Abstract group 663552.ga: $A_4^3.C_2\wr S_3$

• Label: 663552.ga
• Order: \( 2^{13} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$A_4^3.C_2\wr S_3$
Order:	\(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$A_4^3.C_2^6:S_4$, of order \(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \)
Composition factors:	$C_2$ x 13, $C_3$ x 4
Derived length:	$5$

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	18	24
Elements	1	9663	5336	101952	122664	27648	36864	193536	110592	55296	663552
Conjugacy classes	1	52	4	75	79	8	2	41	6	8	276
Divisions	1	52	4	75	75	8	2	37	4	4	262
Autjugacy classes	1	42	4	43	63	4	1	21	2	2	183

Minimal presentations

Permutation degree:	$20$
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	18	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid c^{2}=d^{6}=e^{6}=f^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2,4,6,5,7,10,9,3)(8,11,12)(13,15,16)(18,19), (2,3,5,4)(6,8,10)(7,9)(11,12)(13,14,15,16)(17,18)(19,20)\rangle$
Transitive group:	36T33313	36T33314	36T33315	36T33316	all 8
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$A_4^3$ . $(C_2\wr S_3)$	$C_2^4$ . $(A_4^3:S_4)$	$C_2^7$ . $(C_6^3:S_4)$	$C_2^{10}$ . $(C_3^3:S_4)$	all 27

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$2$

Subgroups

There are 33 normal subgroups (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4^3.C_2^3:S_4$
Commutator:	$G' \simeq$ $A_4^3.C_2^3:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_4^3.C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{10}$	$G/\operatorname{Fit} \simeq$ $C_3^3:S_4$
Radical:	$R \simeq$ $A_4^3.C_2\wr S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_2\times C_3^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2.C_2^6.C_2^6$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$

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 64 (order at least 10368) are shown, as well as all normal subgroups of any index.

Order 663552: $A_4^3.C_2\wr S_3$

Order 331776: $A_4^3.C_2^3:S_4$, $C_2^3.A_4^3:S_4$, $A_4^3.C_2^2\wr C_3$

Order 221184: $A_4^2.A_4.C_2^6.C_2$

Order 165888: $A_4^3.C_2^3:A_4$ x 2, $A_4^3.\GL(2,\mathbb{Z}/4)$ x 2, $A_4:S_4^2.S_4$, $A_4^3.\GL(2,\mathbb{Z}/4)$

Order 110592: $A_4^2.A_4.C_2^5.C_2$ x 6, $A_4^2.A_4.C_2^5.C_2$ x 5, $C_2^2\times C_2^6.C_3^3.C_2^4$, $C_2\times A_4^2.A_4.C_2^4.C_2$, $A_4^2.C_2^4.C_6.C_2^3$, $C_2^6.C_6^3.C_2^3$

Order 82944: $C_2^2\times A_4^3:A_4$ x 3, $C_2\times A_4^3:S_4$ x 2, $A_4^3.A_4:C_4$ x 2, $A_4:S_4^2.A_4$, $C_2\times A_4^3.S_4$, $(C_2\times A_4^3).S_4$, $C_2^2\times A_4^3.A_4$

Order 73728: $A_4^2.C_2^6.C_2^3$

Order 55296: $A_4^3.C_2^2\wr C_2$ x 24, $A_4^2.A_4.C_2^4.C_2$ x 10, $C_2\times C_2^6.C_3^3.C_2^4$ x 9, $A_4^2.(C_2^4\times A_4).C_2$ x 5, $C_2^2\times C_2^6.C_3^3.C_2^3$ x 3, $C_2\times A_4^2.A_4.C_2^3.C_2$ x 2, $A_4^2.A_4.C_2^4.C_2$ x 2, $C_2\times A_4^2.A_4.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2^5$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^2:A_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2\times A_4^2.A_4.C_2^3$, $C_2\times A_4^2.(C_2^3\times A_4).C_2$, $C_2^6.C_6^3.C_2^2$, $A_4^2.C_6.C_2^5.C_2$, $C_2\times C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_6^3.C_2^2$, $C_2^2:A_4^2.\GL(2,\mathbb{Z}/4)$

