Abstract group 746496.bb: $(C_3^3\times C_6).\GL(2,3)\wr C_2$

• Label: 746496.bb
• Order: \( 2^{10} \cdot 3^{6} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $22$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$(C_3^3\times C_6).\GL(2,3)\wr C_2$
Order:	\(746496\)\(\medspace = 2^{10} \cdot 3^{6} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_3^4.Q_8^2.S_3^2.C_2^3$, of order \(1492992\)\(\medspace = 2^{11} \cdot 3^{6} \)
Composition factors:	$C_2$ x 10, $C_3$ x 6
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	12	16	24
Elements	1	4231	6560	55512	135008	113184	241920	93312	96768	746496
Conjugacy classes	1	11	9	14	51	24	20	4	20	154
Divisions	1	11	9	11	51	13	15	1	10	122
Autjugacy classes	1	9	9	10	41	17	14	2	12	115

Dimension	1	2	4	6	8	9	12	16	18	24	32	48	64	96	128	256	512
Irr. complex chars.	8	2	24	8	10	8	12	18	2	4	16	8	18	4	4	8	0	154
Irr. rational chars.	4	4	6	8	13	4	4	15	4	8	11	8	14	4	8	6	1	122

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{4}=c^{3}=e^{12}=g^{12}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,3,5,8,13,14)(2,4,6,10,15,9)(7,12,17,18,16,11)(19,21,22,20), (1,2)(3,4)(5,7,11,16)(6,9)(8,13,17,12)(10,14,18,15)(19,20,22,21)\rangle$
Transitive group:	36T34180	36T34181			more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^4.Q_8^2.S_3^2)$ . $C_4$ (2)	$C_3^4$ . $(Q_8^2.S_3^2:C_4)$	$(C_3^2:S_3^2)$ . $(S_4^2:C_4)$	$(C_3^4:C_2^3)$ . $(S_4\wr C_2)$	all 19

Elements of the group are displayed as permutations of degree 22.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 23 normal subgroups (21 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed

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 128 (order at least 5832) are shown, as well as all normal subgroups of any index.

Order 746496: $(C_3^3\times C_6).\GL(2,3)\wr C_2$

Order 373248: $C_2\times C_3^4.Q_8^2.S_3^2$, $C_2\times C_3^4:(\SL(2,3)^2:C_4)$, $C_3^4:C_2^3.\POPlus(4,3)$

Order 186624: $C_3^4.Q_8^2.S_3^2$ x 3, $C_2\times C_3^4:C_2^2.A_4^2.C_2$, $C_2\times C_3^4.Q_8^2.C_3.S_3$, $C_2.\PU(3,2)\wr C_2$, $C_3^4:(\SL(2,3)^2:C_4)$

Order 124416: $C_2\times C_3:S_3.C_6^2.C_{12}.C_2^3$, $(C_3^3\times C_6).Q_8^2:D_6$

Order 93312: $C_3^4.Q_8^2.C_3.S_3$ x 2, $C_3^4.Q_8.C_6.A_4.C_2$ x 2, $C_2\times C_3^4.Q_8^2.C_3^2$, $C_2\times (C_3\times \PSU(3,2)).S_3^3$

Order 82944: $C_3^4:C_2^3.C_2\wr D_4$

Order 62208: $C_3:S_3.C_6^2.C_{12}.C_2^3$ x 8, $C_2\times C_3:S_3.C_6^2.C_6.D_4$ x 3, $C_2\times C_3:S_3.C_6^2.D_6.C_2^2$ x 2, $C_2\times \ASL(2,3).(C_2\times F_9)$, $C_2\times C_3^4:C_2^2.(D_4\times A_4)$, $C_2\times C_3^4.Q_8^2.S_3$, $(C_3^3\times C_6).Q_8^2:S_3$, $C_3^4.Q_8^2.C_{12}$

