Abstract group 8398080.j: $C_3^3:S_3.\SO(5,3)$

• Label: 8398080.j
• Order: \( 2^{8} \cdot 3^{8} \cdot 5 \)
• Exponent: \( 2^{3} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 1
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{8} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $81$
• Trans deg.: $81$
• Rank: $2$

Group information

Description:	$C_3^3:S_3.\SO(5,3)$
Order:	\(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \)
Exponent:	\(360\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$C_3^3:S_3.\SO(5,3)$, of order \(8398080\)\(\medspace = 2^{8} \cdot 3^{8} \cdot 5 \)
Composition factors:	$C_2$ x 2, $C_3$ x 4, $\SU(4,2)$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	8	9	10	12	18	20	24
Elements	1	10611	64880	98172	419904	1432080	1224720	466560	419904	1555200	466560	839808	1399680	8398080
Conjugacy classes	1	3	9	4	1	15	6	2	1	7	1	2	4	56
Divisions	1	3	9	4	1	15	4	2	1	7	1	1	2	51
Autjugacy classes	1	3	9	4	1	15	6	2	1	7	1	2	4	56

Dimension	1	6	8	10	15	20	24	30	40	60	64	72	80	81	90	120	128	160	240	320	480	640	960	1280	1440
Irr. complex chars.	2	2	1	1	4	5	2	2	2	5	4	1	5	2	1	1	0	3	2	1	3	2	3	1	1	56
Irr. rational chars.	2	2	1	1	4	3	2	2	3	3	2	1	3	2	1	2	1	2	2	2	3	2	3	1	1	51

Minimal presentations

Permutation degree:	$81$
Transitive degree:	$81$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	80	80	80
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $81$ $\langle(1,2,6)(3,7,21)(4,8,17)(5,9,22)(10,35,74)(11,40,55)(12,25,65)(13,26,57) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^3:S_3)$ . $\SO(5,3)$	$(C_3^4:\Sp(4,3))$ . $C_2$	$C_3^4$ . $(C_2.\SU(4,2).C_2)$		more information
Aut. group:	$\Aut(C_3^3:C_3^2)$	$\Aut((C_3\times \He_3):S_3)$	$\Aut(C_6.C_3^4)$	$\Aut(C_3^4:\Sp(4,3))$

Elements of the group are displayed as permutations of degree 81.

Homology

Abelianization:	$C_{2} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 5 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_1$
Commutator:	a subgroup isomorphic to $C_3^4:\Sp(4,3)$
Frattini:	a subgroup isomorphic to $C_1$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
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

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 15552 (order at least 540) are shown, as well as all normal subgroups of any index.

Order 8398080: $C_3^3:S_3.\SO(5,3)$

Order 4199040: $C_3^4:\Sp(4,3)$

Order 311040: $C_3^3:S_3.C_2^4:S_5$

Order 233280: $C_3^2:(C_3:S_3.C_2).A_6.C_2$

Order 209952: $C_3^4.C_3^3.Q_8.D_6$, $C_3^4.C_3^2.Q_8.S_3^2$

Order 186624: $C_3^4:(Q_8^2:S_3^2)$

Order 155520: $C_3^4.(C_4.C_2^3).A_5$

Order 116640: $\ASigmaL(2,9)$ x 2, $C_3^2:(C_3:S_3.C_2).A_6$

Order 104976: $C_3^4.C_3^3.Q_8.S_3$ x 2, $C_3^4.C_3^3.Q_8.C_6$, $C_3^4.C_3^2.C_6.A_4.C_2$, $C_3^4.C_3:S_3.A_4.C_6$, $C_3^4.C_3^3:\GL(2,3)$

Order 103680: $\GSp(4,3)$

Order 93312: $\PU(3,2)\wr C_2$ x 2, $C_3^4.Q_8^2.C_3.S_3$

Order 69984: $C_3^4.C_3^3.C_4.C_2^3$, $C_3^4.C_3^2.Q_8.D_6$, $C_3^4.F_9:D_6$

Order 62208: $C_3^4:(Q_8^2:D_6)$ x 2

Order 58320: $C_3^4.\SL(2,9)$

Order 52488: $C_3^4.C_3^3.Q_8.C_3$, $C_3^4.C_3:S_3.A_4.C_3$, $(C_3^2\times \He_3).S_3^3$

