(2,3)$ x 8, $Q_8\times C_3^2$ x 2, $C_3\times D_{12}$ x 2, $C_3:C_{24}$ x 2, $\PSU(3,2)$, $\SOPlus(4,2)$, $F_9$

Order 54: $C_3^2:C_6$ x 73, $C_3^2:S_3$ x 69, $S_3\times C_3^2$ x 46, $C_3^2:C_6$ x 31, $C_2\times \He_3$ x 5, $C_9:C_6$ x 3, $C_3^2:S_3$ x 3, $C_3\times D_9$ x 2, $C_3^2\times C_6$ x 2

Order 48: $\GL(2,3)$ x 4, $Q_8:S_3$ x 2, $C_3\times \SD_{16}$

Order 36: $S_3^2$ x 15, $C_6\times S_3$ x 9, $C_3\times C_{12}$ x 2, $C_6:S_3$ x 2, $C_3^2:C_4$

Order 27: $C_3^3$ x 78, $\He_3$ x 76, $C_9:C_3$ x 3, $C_3\times C_9$ x 2

Order 24: $\SL(2,3)$ x 5, $C_3\times Q_8$ x 3, $D_{12}$ x 2, $C_3:C_8$ x 2, $C_{24}$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 81, $C_3:S_3$ x 26, $C_3\times C_6$ x 13, $D_9$ x 2

Order 16: $\SD_{16}$

Order 12: $D_6$ x 9, $C_{12}$ x 3, $C_2\times C_6$

Order 9: $C_3^2$ x 82, $C_9$ x 2

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

Order 6: $S_3$ x 22, $C_6$ x 12

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

Order 3: $C_3$ x 17

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 104976: $C_3^4.C_3^3:\GL(2,3)$ ($C_1$)

Order 52488: $C_3^4.C_3:S_3.A_4.C_3$ ($C_2$)

Order 17496: $C_3^4.C_3:S_3.A_4$ ($S_3$) x 2, $C_3^4.C_3:S_3.A_4$ ($S_3$), $C_3^4.C_3:S_3.C_6.C_2$ ($S_3$)

Order 5832: $C_3^4.C_3^2.Q_8$ ($C_3:S_3$)

Order 4374: $C_3^4.C_3^3.C_2$ ($S_4$)

Order 2187: $C_3^4.C_3^3$ ($\GL(2,3)$)

Order 1458: $(C_3^2\times \He_3):S_3$ ($C_3:S_4$)

Order 729: $C_3^4:C_3^2$ ($C_3:\GL(2,3)$)

Order 243: $C_3^2\times \He_3$ ($C_3^2:\GL(2,3)$)

Order 81: $C_3^4$ ($\He_3:\GL(2,3)$) x 3, $C_3^4$ ($C_3^3:\GL(2,3)$)

Order 27: $C_3^3$ ($(C_3\times \He_3).Q_8.S_3$)

Order 9: $C_3^2$ ($C_3^4:C_3:\GL(2,3)$)

Order 3: $C_3$ ($C_3^3.C_3^3.Q_8.S_3$)

Order 1: $C_1$ ($C_3^4.C_3^3:\GL(2,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 104976: $C_3^4.C_3^3:\GL(2,3)$ ($C_1$)

Order 52488: $C_3^4.C_3:S_3.A_4.C_3$ ($C_2$)

Order 17496: $C_3^4.C_3:S_3.A_4$ ($S_3$), $C_3^4.C_3:S_3.A_4$ ($S_3$), $C_3^4.C_3:S_3.C_6.C_2$ ($S_3$)

Order 5832: $C_3^4.C_3^2.Q_8$ ($C_3:S_3$)

Order 4374: $C_3^4.C_3^3.C_2$ ($S_4$)

Order 2187: $C_3^4.C_3^3$ ($\GL(2,3)$)

Order 1458: $(C_3^2\times \He_3):S_3$ ($C_3:S_4$)

Order 729: $C_3^4:C_3^2$ ($C_3:\GL(2,3)$)

Order 243: $C_3^2\times \He_3$ ($C_3^2:\GL(2,3)$)

Order 81: $C_3^4$ ($C_3^3:\GL(2,3)$), $C_3^4$ ($\He_3:\GL(2,3)$)

Order 27: $C_3^3$ ($(C_3\times \He_3).Q_8.S_3$)

Order 9: $C_3^2$ ($C_3^4:C_3:\GL(2,3)$)

Order 3: $C_3$ ($C_3^3.C_3^3.Q_8.S_3$)

Order 1: $C_1$ ($C_3^4.C_3^3:\GL(2,3)$)

Series

Derived series

$C_3^4.C_3^3:\GL(2,3)$

$\rhd$

$C_3^4.C_3^3:\GL(2,3)$

$\rhd$

$C_3^4.C_3:S_3.A_4.C_3$

$\rhd$

$C_3^4.C_3:S_3.A_4.C_3$

$\rhd$

$C_3^4.C_3^2.Q_8$

$\rhd$

$C_3^4.C_3^2.Q_8$

$\rhd$

$(C_3^2\times \He_3):S_3$

$\rhd$

$(C_3^2\times \He_3):S_3$

$\rhd$

$C_3^4:C_3^2$

$\rhd$

$C_3^4:C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_3^4.C_3^3:\GL(2,3)$

$\rhd$

$C_3^4.C_3^3:\GL(2,3)$

$\rhd$

$C_3^4.C_3:S_3.A_4.C_3$

$\rhd$

$C_3^4.C_3:S_3.A_4.C_3$

$\rhd$

$C_3^4.C_3:S_3.C_6.C_2$

$\rhd$

$C_3^4.C_3:S_3.C_6.C_2$

$\rhd$

$C_3^4.C_3^3.C_2$

$\rhd$

$C_3^4.C_3^3.C_2$

$\rhd$

$C_3^4.C_3^3$

$\rhd$

$C_3^4.C_3^3$

$\rhd$

$C_3^4:C_3^2$

$\rhd$

$C_3^4:C_3^2$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^4.C_3^3:\GL(2,3)$

$\rhd$

$C_3^4.C_3^3:\GL(2,3)$

$\rhd$

$C_3^4.C_3:S_3.A_4.C_3$

$\rhd$

$C_3^4.C_3:S_3.A_4.C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_3$

$\lhd$

$C_3$

Character theory

Complex character table

See the $76 \times 76$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $66 \times 66$ rational character table.