Order 46656: $C_3:S_3.C_6^2.S_3^2.C_2$ x 4, $(C_3\times \PSU(3,2)).S_3^3$ x 4, $C_3^4.Q_8^2.C_3^2$, $C_3:S_3.(A_4\times \He_3).C_2^3$, $C_2\times \PSU(3,2).C_3^3.C_6.C_2$, $C_2\times \ASL(2,3).\He_3.C_2^2$, $C_2\times C_3^2.Q_8:\He_3.D_6$, $C_2\times C_3^2.(Q_8\times \He_3).C_6.C_2$, $C_2\times C_3:S_3.C_6^2.S_3^2$, $C_2\times C_3:S_3.C_6^2.C_3.D_6$, $C_3^2.S_3^4:C_4$

Order 41472: $C_3^2.Q_8^2.C_3^2.C_2^3$, $C_2\times C_3^4.D_4^2.C_2^2$, $S_3^4:C_2^3.C_4$, $S_3^4:C_2^3.C_2^2$, $C_2\times C_3^4:(Q_8^2:C_4)$, $C_2.F_9\wr C_2:C_2$, $(C_3^3\times C_6).Q_8^2:C_2^2$, $C_2\times C_3^4:(C_8^2:C_4)$

Order 31104: $C_3:S_3.C_6^2.C_6.D_4$ x 12, $C_3^4:(D_4\times \GL(2,3))$ x 6, $\ASL(2,3).(C_2\times F_9)$ x 4, $C_3^4:C_2^2.(D_4\times A_4)$ x 4, $C_3:S_3.C_6^2.D_6.C_2^2$ x 4, $C_3^4.Q_8^2.S_3$ x 2, $C_2\times C_3^4:C_2^2.A_4.C_2^2$ x 2, $C_2\times \PSU(3,2).S_3^3$, $C_2\times \ASL(2,3).F_9$, $C_2\times \ASL(2,3).D_6\times S_3$, $C_2\times \AGL(2,3)\times C_3:S_3.C_2$, $C_2\times C_3^4.Q_8^2.C_3$, $C_2\times C_3^4.C_{12}.C_2^4$, $C_2\times C_3:S_3.C_6^2.D_{12}$, $C_2\times C_3:S_3.C_6^2.D_6.C_2$, $C_2\times C_3:S_3.C_6^2.C_{24}$, $C_2\times C_3:S_3.C_6^2.C_3.Q_8$, $C_3^4:(C_2\times \GL(2,3)):C_4$

Order 23328: $C_3:S_3.(C_3^2\times A_4).D_6$ x 9, $\ASL(2,3).\He_3.C_2^2$ x 5, $C_3:S_3.C_6^2.S_3^2$ x 5, $C_3^2.(Q_8\times \He_3).C_6.C_2$ x 4, $C_3:S_3.C_6^2.C_3.D_6$ x 2, $C_3:S_3.C_6^2.(C_6\times S_3)$ x 2, $C_2\times C_3:S_3.C_6^2.C_3.C_6$ x 2, $\He_3^2:C_2^3:C_4$ x 2, $\He_3^2:C_2^3:C_4$ x 2, $\PSU(3,2).C_3^3.C_6.C_2$, $\ASL(2,3).C_3.C_6^2$, $C_3^2.Q_8:\He_3.D_6$, $C_2\times \PSU(3,2).C_3^3.C_6$, $C_2\times \ASL(2,3).\He_3.C_2$, $C_2\times C_3^2.(Q_8\times \He_3).C_6$, $C_2\times (C_3\times \He_3).D_6^2$, $\He_3^2:(C_2^3:C_4)$, $\He_3^2:(C_2^3:C_4)$