Order 41472: $C_2\times A_4^3:A_4$ x 7, $C_2\times A_4^3.A_4$ x 2, $A_4^3:S_4$ x 2, $A_4^3.S_4$, $A_4\wr C_3:D_4$

Order 36864: $A_4^2.C_2^6.C_2^2$ x 6, $A_4^2.C_2^6.C_2^2$ x 6, $A_4^2.C_2^6.C_2^2$ x 6, $A_4^2.C_2^6.C_2^2$ x 5, $A_4^2.C_2^6.C_2^2$, $C_2\times A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^6.C_2^2$, $C_2^6.C_6^2.C_2^4$, $A_4^2.C_2^6.C_2^2$, $C_2^2\times A_4^2.C_2^5.C_2$

Order 27648: $C_2\times C_2^6.C_3^3.C_2^3$ x 52, $C_2^6.C_3^3.C_2^4$ x 51, $(C_2\times A_4^3).D_4$ x 24, $C_2\times A_4^3:D_4$ x 24, $C_2\times A_4^2.A_4.C_2^3$ x 14, $A_4^2.(C_2\times \GL(2,\mathbb{Z}/4))$ x 12, $A_4^3:C_2^2:C_4$ x 12, $A_4^2.C_6.C_2^4.C_2$ x 6, $C_2^2\times C_2^6.C_3^2.D_6$ x 6, $A_4^2.C_6.C_2^4.C_2$ x 5, $C_2^2\times A_4^2.A_4.C_2^2$ x 4, $C_2^6.C_6^3.C_2$ x 3, $C_2\times A_4^2.A_4.C_2^2.C_2$ x 2, $A_4^2.A_4.C_2^3.C_2$ x 2, $C_2^6.C_6^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2\wr C_9.C_6$ x 2, $C_2\wr C_9.C_6$ x 2, $C_2\wr C_9.C_6$ x 2, $C_2\times C_2^6.C_3^3.C_2^3$, $C_2^2\times A_4^2.D_6.C_2^2$, $C_2\times C_2^6.(C_3\times C_6^2).C_2$, $C_2\times C_2^6.C_3^3.D_4$, $C_2\times C_2^6.C_3^3.C_2^2.C_2$, $C_2\times A_4^2.C_2^2:D_{12}$, $C_2^4.C_6^3.C_2^3$, $C_2^6.C_3^2.D_6:C_4$, $A_4^2.(C_2^4\times C_6).C_2$, $C_2^2\times C_2^6.C_3^3.C_2^2$, $C_2^3:A_4^2.S_4$, $C_2^3:A_4^2.S_4$, $C_2^{10}:\He_3$

Order 24576: $(C_2^4\times A_4).C_2^6.C_2$, $C_2^7.A_4.C_2^3.C_2$

Order 20736: $A_4^3:A_4$ x 3, $A_4^3.A_4$, $A_4\wr C_3:C_4$, $C_2^2\times A_4\wr C_3$, $A_4^3:D_6$

