Abstract group 663552.cp: $A_4^2.S_4\wr C_2.C_4$

• Label: 663552.cp
• Order: \( 2^{13} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $20$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$A_4^2.S_4\wr C_2.C_4$
Order:	\(663552\)\(\medspace = 2^{13} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3\times C_2^8.C_3^4.D_4.C_2^2$, of order \(5308416\)\(\medspace = 2^{16} \cdot 3^{4} \)
Composition factors:	$C_2$ x 13, $C_3$ x 4
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	12
Elements	1	5983	6560	146592	98912	248832	156672	663552
Conjugacy classes	1	23	8	46	50	22	26	176
Divisions	1	23	8	39	50	13	23	157
Autjugacy classes	1	19	7	28	40	8	12	115

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid d^{3}=e^{6}=f^{6}=g^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,4)(2,6)(3,5)(7,12,14,16)(8,10)(9,13,15,11)(17,18,19,20), (1,2,5,10,9,12) \!\cdots\! \rangle$
Transitive group:	24T20601	24T20609	36T33426	36T33434	more information
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:	$(A_4^2.S_4\wr C_2)$ . $C_4$	$(C_2^9.C_3^4:Q_8)$ . $C_2$	$(C_2^9.C_3^4:D_4)$ . $C_2$	$(C_2^9.C_3^4:D_4)$ . $C_2$	all 24

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

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_{2}^{3} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 36 normal subgroups (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4^2:\POPlus(4,3).C_2^2$
Commutator:	$G' \simeq$ $C_2^8.C_3^4.C_2$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_4^2:\POPlus(4,3).C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_3^4:D_4:C_2$
Radical:	$R \simeq$ $A_4^2.S_4\wr C_2.C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_3^4:D_4:C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^6.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 5184) are shown, as well as all normal subgroups of any index.

Order 663552: $A_4^2.S_4\wr C_2.C_4$

Order 331776: $A_4^2.S_4^2:C_2^2$, $C_2^9.C_3^4:D_4$, $A_4^2.(A_4^2:C_4\times C_2^2)$, $C_2^9.C_3^4:Q_8$, $C_2^9.C_3^4:D_4$, $C_2^8.C_3^4:C_4^2$, $C_2\times A_4^2\wr C_2.C_4$

Order 165888: $C_2^9.C_3^4:C_4$ x 4, $A_4^2.S_4\wr C_2$ x 4, $A_4^2\wr C_2.C_4$ x 2, $A_4^2.(C_2\times A_4^2:C_4)$ x 2, $C_2\times C_2^8.C_3^3.D_6$, $A_4^2\wr C_2.C_2^2$, $C_2^9.C_3^2:S_3^2$

Order 82944: $A_4^2:(A_4^2:C_4)$ x 6, $C_2^8.C_3^3.D_6$ x 3, $A_4^2:\POPlus(4,3)$ x 3, $C_2\times A_4^2.A_4^2.C_2$, $C_2\times C_2^8.C_3^4.C_2$, $C_2\times A_4^2\wr C_2$, $C_2.A_4^2\wr C_2$

Order 55296: $C_2^6.C_3^3.C_2^4.C_2$, $C_2\times C_2^8.C_3^2.D_6$, $C_2^5.A_4^2:D_6$, $C_2^5.A_4^2:D_6$, $A_4^2.C_2^4:S_4$, $C_2^3.A_4^3:C_2^2$

Order 41472: $A_4^2.A_4^2.C_2$ x 2, $C_2^8.C_3^4.C_2$ x 2, $A_4^2\wr C_2$ x 2, $C_2\times C_2^8.C_3^4$, $C_2\times C_2^6.C_3^3.D_6$

Order 36864: $C_2^9.\PSU(3,2)$ x 2, $A_4^2.C_2^3.C_2^4.C_2$, $C_2\times A_4^2.D_4^2.C_2$, $A_4^2.C_2^3.C_2^4.C_2$, $C_2^6.A_4^2:C_4$, $C_2^4.A_4^2:C_4^2$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$