Order 20736: $C_3^4.D_4^2.C_2^2$ x 10, $C_2\times C_3^4.C_4^2.C_2^3$ x 6, $C_3^2:\GL(2,3)\times \GL(2,3)$ x 4, $C_2\times C_3^4.D_4^2.C_2$ x 2, $C_3^4:(Q_8^2:C_4)$ x 2, $C_3^4:(C_8^2:C_4)$ x 2, $\SOPlus(4,2)^2.C_4$ x 2, $C_3^2.Q_8^2.C_3.D_6$, $C_3:S_3.(Q_8\times A_4).C_6.C_2$, $C_2\times C_3^4.(C_4\times \SD_{16}).C_2$, $(C_3\times C_6).(Q_8\times A_4).C_6.C_2$, $(C_2\times C_3^4:C_4^2).D_4$, $C_3^4:C_4^2.(C_2^2\times C_4)$, $C_3^2:(C_4\times F_9).D_4$, $C_3^4:(Q_8^2:C_4)$, $C_2\times C_3^2:F_9.D_8$, $C_3^4:(C_2\times D_4).C_4^2$, $C_3^4:C_4^2.(C_2^2\times C_4)$, $C_3^3:D_6.D_4^2$, $C_2.F_9\wr C_2$, $S_3^4:(C_2^2:C_4)$

Order 15552: $C_3^4.C_{12}.C_2^4$ x 16, $S_3^2\times C_3^2:\GL(2,3)$ x 14, $C_2\times C_3^4.C_{12}.C_2^3$ x 8, $C_3^4:(C_4\times \GL(2,3))$ x 4, $(C_3^2\times S_3^2):\GL(2,3)$ x 4, $C_3:S_3.C_6^2.C_3.Q_8$ x 3, $C_2\times C_3^4.D_{12}.C_2^2$ x 3, $C_3^3:C_{12}:\GL(2,3)$ x 3, $C_3^4:(D_4\times \SL(2,3))$ x 3, $\ASL(2,3).F_9$ x 2, $C_3^4:C_2^2.A_4.C_2^2$ x 2, $C_3:S_3.C_6^2.C_{24}$ x 2, $C_2\times \ASL(2,3).S_3^2$ x 2, $C_2\times C_3:S_3.(A_4\times S_3^2)$ x 2, $\PSU(3,2).C_3^3.C_2^3$, $\ASL(2,3).C_6^2.C_2$, $C_6\times \ASL(2,3).D_6$, $C_3^4.Q_8^2.C_3$, $C_2\times \PSU(3,2):C_2\times \He_3:C_2$, $C_2\times \PSU(3,2).C_3^2.D_6$, $C_2\times \He_3.(C_2^2\times F_9)$, $C_2\times \ASL(2,3)\times C_3:S_3.C_2$, $C_2\times \ASL(2,3).S_3\times S_3$, $C_2\times \ASL(2,3).C_6\times S_3$, $C_2\times \ASL(2,3).C_3^2.C_4$, $C_2\times \ASL(2,3).C_3.D_6$, $C_2\times \AGL(2,3)\times C_3:S_3$, $C_2\times C_3^4.D_6.C_2^3$, $C_2\times C_3^3.D_6^2.C_2$, $C_2\times C_3:S_3.C_6^2.D_6$, $C_2\times C_3:S_3.C_3:S_3:S_4$, $C_2\times C_3:S_3.A_4:S_3^2$, $C_3^4:\GL(2,3):C_4$, $C_3^4:\GL(2,3):C_4$, $C_3^4:(C_2^2\times \GL(2,3))$, $C_3^4:(C_4\times \GL(2,3))$, $C_3^4:C_2^3.S_4$, $C_3^4:(C_4\times \GL(2,3))$, $C_3^4:(C_2\times Q_8):C_{12}$