Order 51840: $C_3^4.(C_4.C_2^3).F_5$, $\Sp(4,3)$

Order 46656: $C_3^4.Q_8^2.C_3^2$

Order 38880: $C_3^4.(C_4.A_5).C_2$ x 2

Order 34992: $C_3^4.C_3^3.\SD_{16}$ x 5, $C_3^4.C_3^2.D_6.C_2^2$ x 2, $C_3^2.C_3^4.Q_8.S_3$ x 2, $C_3^4.F_9:S_3$, $C_3^4.C_3^3.D_4.C_2$, $C_3^4.C_3^2.C_4.C_6.C_2$, $C_3^4.C_3:S_3.S_4$, $C_3^4.(S_3\times F_9)$, $C_3^4.(C_3\times F_9).C_2$, $C_3^3.C_3^4.C_4.C_2^2$, $C_3^2.C_3^4.Q_8.C_6$, $\He_3^2:(S_3\times D_4)$

Order 31104: $C_3^4.Q_8^2.S_3$ x 2, $C_3^4:(Q_8^2:S_3)$ x 2, $C_3^4:(Q_8^2:C_6)$ x 2, $C_3:S_3.C_6^2.C_6.D_4$, $C_3^4:\GL(2,3):D_4$

Order 26244: $C_3^4.C_3^3.D_6$ x 4, $C_3^4.C_3^3.C_6.C_2$ x 2, $C_3^5.\He_3.C_2^2$

Order 25920: $C_3^4.(C_4.C_2^3).D_5$

Order 23328: $C_3^4.S_3\wr C_2.C_2^2$, $C_3^4.C_3:S_3.D_4.C_2$, $C_3:S_3.(C_3^2\times A_4).D_6$, $C_3^5:(C_2\times \GL(2,3))$

Order 20736: $C_3^4:(Q_8^2:C_2^2)$

Order 19440: $C_3^4.(C_4.A_5)$ x 2, $C_3^4.(C_2.S_5)$ x 2, $C_3^4:\SL(2,5):C_2$ x 2

Order 17496: $C_3^4.C_3^3.D_4$ x 3, $C_3^4:C_3.C_6.C_6.C_2$ x 2, $C_3^4.S_3^3$ x 2, $C_3^4.C_3^3.Q_8$ x 2, $C_3^4.C_3^3.C_8$ x 2, $C_3^4.C_3:S_3.C_6.C_2$ x 2, $C_3^4.C_3:F_9$ x 2, $C_3^4:C_3.C_6^2.C_2$, $C_3^4.C_3^3.C_2^2.C_2$, $C_3^4.C_3^2.D_6.C_2$, $C_3^4.C_3:S_3.A_4$, $C_3^3.C_3^4.C_2^3$, $C_3^2.C_3^4.Q_8.C_3$

Order 15552: $C_3^4.Q_8^2.C_3$ x 2, $C_3^4:\GL(2,3):C_2^2$ x 2, $\ASL(2,3).F_9$, $C_3^4:C_2^2.A_4.C_2^2$, $C_3^3:C_{12}:\GL(2,3)$, $C_3^2:F_9.S_4$, $C_3^3:S_3.\GL(2,\mathbb{Z}/4)$, $C_3^4:\SL(2,3).D_4$, $C_3^4:\GL(2,3):C_2^2$, $C_3^4:\GL(2,3):C_4$, $C_3^4:\SL(2,3):D_4$

Order 13122: $C_3^4.C_3^3.C_6$ x 4, $C_3^4.\He_3.C_6$ x 2, $C_3^4.\He_3:S_3$

Order 12960: $C_3^4.(C_4.C_2^3).C_5$

Order 11664: $C_3^4.C_3:S_3.D_4$ x 4, $C_3^2.C_3^4.\SD_{16}$ x 3, $(C_3\times \He_3).F_9.C_2$ x 2, $\ASL(2,3).\He_3.C_2$, $C_3^4:C_3.D_6.C_2^2$, $C_3^4.S_3\wr C_2.C_2$, $C_3^4.C_4:S_3^2$, $C_3^4.C_3^2.C_4.C_2^2$, $C_3^2.C_3^4.Q_8.C_2$, $C_3:S_3.C_6^2.C_3.C_6$, $C_3:S_3.(A_4\times \He_3).C_2$, $C_3^5:(C_2\times \SL(2,3))$, $C_3^5:\GL(2,3)$, $C_3^5:\GL(2,3)$, $\He_3^2:(C_2\times D_4)$