Order 18432: $A_4^2.C_2^5.C_2^2$ x 48, $C_2\times A_4^2.C_2^5.C_2$ x 39, $A_4^2.C_2^5.C_2^2$ x 24, $A_4^2.C_2^6.C_2$ x 16, $A_4^2.C_2^5.C_2^2$ x 12, $A_4^2.C_2^6.C_2$ x 11, $C_2^2\times A_4^2.C_2^4.C_2$ x 11, $A_4^2.C_2^5.C_2^2$ x 10, $A_4^2.C_2^5.C_2^2$ x 10, $A_4^2.C_2^5.C_2^2$ x 10, $(C_2^2\times S_4^2).D_4$ x 8, $A_4^2.C_2^5.C_2^2$ x 7, $A_4^2.C_2^5.C_2^2$ x 6, $A_4^2.C_2^5.C_2^2$ x 6, $A_4^2.C_2^5.C_2^2$ x 6, $A_4^2.C_2^4.C_2^3$ x 6, $A_4^2.C_2^5.C_2^2$ x 5, $A_4^2.C_2^4.C_2^3$ x 5, $A_4^2.C_2^5.C_2^2$ x 5, $A_4^2.C_2^5.C_2^2$ x 5, $C_2^6.C_6^2.C_2^3$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $C_2\times A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2\times A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2\times A_4^2.C_2^4.C_2^2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $C_2\times A_4^2.C_2^4.C_2^2$ x 2, $C_2^9.D_{18}$ x 2, $C_2\times C_2^4.D_6\wr C_2.C_2$, $A_4^2.C_2^6.C_2$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^2\times C_2^2.A_4^2.C_2^3$, $C_2\times A_4^2.C_2^5.C_2$, $C_2^2\times A_4^2.C_2^5$, $C_2^6.C_6^2.C_2^3$, $C_2\times A_4^2.C_2^3.C_2^3$, $C_2^2\times A_4^2.C_2^4.C_2$, $C_2^6.C_6^2.C_2^3$, $(C_2\times A_4^2).C_2^6$, $C_2^2\times A_4^2.C_2^4.C_2$, $A_4.C_2^6.C_6.C_2^2$, $C_2^2\times A_4^2.C_2^5$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2\times C_2^6.(S_3\times C_3:D_4)$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^4.(C_2\times C_6^2).C_2^4$, $C_2^6.\PSOPlus(4,3)$

Order 13824: $A_4^3.C_2^3$ x 176, $C_2\times A_4^2.A_4.C_2^2$ x 39, $A_4^2.A_4.C_2^3$ x 36, $(C_2^2\times A_4^2):D_{12}$ x 24, $A_4^3:D_4$ x 16, $C_2\times (C_3\times A_4^2).C_2^4$ x 15, $C_2\times A_4^3:C_4$ x 12, $A_4^2.C_6.C_2^3.C_2$ x 10, $(C_3\times A_4^2).C_2^5$ x 6, $A_4^2:\GL(2,\mathbb{Z}/4)$ x 6, $A_4^3:C_2^3$ x 6, $A_4^3:(C_2\times C_4)$ x 6, $A_4^2.\GL(2,\mathbb{Z}/4)$ x 6, $A_4^2.C_6.C_2^3.C_2$ x 5, $C_2\times A_4^2.D_6.C_2^2$ x 5, $C_2\times C_2^6.C_3^3.C_2^2$ x 4, $C_2^2\times (C_3\times A_4^2).C_2^3$ x 4, $C_2\wr C_9.C_3$ x 4, $C_2^2\times A_4^2:D_{12}$ x 4, $C_2^2\times A_4^2.A_4.C_2$ x 3, $C_2^2\times A_4^2.D_6.C_2$ x 3, $C_2\times A_4^2.D_6:C_4$ x 2, $C_2^6.C_3^3.C_2^2.C_2$ x 2, $A_4^2.C_6.C_2^3.C_2$ x 2, $C_2^2:A_4^2.S_4$ x 2, $C_2^9:\He_3$ x 2, $C_2^3:A_4^2.D_6$ x 2, $C_2\times C_2^6.C_3^3.C_4$, $(C_3^2\times A_4).C_2^6.C_2$, $C_2^4.C_6^3.C_2^2$, $C_2\times C_2^6.C_3^3.C_2^2$, $C_2\times A_4^2.C_6.C_2^3$, $A_4^2.C_6.C_2^4$, $(C_3\times A_4^2).C_2^4.C_2$, $C_2^2\times A_4^2.C_6.C_2^2$, $C_2^2\times C_2^6.C_3^3.C_2$, $C_2^2\times (C_3\times A_4^2).C_2^3$, $C_2^4.C_6^3.C_2^2$, $C_2\times A_4^2.C_6.C_2^2.C_2$, $C_2^4.C_6^3.C_2^2$, $A_4^2.C_2^4:S_3$, $(C_3\times A_4^2).C_2^4.C_2$, $C_2^2:A_4^2.S_4$, $C_2^4:A_4^2.C_6$