Order 13824: $\ASL(2,3).C_4.C_2^4$, $(C_3^2\times Q_8).C_{12}.C_2^4$

Order 11664: $C_3^4.D_6^2$ x 6, $C_3:S_3.C_6^2.C_3.C_6$ x 5, $(C_3\times \He_3).D_6^2$ x 5, $\PSU(3,2).C_3^3.C_6$ x 4, $C_2\times C_3^3.S_3^3$ x 4, $C_3^4:(C_6^2:C_4)$ x 4, $\ASL(2,3).\He_3.C_2$ x 2, $C_3^2.(Q_8\times \He_3).C_6$ x 2, $C_3:S_3.(A_4\times \He_3).C_2$ x 2, $C_2\times C_3^4.C_3^2.C_2^3$ x 2, $(C_2\times \He_3^2).D_4$ x 2, $\He_3^2:(C_2^2\times C_4)$ x 2, $(C_2\times \He_3^2).D_4$ x 2, $C_3^4:(S_3^2:C_4)$ x 2, $C_2\times \He_3.\He_3.C_2^3$, $C_2\times C_3^3.C_3^3.C_2^3$, $(C_2\times \He_3^2).D_4$, $(C_2\times \He_3^2).D_4$, $\He_3^2:(C_2^2\times C_4)$, $\He_3^2:(C_2^2\times C_4)$

Order 10368: $C_3^4.C_4^2.C_2^3$ x 44, $\SOPlus(4,2)^2.C_2$ x 12, $C_3^4.(C_4\times \SD_{16}).C_2$ x 8, $C_3^4.D_4^2.C_2$ x 6, $C_3^2.Q_8^2.C_3.S_3$ x 2, $C_3:S_3.(Q_8\times A_4).C_6$ x 2, $C_2\times C_3^4.C_8:Q_8$ x 2, $C_2\times C_3^4.C_4:\SD_{16}$ x 2, $\GL(2,3).C_2^6$ x 2, $C_3^2:(C_4\times F_9):C_4$ x 2, $C_3^2:F_9.D_8$ x 2, $C_3^2:(C_4\times F_9).C_4$ x 2, $C_3^2.Q_8^2.C_3.C_6$, $C_2\times C_3^4:C_2^2.C_2^3.C_2$, $C_2\times C_3^4.Q_8^2$, $C_2\times C_3^4.Q_8:Q_8$, $C_2\times C_3^4.Q_8:C_8$, $C_2\times C_3^4.D_4:C_8$, $C_2\times C_3^4.D_4:C_4.C_2$, $C_2\times C_3^4.D_4.C_2^3$, $C_2\times C_3^4.C_8^2$, $C_2\times C_3^4.C_4^2.C_2^2$, $C_2\times C_3^4.C_4:Q_8.C_2$, $C_2\times C_3^4.C_4:C_8.C_2$, $C_2\times C_3^4.C_2^2:\SD_{16}$, $C_2\times C_3^4.(D_4\times Q_8)$, $C_2\times C_3^4.(C_8\times Q_8)$, $C_2\times C_3^4.(C_8\times D_4)$, $C_2\times C_3^4.(C_4\times D_4).C_2$, $C_2\times C_3^4.(C_4:D_4.C_2)$, $C_2\times C_3^4.(C_4.D_8)$, $C_2\times C_3^4.(C_2^2\times \SD_{16})$, $C_2\times C_3^4.(C_2\times \SD_{16}).C_2$, $C_2\times C_3:S_3.C_2\times \PSU(3,2):C_2$, $C_2.(A_4\times \He_3).C_2^4$, $(C_3\times C_6).(Q_8\times A_4).C_6$, $(C_2\times C_3:S_3).A_4^2.C_2$, $S_3^4:C_2^3$, $C_3^4:C_4.(C_4\times D_4)$, $C_2\times C_3^4:C_4^2.C_4$, $C_3^4:\OD_{16}.D_4$, $C_3^2:F_9.C_4^2$, $C_3^4:\OD_{32}:C_4$, $C_3^4:C_4^2.C_2^3$, $C_3^4:C_2^3.\OD_{16}$, $C_3^2:F_9.C_4^2$, $C_6:S_3^3.D_4$, $C_6:S_3^3.D_4$, $C_2\times S_3\wr C_4$, $S_3^4:(C_2\times C_4)$