Order 10368: $C_3^4:C_4^2.D_4$ x 2, $\PSU(3,2)\wr C_2$ x 2, $C_3^4.C_4^2.C_2^3$, $C_3^2.Q_8^2.C_3.S_3$, $C_3^2:(C_4\times F_9).C_4$, $C_3^4:D_8:D_4$

Order 9720: $C_3^4:\SL(2,5)$ x 2

Order 8748: $C_3^4.C_3^2.D_6$ x 7, $C_3^4.C_3^3.C_2^2$ x 4, $C_3^4.C_3^3.C_4$ x 3, $C_3^4.\He_3.C_2^2$ x 2, $C_3^5.S_3^2$, $C_3^5.C_3.D_6$, $C_3^5.(C_6\times S_3)$, $C_3^4:C_3.C_6^2$, $C_3^3.C_3^4.C_4$, $C_3^3.C_3^4.C_2^2$, $C_3^3.C_3^3.C_6.C_2$, $C_3^5:S_3^2$

Order 7776: $C_3^4:(C_2\times \GL(2,3))$ x 3, $C_3^4:\GL(2,3):C_2$ x 3, $C_3^4:\GL(2,3):C_2$ x 3, $C_3^4:C_4.S_4$ x 2, $C_3^4:\GL(2,3):C_2$ x 2, $C_3^4:D_4.A_4$ x 2, $\SU(3,2).S_3^2$, $C_3^4.D_{12}.C_2^2$, $C_3:S_3.C_6^2.D_6$, $C_3:S_3.C_3:S_3:S_4$, $C_3^4:(C_4\times \SL(2,3))$, $S_3\times C_3^3:\GL(2,3)$, $C_3^3:(S_3\times \GL(2,3))$, $C_3^2:F_9.A_4$, $C_3^4:\Unitary(2,3)$, $C_3^4:\SL(2,3):C_2^2$, $C_3^4:(S_3\times \SD_{16})$, $C_3^4:(S_3\times \SD_{16})$

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

Order 6480: $C_3^4:C_5.\OD_{16}$

Order 5832: $C_3^3.S_3^3$ x 3, $C_3^2.C_3^4.D_4$ x 3, $C_3^4.D_6:S_3$ x 2, $C_3^4.C_3^2.Q_8$ x 2, $C_3^4.C_3^2.C_8$ x 2, $C_3^2.C_3^4.Q_8$ x 2, $(C_3\times \He_3).F_9$ x 2, $\He_3^2:D_4$ x 2, $\He_3^2:D_4$ x 2, $C_3^4:C_3.D_6.C_2$, $C_3^4:C_3.C_{12}.C_2$, $C_3^4.C_3:S_3.C_2^2$, $C_3^4.(C_3\times D_{12})$, $C_3^2.C_3^4.C_4.C_2$, $C_3:S_3.(A_4\times \He_3)$, $\He_3^2:C_2^3$, $C_3^3.S_3^3$, $C_3^5:\SL(2,3)$, $\He_3^2:(C_2\times C_4)$, $\He_3^2:C_2^3$, $C_3^3.S_3^3$

Order 5184: $C_3^4:D_4.D_4$ x 2, $C_3^4:C_4^2.C_4$ x 2, $C_3^2:F_9.D_4$ x 2, $C_3^4:D_8:C_2^2$ x 2, $C_3^4.Q_8^2$, $C_3^4.Q_8:C_8$, $C_3^4.C_4:C_8.C_2$, $C_3^2.Q_8^2.C_3^2$, $C_3^3:C_{12}:\SD_{16}$, $C_3^2:F_9:D_4$, $C_3^4:D_8:C_4$, $C_3^4:D_4:D_4$

Order 4374: $C_3^4.C_3^3.C_2$ x 7, $C_3^5.C_3.C_6$ x 5, $C_3^4.\He_3.C_2$ x 5, $C_3^3.\He_3.C_6$ x 4, $C_3^5.C_3.S_3$ x 2, $C_3^4.C_9.C_6$, $C_3^3.\He_3.S_3$, $C_3^3.C_3^4.C_2$, $C_3^3.C_3^3.C_6$