Order 12288: $(C_2^4\times A_4).C_2^5.C_2$ x 12, $(C_2^4\times A_4).C_2^5.C_2$ x 11, $C_2^7.A_4.D_4$, $C_2^2\times A_4.C_2^6.C_2^2$, $A_4.C_2^6.C_2^4$, $A_4.C_2^6.C_2^4$, $C_2^7.C_6.C_2^4$, $(C_2^4\times A_4).C_2^5.C_2$, $A_4.C_2^5.C_2^4.C_2$, $C_2^7.A_4.D_4$, $C_2^5.C_2^4.C_3.D_4$, $C_2\times (C_2^4\times A_4).C_2^4.C_2$, $C_2^2\times C_2^7.A_4.C_2$, $(C_2^4\times A_4).C_2^5.C_2$, $C_2^4:C_3.C_2^6.C_2^2$, $C_2^9.S_4$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $C_2^9.S_4$

Order 10368: $A_4^3:C_6$ x 2, $(C_2\times C_6^3):S_4$, $A_4^3:S_3$

Order 8192: $C_2.C_2^6.C_2^6$

Order 6912: $A_4:S_4^2$ x 2, $C_2^6.C_3^3.C_2^2$, $C_2^2\times C_2^6.C_3^3$

Order 3456: $C_2\times A_4^3$

Order 1728: $A_4^3$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$ x 2

Order 128: $C_2^7$

Order 81: $C_3\wr C_3$

Order 64: $C_2^6$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 64 (order at least 10368) are shown, as well as all normal subgroups of any index.

Order 663552: $A_4^3.C_2\wr S_3$

Order 331776: $A_4^3.C_2^3:S_4$, $C_2^3.A_4^3:S_4$, $A_4^3.C_2^2\wr C_3$

Order 221184: $A_4^2.A_4.C_2^6.C_2$

Order 165888: $A_4^3.C_2^3:A_4$ x 2, $A_4:S_4^2.S_4$, $A_4^3.\GL(2,\mathbb{Z}/4)$, $A_4^3.\GL(2,\mathbb{Z}/4)$

Order 110592: $A_4^2.A_4.C_2^5.C_2$ x 4, $A_4^2.A_4.C_2^5.C_2$ x 3, $C_2^2\times C_2^6.C_3^3.C_2^4$, $C_2\times A_4^2.A_4.C_2^4.C_2$, $A_4^2.C_2^4.C_6.C_2^3$, $C_2^6.C_6^3.C_2^3$

Order 82944: $C_2^2\times A_4^3:A_4$ x 2, $A_4:S_4^2.A_4$, $C_2\times A_4^3:S_4$, $C_2\times A_4^3.S_4$, $A_4^3.A_4:C_4$, $(C_2\times A_4^3).S_4$, $C_2^2\times A_4^3.A_4$

Order 73728: $A_4^2.C_2^6.C_2^3$

Order 55296: $A_4^3.C_2^2\wr C_2$ x 11, $C_2\times C_2^6.C_3^3.C_2^4$ x 7, $A_4^2.A_4.C_2^4.C_2$ x 5, $A_4^2.(C_2^4\times A_4).C_2$ x 3, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^2\times C_2^6.C_3^3.C_2^3$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^2\times A_4^2.A_4.C_2^3$, $C_2\times A_4^2.A_4.C_2^3.C_2$, $A_4^2.A_4.C_2^4.C_2$, $C_2\times A_4^2.(C_2^3\times A_4).C_2$, $C_2^6.C_6^3.C_2^2$, $C_2\times A_4^2.A_4.C_2^3.C_2$, $A_4^2.C_6.C_2^5.C_2$, $C_2\times C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_6^3.C_2^2$, $C_2^6.C_3^3.C_2^5$, $C_2^6.C_6^3.C_2^2$, $C_2^2:A_4^2.\GL(2,\mathbb{Z}/4)$, $C_2^2:A_4^2.\GL(2,\mathbb{Z}/4)$