Order 9216: $Q_8^2.S_3^2:C_4$

Order 7776: $C_3^4.C_{12}.C_2^3$ x 70, $C_3^4.D_{12}.C_2^2$ x 22, $S_3\times C_3^3:\GL(2,3)$ x 16, $S_3\times C_3^3:\GL(2,3)$ x 12, $C_3^4:(D_4\times D_6)$ x 12, $C_3^3:(S_3\times \GL(2,3))$ x 8, $C_2\times C_3^4.D_6.C_2^2$ x 7, $S_3^2\times \PU(3,2)$ x 7, $C_3^3:(S_3\times \GL(2,3))$ x 7, $\He_3.(C_2^2\times F_9)$ x 6, $C_3^4.D_6.C_2^3$ x 6, $C_3:S_3.C_6^2.D_6$ x 6, $\PSU(3,2):C_2\times \He_3:C_2$ x 5, $C_2\times C_3^4.C_{12}.C_2^2$ x 5, $C_2\times \He_3.(C_2\times F_9)$ x 4, $C_3^4:(C_2\times \GL(2,3))$ x 4, $C_3:S_3.C_3:S_3:S_4$ x 3, $C_2\times C_3^4.Q_8:S_3$ x 3, $C_2\times C_3^4.C_3:\SD_{16}$ x 3, $C_2\times C_3:S_3.C_6^2.C_6$ x 3, $\ASL(2,3).(C_6\times S_3)$ x 2, $C_6\times \ASL(2,3).C_6$ x 2, $C_6\times C_3:S_3.(S_3\times A_4)$ x 2, $C_3^4.C_8:S_3.C_2$ x 2, $C_3^2.Q_8:\He_3.C_2^2$ x 2, $C_3.S_3^2\times \PSU(3,2)$ x 2, $C_2\times C_3^4.(C_2\times C_3:Q_8)$ x 2, $C_2\times C_3^2.Q_8:\He_3.C_2$ x 2, $C_2\times C_3:S_3.C_3^2:S_4$ x 2, $C_3^4:(C_4\times \SL(2,3))$ x 2, $C_3^2\wr C_2.\GL(2,3)$ x 2, $C_3^2.S_3^3:C_4$ x 2, $C_6\times \ASL(2,3).S_3$, $C_3^4.Q_8:S_3.C_2$, $C_3^3.(C_3\times C_6\times \SD_{16})$, $C_3\times C_6\times \AGL(2,3)$, $C_2\times \He_3:C_2\times \PSU(3,2)$, $C_2\times \He_3:C_2\times F_9$, $C_2\times \He_3.D_6^2$, $C_2\times \ASL(2,3)\times C_3:S_3$, $C_2\times \ASL(2,3).C_3\times S_3$, $C_2\times \ASL(2,3).C_3.C_6$, $C_2\times S_3\wr C_2\times \He_3:C_2$, $C_2\times C_3^4.Q_8.S_3$, $C_2\times C_3^4.C_{24}.C_2$, $C_2\times C_3^4.C_8:S_3$, $C_2\times C_3^4.(S_3\times D_4)$, $C_2\times C_3^4.(C_6\times Q_8)$, $C_2\times C_3^4.(C_6\times D_4)$, $C_2\times C_3^3.D_6:D_6$, $C_2\times C_3^2:S_3\times \PSU(3,2)$, $C_2\times C_3^2.(C_6\times F_9)$, $C_2\times C_3\times S_3\times \ASL(2,3)$, $C_2\times (C_3\times \He_3):C_2^3\times S_3$, $C_2\times C_3^4:\GL(2,3)$, $(C_3^3\times C_6).\GL(2,3)$, $C_3^4:\SL(2,3):C_4$, $C_3^4:(C_4\times \SL(2,3))$, $C_3^4:(C_4\times \SL(2,3))$, $C_3^4:(C_2^2\times \SL(2,3))$