Order 3888: $C_3^2.Q_8:\He_3.C_2$ x 6, $C_3^4:\GL(2,3)$ x 3, $C_3^4:\GL(2,3)$ x 3, $C_3^3:S_3.S_4$ x 3, $C_3^4:\SL(2,3):C_2$ x 3, $C_3^4:(C_2\times \SL(2,3))$ x 3, $C_3:S_3.C_6^2.C_6$ x 2, $C_3^4:\GL(2,3)$ x 2, $C_3^4:(S_3\times D_4)$ x 2, $\SU(3,2).C_3.S_3$, $\SU(3,2).C_3.C_6$, $\He_3.(C_3\times Q_8).C_6$, $\He_3.(C_2\times F_9)$, $C_3^4.Q_8:S_3$, $C_3^4.C_{12}.C_2^2$, $C_3^4.C_8:S_3$, $C_3^4.C_3:\SD_{16}$, $C_3^4.(C_6\times Q_8)$, $C_3:S_3.C_6^2.S_3$, $S_3\times C_3^3:\SL(2,3)$, $C_3^3:(S_3\times \SL(2,3))$, $C_3^4:\GL(2,3)$, $C_3^4:(C_2\times D_{12})$, $C_3^2:F_9:S_3$, $C_3^3:F_9:C_2$, $C_3^4:(C_3\times \SD_{16})$, $C_3^4:(S_3\times Q_8)$, $C_3^3:(C_2\times F_9)$, $C_3^3:F_9:C_2$, $C_3^2:(S_3\times F_9)$, $C_3^4:(S_3\times Q_8)$, $C_3\wr C_4:D_6$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$, $C_3^5:\SD_{16}$

Order 3840: $C_2.C_2^4:S_5$

Order 3456: $C_3^2.Q_8^2.S_3$ x 2, $\ASL(2,3).\SD_{16}$, $(C_3^2\times Q_8).Q_8:S_3$

Order 3240: $C_3^4:C_5:C_8$ x 2, $C_3^4:(C_4\times D_5)$

Order 2916: $C_3^4:C_3.D_6$ x 7, $C_3^4.(C_6\times S_3)$ x 7, $C_3^4.S_3^2$ x 5, $C_3^4:C_3.C_{12}$ x 3, $C_3^4:C_3.C_6.C_2$ x 3, $C_3^4:S_3^2$ x 3, $C_3^4.C_3:S_3.C_2$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^2.C_3^4.C_4$ x 2, $C_3^4:S_3^2$ x 2, $C_3^5:D_6$ x 2, $C_3^4:S_3^2$ x 2, $\He_3^2:C_4$ x 2, $\He_3.\He_3.C_2^2$, $C_3^5.D_6$, $C_3^5.C_6.C_2$, $C_3^4.C_6^2$, $C_3^4.C_3^2.C_4$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4:D_{18}$, $C_3^4.S_3^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:(C_2\times C_6)$, $C_3^5:D_6$

Order 2880: $C_4.S_6$

Order 2592: $S_3\times C_3^2:\GL(2,3)$ x 3, $C_3^4:(C_2\times \SD_{16})$ x 3, $C_3^4:Q_{16}:C_2$ x 3, $C_3^4:D_8:C_2$ x 3, $C_3^4:C_4.D_4$ x 2, $C_3^4:D_8:C_2$ x 2, $C_3^4:\OD_{16}:C_2$ x 2, $C_3^4:C_4.C_2^3$ x 2, $C_3^4:C_4\wr C_2$ x 2, $\He_3.Q_8:S_3.C_2$, $C_6.C_3^2:S_4.C_2$, $C_3^3:C_{12}:C_8$, $C_3^4:(C_4\times Q_8)$, $C_2\times C_3^3:\GL(2,3)$, $\PSU(3,2):S_3^2$, $\PSU(3,2):S_3^2$, $C_3^3:C_{12}:Q_8$, $C_3^2:(C_4\times F_9)$, $C_3^3:C_{12}:D_4$, $C_3^2\wr C_2.\SD_{16}$, $C_3^4:(C_2\times \SD_{16})$, $C_3^2:F_9.C_4$, $C_3:S_3^3.C_4$, $C_3:S_3^3:C_2^2$, $C_3^3:(C_2\times \GL(2,3))$, $\SU(3,2):D_6$