Order 41472: $C_2\times A_4^3:A_4$ x 4, $C_2\times A_4^3.A_4$ x 2, $A_4^3.S_4$, $A_4^3:S_4$, $A_4\wr C_3:D_4$

Order 36864: $A_4^2.C_2^6.C_2^2$ x 4, $A_4^2.C_2^6.C_2^2$ x 4, $A_4^2.C_2^6.C_2^2$ x 4, $A_4^2.C_2^6.C_2^2$ x 3, $A_4^2.C_2^6.C_2^2$, $C_2\times A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^6.C_2^2$, $A_4^2.C_2^6.C_2^2$, $C_2^6.C_6^2.C_2^4$, $A_4^2.C_2^6.C_2^2$, $C_2^2\times A_4^2.C_2^5.C_2$

Order 27648: $C_2\times C_2^6.C_3^3.C_2^3$ x 24, $C_2^6.C_3^3.C_2^4$ x 21, $C_2\times A_4^2.A_4.C_2^3$ x 10, $(C_2\times A_4^3).D_4$ x 10, $C_2\times A_4^3:D_4$ x 10, $A_4^2.(C_2\times \GL(2,\mathbb{Z}/4))$ x 7, $A_4^3:C_2^2:C_4$ x 7, $A_4^2.C_6.C_2^4.C_2$ x 4, $C_2^2\times C_2^6.C_3^2.D_6$ x 4, $A_4^2.C_6.C_2^4.C_2$ x 3, $C_2^6.C_6^3.C_2$ x 3, $C_2^2\times A_4^2.A_4.C_2^2$ x 3, $C_2^6.C_6^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2\times C_2^6.C_3^3.C_2^3$, $C_2\times A_4^2.A_4.C_2^2.C_2$, $A_4^2.A_4.C_2^3.C_2$, $C_2^2\times A_4^2.D_6.C_2^2$, $C_2\times C_2^6.(C_3\times C_6^2).C_2$, $C_2\times C_2^6.C_3^3.D_4$, $C_2\times C_2^6.C_3^3.C_2^2.C_2$, $C_2\times A_4^2.C_2^2:D_{12}$, $C_2^4.C_6^3.C_2^3$, $C_2^6.C_3^2.D_6:C_4$, $A_4^2.(C_2^4\times C_6).C_2$, $C_2^2\times C_2^6.C_3^3.C_2^2$, $C_2^3:A_4^2.S_4$, $C_2\wr C_9.C_6$, $C_2^3:A_4^2.S_4$, $C_2\wr C_9.C_6$, $C_2^{10}:\He_3$, $C_2\wr C_9.C_6$

Order 24576: $(C_2^4\times A_4).C_2^6.C_2$, $C_2^7.A_4.C_2^3.C_2$

Order 20736: $A_4^3:A_4$ x 2, $A_4^3.A_4$, $A_4\wr C_3:C_4$, $C_2^2\times A_4\wr C_3$, $A_4^3:D_6$