Order 6912: $C_3:S_3.(D_4\times S_4).C_2$ x 8, $(C_3\times C_6).(D_4\times S_4).C_2$ x 8, $\ASL(2,3).C_4.C_2^3$ x 4, $(C_3^2\times Q_8).C_{12}.C_2^3$ x 3, $(C_3^2\times Q_8).D_6.C_2^3$ x 2, $\SL(2,3).(C_2^2\times F_9)$, $\PSU(3,2).D_6.C_2^3$, $C_3^2.Q_8^2.D_6$, $C_3:S_3.(D_4\times A_4).C_2^2$, $(C_3\times C_6).(D_4\times A_4).C_2^2$, $C_2\times C_2^4.\SL(3,3)$

Order 5832: $C_3^3.S_3^3$ x 22, $\He_3^2:C_2^3$ x 20, $C_3^4.C_3^2.C_2^3$ x 12, $C_3^3.C_3^3.C_2^3$ x 12, $\He_3.\He_3.C_2^3$ x 5, $\He_3^2:(C_2\times C_4)$ x 4, $C_2\times C_3^4.C_6^2$ x 3, $C_2\times C_3^4.(C_6\times S_3)$ x 3, $C_2\times C_3^3.C_3^3.C_2^2$ x 3, $C_2\times (C_3^2\times \He_3).D_6$ x 3, $C_2\times C_3^4.S_3^2$ x 2, $C_2\times \He_3^2:C_4$ x 2, $\He_3^2:(C_2\times C_4)$ x 2, $\He_3^2:(C_2\times C_4)$ x 2, $C_3^4.S_3^2.C_2$, $C_3:S_3.(A_4\times \He_3)$, $C_2\times \He_3.\He_3.C_2^2$, $\He_3^2:(C_2\times C_4)$, $C_2\times \He_3^2:C_4$

Order 5184: $C_3^4.Q_8^2$

Order 648: $C_3^4:C_2^3$

Order 324: $C_3^2:S_3^2$ x 2, $C_3^3:D_6$

Order 162: $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 746496: $(C_3^3\times C_6).\GL(2,3)\wr C_2$ ($C_1$)

Order 373248: $C_2\times C_3^4.Q_8^2.S_3^2$ ($C_2$), $C_2\times C_3^4:(\SL(2,3)^2:C_4)$ ($C_2$), $C_3^4:C_2^3.\POPlus(4,3)$ ($C_2$)

Order 186624: $C_3^4.Q_8^2.S_3^2$ ($C_4$) x 2, $C_2\times C_3^4.Q_8^2.C_3.S_3$ ($C_2^2$)

Order 93312: $C_3^4.Q_8^2.C_3.S_3$ ($D_4$), $C_3^4.Q_8^2.C_3.S_3$ ($C_2\times C_4$), $C_2\times C_3^4.Q_8^2.C_3^2$ ($D_4$)

Order 46656: $C_3^4.Q_8^2.C_3^2$ ($C_2^2:C_4$)

Order 10368: $C_2\times C_3^4.Q_8^2$ ($\SOPlus(4,2)$)

Order 5184: $C_3^4.Q_8^2$ ($S_3^2:C_4$)

Order 648: $C_3^4:C_2^3$ ($S_4\wr C_2$)

Order 324: $C_3^3:D_6$ ($\GOrthPlus(4,3)$), $C_3^2:S_3^2$ ($\SL(2,3):A_4.D_4$), $C_3^2:S_3^2$ ($S_4^2:C_4$)

Order 162: $C_3^3\times C_6$ ($\GL(2,3)\wr C_2$), $C_3^3:S_3$ ($Q_8:C_2^2.S_3^2:C_4$), $C_3^3:S_3$ ($\GL(2,3)\wr C_2$)

Order 81: $C_3^4$ ($Q_8^2.S_3^2:C_4$)

Order 2: $C_2$ ($C_3^4:\GL(2,3)\wr C_2$)

Order 1: $C_1$ ($(C_3^3\times C_6).\GL(2,3)\wr C_2$)

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

Supergroups

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

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $154 \times 154$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $122 \times 122$ rational character table.