Order 2304: $Q_8^2:S_3^2$

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: $C_3^2.S_3^3$ x 7, $C_3^4:\SL(2,3)$ x 5, $C_3^4:\SL(2,3)$ x 3, $C_3^3:\SOPlus(4,2)$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3^4:\SL(2,3)$ x 2, $C_3^2:S_3^3$ x 2, $C_3^3:\PSU(3,2)$ x 2, $C_3^4:D_{12}$ x 2, $C_3^3:\SOPlus(4,2)$ x 2, $C_3^2:S_3^3$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^5:D_4$, $C_3\wr C_4:S_3$, $C_3^3:F_9$, $C_3^4:C_{24}$, $C_3^2:S_3^3$, $\He_3:\PSU(3,2)$, $\He_3\times \PSU(3,2)$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^4:D_{12}$, $C_3^4:(C_2\times C_{12})$, $\He_3:F_9$, $C_3^4:C_{24}$, $C_3\times \Unitary(3,2)$, $C_3^2.S_3^3$, $C_3^3:\PSU(3,2)$, $C_3^3:F_9$, $C_3^4:C_{24}$

Order 1920: $C_2.C_2^4:A_5$

Order 1728: $\PSU(3,2):\SL(2,3)$ x 2, $\SL(2,3)\times \PSU(3,2)$, $\SL(2,3):\SOPlus(4,2)$, $Q_8\times \PU(3,2)$, $\SL(2,3):F_9$, $\PSU(3,2):D_{12}$, $(C_3\times C_{12}).\GL(2,3)$

Order 1620: $C_3^4:D_{10}$, $C_3^4:C_{20}$, $C_3^4:C_5:C_4$

Order 1458: $C_3^4:(C_3\times S_3)$ x 6, $C_3^4:(C_3\times S_3)$ x 4, $\He_3\wr C_2$ x 4, $C_3^4:(C_3\times S_3)$ x 4, $C_3^4:(C_3\times C_6)$ x 4, $(C_3^2\times \He_3):S_3$ x 4, $C_3^4.(C_3\times S_3)$ x 3, $\He_3^2:C_2$ x 3, $C_3^4:(C_3\times S_3)$ x 3, $(C_3^2\times \He_3):S_3$ x 2, $C_3^4:D_9$ x 2, $C_3^3:C_9:S_3$ x 2, $C_3^4:C_3:S_3$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $\He_3^2:C_2$ x 2, $\He_3^2:C_2$ x 2, $\He_3^2:C_2$ x 2, $C_3^4.(C_3\times S_3)$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $(C_3^2\times \He_3):S_3$, $(C_3^2\times \He_3):C_6$, $C_3^4:D_9$, $C_3^4:C_{18}$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times C_6)$, $C_3^5:S_3$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1440: $\SL(2,9):C_2$ x 2, $\SL(2,9):C_2$

Order 1296: $C_3^3:\GL(2,3)$ x 7, $C_3^3:\GL(2,3)$ x 6, $C_3^4:D_4:C_2$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:(C_2\times Q_8)$ x 3, $C_3^4:\SD_{16}$ x 3, $C_3^4:\SD_{16}$ x 3, $C_3^4:\OD_{16}$ x 3, $S_3\times \PU(3,2)$ x 3, $\He_3:\GL(2,3)$ x 3, $C_3^4:(C_2\times D_4)$ x 2, $C_3^3:(S_3\times D_4)$ x 2, $C_3^4:Q_{16}$ x 2, $C_3^4:D_8$ x 2, $C_3^2:(C_2\times F_9)$ x 2, $(C_3\times \He_3):\SD_{16}$ x 2, $C_3^3:(S_3\times Q_8)$, $C_3^3:(S_3\times Q_8)$, $S_3\times C_3^3:Q_8$, $S_3\times C_3^2:D_{12}$, $S_3^2:S_3^2$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^2:(C_2\times F_9)$, $C_3:(S_3\times F_9)$, $S_3\times C_3:F_9$, $C_3^2\wr C_2.Q_8$, $C_3^4:C_4^2$, $C_6\times \PU(3,2)$, $C_3^3:(C_2\times \SL(2,3))$, $C_3^3:\GL(2,3)$, $C_3^3:\GL(2,3)$, $S_3\times \SU(3,2)$, $S_3\times \He_3:D_4$, $(C_3\times \He_3):\SD_{16}$, $\SU(3,2):S_3$, $\SU(3,2):C_6$, $S_3\times \He_3:C_8$, $C_2\times \Unitary(3,2)$, $C_3^2:C_6\times \SL(2,3)$, $\He_3:\GL(2,3)$, $\He_3:\GL(2,3)$