Order 18432: $A_4^2.C_2^5.C_2^2$ x 22, $C_2\times A_4^2.C_2^5.C_2$ x 21, $A_4^2.C_2^5.C_2^2$ x 12, $A_4^2.C_2^6.C_2$ x 9, $A_4^2.C_2^5.C_2^2$ x 8, $A_4^2.C_2^6.C_2$ x 7, $A_4^2.C_2^5.C_2^2$ x 7, $A_4^2.C_2^5.C_2^2$ x 6, $A_4^2.C_2^5.C_2^2$ x 6, $C_2^2\times A_4^2.C_2^4.C_2$ x 6, $(C_2^2\times S_4^2).D_4$ x 6, $A_4^2.C_2^5.C_2^2$ x 5, $A_4^2.C_2^5.C_2^2$ x 5, $A_4^2.C_2^5.C_2^2$ x 5, $A_4^2.C_2^5.C_2^2$ x 4, $A_4^2.C_2^4.C_2^3$ x 4, $A_4^2.C_2^5.C_2^2$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $A_4^2.C_2^4.C_2^3$ x 3, $C_2^6.C_6^2.C_2^3$ x 3, $A_4^2.C_2^5.C_2^2$ x 3, $A_4^2.C_2^5.C_2^2$ x 3, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^6.C_6^2.C_2^3$ x 2, $C_2\times C_2^4.D_6\wr C_2.C_2$, $C_2\times A_4^2.C_2^4.C_2^2$, $A_4^2.C_2^5.C_2^2$, $A_4^2.C_2^6.C_2$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^2\times C_2^2.A_4^2.C_2^3$, $C_2\times A_4^2.C_2^5.C_2$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^2\times A_4^2.C_2^5$, $C_2^6.C_6^2.C_2^3$, $C_2\times A_4^2.C_2^3.C_2^3$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^2\times A_4^2.C_2^4.C_2$, $C_2^6.C_6^2.C_2^3$, $(C_2\times A_4^2).C_2^6$, $C_2^2\times A_4^2.C_2^4.C_2$, $A_4.C_2^6.C_6.C_2^2$, $C_2^2\times A_4^2.C_2^5$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2\times C_2^6.(S_3\times C_3:D_4)$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^4.(C_2\times C_6^2).C_2^4$, $C_2\times A_4^2.C_2^4.C_2^2$, $C_2^9.D_{18}$, $C_2^6.\PSOPlus(4,3)$

Order 13824: $A_4^3.C_2^3$ x 68, $C_2\times A_4^2.A_4.C_2^2$ x 20, $A_4^2.A_4.C_2^3$ x 18, $(C_2^2\times A_4^2):D_{12}$ x 11, $C_2\times (C_3\times A_4^2).C_2^4$ x 9, $C_2\times A_4^3:C_4$ x 6, $A_4^3:D_4$ x 6, $A_4^2.C_6.C_2^3.C_2$ x 5, $A_4^2.\GL(2,\mathbb{Z}/4)$ x 5, $C_2\times C_2^6.C_3^3.C_2^2$ x 4, $A_4^2:\GL(2,\mathbb{Z}/4)$ x 4, $A_4^3:C_2^3$ x 4, $A_4^3:(C_2\times C_4)$ x 4, $A_4^2.C_6.C_2^3.C_2$ x 3, $C_2\times A_4^2.D_6.C_2^2$ x 3, $C_2^6.C_3^3.C_2^2.C_2$ x 2, $A_4^2.C_6.C_2^3.C_2$ x 2, $C_2^2\times A_4^2.A_4.C_2$ x 2, $C_2^2\times A_4^2.D_6.C_2$ x 2, $C_2^2\times (C_3\times A_4^2).C_2^3$ x 2, $(C_3\times A_4^2).C_2^5$ x 2, $C_2\wr C_9.C_3$ x 2, $C_2^9:\He_3$ x 2, $C_2^2\times A_4^2:D_{12}$ x 2, $C_2\times A_4^2.D_6:C_4$, $C_2\times C_2^6.C_3^3.C_4$, $(C_3^2\times A_4).C_2^6.C_2$, $C_2^4.C_6^3.C_2^2$, $C_2\times C_2^6.C_3^3.C_2^2$, $C_2\times A_4^2.C_6.C_2^3$, $A_4^2.C_6.C_2^4$, $(C_3\times A_4^2).C_2^4.C_2$, $C_2^2\times A_4^2.C_6.C_2^2$, $C_2^2\times C_2^6.C_3^3.C_2$, $C_2^2\times (C_3\times A_4^2).C_2^3$, $C_2^4.C_6^3.C_2^2$, $C_2\times A_4^2.C_6.C_2^2.C_2$, $C_2^4.C_6^3.C_2^2$, $A_4^2.C_2^4:S_3$, $(C_3\times A_4^2).C_2^4.C_2$, $C_2^2:A_4^2.S_4$, $C_2^2:A_4^2.S_4$, $C_2^4:A_4^2.C_6$, $C_2^3:A_4^2.D_6$