Order 1152: $\SL(2,3)\wr C_2$ x 2, $\SL(2,3):\GL(2,3)$, $C_3:S_3.D_4:D_4$

Order 972: $C_3^3:S_3^2$ x 30, $C_3^4:D_6$ x 12, $C_3^3:S_3^2$ x 10, $C_3^3:S_3^2$ x 8, $C_3^3:S_3^2$ x 7, $C_3^4:D_6$ x 6, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ 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, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^3.S_3^2$, $C_3^3:D_{18}$, $C_3^3.S_3^2$

Order 864: $C_2\times C_3^2:\GL(2,3)$ x 6, $F_9:D_6$ x 2, $C_3^2:(C_2\times \GL(2,3))$ x 2, $\PSU(3,2):D_6$, $\SL(2,3):S_3^2$, $\SL(2,3):S_3^2$, $C_3^2:C_4\times \SL(2,3)$, $C_3^2:(S_3\times \SD_{16})$, $C_4\times \PU(3,2)$, $\SU(3,2):C_2^2$, $(Q_8\times C_3^2):D_6$

Order 810: $C_3^4:C_{10}$, $C_3^4:D_5$

Order 768: $Q_8^2:D_6$ x 2

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

Order 720: $\SL(2,9)$

Order 648: $C_3^3:\SL(2,3)$ x 10, $C_3.S_3^3$ x 6, $C_3:S_3^3$ x 4, $C_3^4:D_4$ x 4, $C_3^3:D_{12}$ x 4, $C_3^4:Q_8$ x 3, $C_3^4:Q_8$ x 3, $C_3^3:D_{12}$ x 3, $C_3^4:D_4$ x 3, $C_3^3:C_{24}$ x 3, $C_3:S_3^3$ x 2, $C_3^4:Q_8$ x 2, $C_3^4:Q_8$ x 2, $C_3\wr D_4$ x 2, $C_3^4:(C_2\times C_4)$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^2:F_9$ x 2, $C_3^2:F_9$ x 2, $C_3:\SU(3,2)$ x 2, $C_3^2.\SOPlus(4,2)$ x 2, $C_3\times \He_3:D_4$ x 2, $C_3^2.F_9$ x 2, $\Unitary(3,2)$ x 2, $C_3:S_3^3$, $C_3^4:Q_8$, $C_3^3:D_{12}$, $S_3\times C_3^2:C_{12}$, $C_3^2:F_9$, $C_3^2:F_9$, $C_3^3:\SL(2,3)$, $C_2\times C_3^2:S_3^2$, $C_3\times \SU(3,2)$, $\He_3:D_{12}$, $S_3\times \He_3:C_4$, $\He_3:C_{24}$, $\He_3\times \SL(2,3)$, $C_2\times C_3^3:D_6$

Order 640: $C_2.C_2^4:F_5$

Order 576: $C_3^2:Q_8^2$, $\PSU(3,2):D_4$, $F_9:D_4$, $Q_8:\SOPlus(4,2)$, $Q_8:F_9$, $\SL(2,3)^2$, $(C_3\times C_{12}).\SD_{16}$, $\PSU(3,2):C_8$

Order 162: $C_3^3:S_3$

Order 81: $C_3^4$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 8398080: $C_3^3:S_3.\SO(5,3)$ ($C_1$)

Order 4199040: $C_3^4:\Sp(4,3)$ ($C_2$)

Order 162: $C_3^3:S_3$ ($\SO(5,3)$)

Order 81: $C_3^4$ ($C_2.\SU(4,2).C_2$)

Order 1: $C_1$ ($C_3^3:S_3.\SO(5,3)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Character theory

Complex character table

See the $56 \times 56$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $51 \times 51$ rational character table.