Order 12288: $(C_2^4\times A_4).C_2^5.C_2$ x 8, $(C_2^4\times A_4).C_2^5.C_2$ x 7, $C_2^7.A_4.D_4$, $C_2^2\times A_4.C_2^6.C_2^2$, $A_4.C_2^6.C_2^4$, $A_4.C_2^6.C_2^4$, $C_2^7.C_6.C_2^4$, $(C_2^4\times A_4).C_2^5.C_2$, $A_4.C_2^5.C_2^4.C_2$, $C_2^7.A_4.D_4$, $C_2^5.C_2^4.C_3.D_4$, $C_2\times (C_2^4\times A_4).C_2^4.C_2$, $C_2^2\times C_2^7.A_4.C_2$, $(C_2^4\times A_4).C_2^5.C_2$, $C_2^4:C_3.C_2^6.C_2^2$, $C_2^9.S_4$, $C_2^7.\GL(2,\mathbb{Z}/4)$, $C_2^9.S_4$

Order 10368: $A_4^3:C_6$ x 2, $(C_2\times C_6^3):S_4$, $A_4^3:S_3$

Order 8192: $C_2.C_2^6.C_2^6$

Order 6912: $C_2^6.C_3^3.C_2^2$, $C_2^2\times C_2^6.C_3^3$, $A_4:S_4^2$

Order 3456: $C_2\times A_4^3$

Order 1728: $A_4^3$

Order 1024: $C_2^{10}$

Order 512: $C_2^9$

Order 256: $C_2^8$ x 2

Order 128: $C_2^7$

Order 81: $C_3\wr C_3$

Order 64: $C_2^6$

Order 16: $C_2^4$

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 663552: $A_4^3.C_2\wr S_3$ ($C_1$)

Order 331776: $A_4^3.C_2^3:S_4$ ($C_2$), $C_2^3.A_4^3:S_4$ ($C_2$), $A_4^3.C_2^2\wr C_3$ ($C_2$)

Order 165888: $A_4^3.C_2^3:A_4$ ($C_2^2$)

Order 110592: $C_2^2\times C_2^6.C_3^3.C_2^4$ ($S_3$)

Order 82944: $A_4:S_4^2.A_4$ ($D_4$)

Order 55296: $C_2\times C_2^6.C_3^3.C_2^4$ ($D_6$)

Order 27648: $C_2^2\times C_2^6.C_3^2.D_6$ ($S_4$) x 2, $C_2^6.C_3^3.C_2^4$ ($C_3:D_4$), $C_2^2\times C_2^6.C_3^3.C_2^2$ ($S_4$)

Order 13824: $A_4^3.C_2^3$ ($C_2\times S_4$) x 2, $C_2\times C_2^6.C_3^3.C_2^2$ ($C_2\times S_4$)

Order 6912: $A_4:S_4^2$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^6.C_3^3.C_2^2$ ($\GL(2,\mathbb{Z}/4)$), $C_2^2\times C_2^6.C_3^3$ ($C_2^2:S_4$)

Order 3456: $C_2\times A_4^3$ ($C_2^3:S_4$)

Order 1728: $A_4^3$ ($C_2\wr S_3$)

Order 1024: $C_2^{10}$ ($C_3^3:S_4$)

Order 512: $C_2^9$ ($C_2\times C_3^3:S_4$)

Order 256: $C_2^8$ ($C_3^3:\GL(2,\mathbb{Z}/4)$), $C_2^8$ ($(C_3\times C_6^2):S_4$)

Order 128: $C_2^7$ ($C_6^3:S_4$)

Order 64: $C_2^6$ ($(C_2\times C_6^3):S_4$)

Order 16: $C_2^4$ ($A_4^3:S_4$)

Order 8: $C_2^3$ ($C_2\times A_4^3:S_4$)

Order 4: $C_2^2$ ($A_4:S_4^2.S_4$), $C_2^2$ ($A_4^3.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($A_4^3.C_2^3:S_4$)

Order 1: $C_1$ ($A_4^3.C_2\wr S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 663552: $A_4^3.C_2\wr S_3$ ($C_1$)

Order 331776: $A_4^3.C_2^3:S_4$ ($C_2$), $C_2^3.A_4^3:S_4$ ($C_2$), $A_4^3.C_2^2\wr C_3$ ($C_2$)

Order 165888: $A_4^3.C_2^3:A_4$ ($C_2^2$)

Order 110592: $C_2^2\times C_2^6.C_3^3.C_2^4$ ($S_3$)

Order 82944: $A_4:S_4^2.A_4$ ($D_4$)

Order 55296: $C_2\times C_2^6.C_3^3.C_2^4$ ($D_6$)

Order 27648: $C_2^6.C_3^3.C_2^4$ ($C_3:D_4$), $C_2^2\times C_2^6.C_3^2.D_6$ ($S_4$), $C_2^2\times C_2^6.C_3^3.C_2^2$ ($S_4$)

Order 13824: $C_2\times C_2^6.C_3^3.C_2^2$ ($C_2\times S_4$), $A_4^3.C_2^3$ ($C_2\times S_4$)

Order 6912: $C_2^6.C_3^3.C_2^2$ ($\GL(2,\mathbb{Z}/4)$), $C_2^2\times C_2^6.C_3^3$ ($C_2^2:S_4$), $A_4:S_4^2$ ($\GL(2,\mathbb{Z}/4)$)

Order 3456: $C_2\times A_4^3$ ($C_2^3:S_4$)

Order 1728: $A_4^3$ ($C_2\wr S_3$)

Order 1024: $C_2^{10}$ ($C_3^3:S_4$)

Order 512: $C_2^9$ ($C_2\times C_3^3:S_4$)

Order 256: $C_2^8$ ($C_3^3:\GL(2,\mathbb{Z}/4)$), $C_2^8$ ($(C_3\times C_6^2):S_4$)

Order 128: $C_2^7$ ($C_6^3:S_4$)

Order 64: $C_2^6$ ($(C_2\times C_6^3):S_4$)

Order 16: $C_2^4$ ($A_4^3:S_4$)

Order 8: $C_2^3$ ($C_2\times A_4^3:S_4$)

Order 4: $C_2^2$ ($A_4:S_4^2.S_4$), $C_2^2$ ($A_4^3.\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($A_4^3.C_2^3:S_4$)

Order 1: $C_1$ ($A_4^3.C_2\wr S_3$)

Series

Derived series

$A_4^3.C_2\wr S_3$

$\rhd$

$A_4^3.C_2^3:A_4$

$\rhd$

$C_2^6.C_3^3.C_2^4$

$\rhd$

$A_4^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$A_4^3.C_2\wr S_3$

$\rhd$

$A_4^3.C_2^2\wr C_3$

$\rhd$

$A_4^3.C_2^3:A_4$

$\rhd$

$A_4:S_4^2.A_4$

$\rhd$

$C_2^6.C_3^3.C_2^4$

$\rhd$

$A_4:S_4^2$

$\rhd$

$A_4^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Lower central series

$A_4^3.C_2\wr S_3$

$\rhd$

$A_4^3.C_2^3:A_4$

$\rhd$

$A_4:S_4^2.A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 12 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

The $276 \times 276$ character table is not available for this group.

Rational character table

The $262 \times 262$ rational character table is not available for this group.