Order 27648: $C_2^6.C_3^3.C_2^3.C_2$ x 8, $C_2^8.C_3^2.D_6$ x 4, $A_4^2.C_2^3:S_4$ x 4, $C_2^4.A_4^2:D_6$ x 4, $C_2\times C_2^8.C_3^3.C_2$ x 3, $C_2\times C_2^8.C_3^3.C_2$ x 2, $C_2\times C_2^8.C_3^3.C_2$ x 2, $C_2^2\times C_2^6.C_3^2.D_6$, $A_4.C_2^5.C_6^2.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^3\times A_4^2.A_4.C_2$, $C_2\times A_4^2.C_2^4.C_6$, $C_2\times C_2^4:C_3.A_4^2.C_2$, $C_2^4:A_4^2.C_{12}$, $C_2^4.A_4^2:D_6$, $(C_2^3\times A_4^2).S_4$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 20736: $A_4^3.D_6$ x 4, $C_2\times (C_3\times A_4).A_4^2.C_2$, $C_2\times A_4^2.A_4.C_6$, $C_2\times C_2^6.C_3^4.C_2$, $C_2^8.C_3^4$, $C_3^2:S_4^2:C_2^2$, $C_3^2:S_4^2:C_4$, $A_4^2:(C_6^2:C_4)$

Order 18432: $A_4^2.C_2^5:C_4$ x 8, $(C_2^2\times S_4^2).D_4$ x 8, $C_2^5.A_4^2:C_4$ x 8, $C_2^5.A_4^2:C_4$ x 7, $C_2^8.\SOPlus(4,2)$ x 4, $C_2^5.A_4^2:C_4$ x 4, $A_4^2.C_2^3.C_2^3.C_2$ x 3, $C_2^6.D_6^2.C_2$ x 2, $A_4^2.C_2^3.C_2^3.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^5.S_4^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^9.S_3^2$ x 2, $C_2^9.S_3^2$ x 2, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^2.C_2^4.C_2$, $C_2\times A_4^2.C_2\wr C_2^2$, $C_2\times A_4^2.D_4:D_4$, $C_2\times A_4^2.C_2\wr C_2^2$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2\wr C_4$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_4^2:C_4$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2\wr C_4$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^4.C_2^3$, $C_2^6.\PSOPlus(4,3)$, $C_2^6.C_{12}:S_4$, $C_2^5.S_4^2$, $C_2^5.S_4^2$

Order 13824: $A_4^3.C_2^3$ x 11, $C_2^8.C_3^3.C_2$ x 6, $C_2^2\times A_4^2.A_4.C_2$ x 6, $C_2^6.C_3^3.D_4$ x 4, $C_2^6.C_3^3.D_4$ x 4, $C_2^6.C_3^2.D_6.C_2$ x 4, $A_4.C_2^5.C_6^2$ x 4, $C_2^6.C_3^3.D_4$ x 4, $C_2^2.A_4^2:S_4$ x 4, $C_2^2:A_4^2.S_4$ x 4, $C_2\times C_2^8.C_3^3$ x 3, $A_4^2.C_2^4.C_6$ x 2, $C_2\times C_2^8.C_3^3$ x 2, $C_2\times C_2^8.C_3^3$ x 2, $C_2^4:C_3.A_4^2.C_2$ x 2, $C_2\times C_2^6.C_3^2.D_6$ x 2, $C_2^4:A_4^2.C_6$ x 2, $C_2^8.(S_3\times C_3^2)$ x 2, $(C_2^2\times A_4^2).S_4$ x 2, $C_2^8:(S_3\times C_3^2)$ x 2, $C_2^2\times A_4.A_4^2.C_2$, $C_2^3\times C_2^6.C_3^3$, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^2\times C_2^6.C_3^3.C_2$, $A_4^2.C_6.C_2^4$, $(C_3\times A_4^2).C_2^4.C_2$, $A_4.A_4^2.C_2^2.C_2$, $A_4^2.C_6.C_2^4$, $A_4^3.C_2^3$, $A_4^2:(C_4\times S_4)$

Order 10368: $C_2^5:(C_3^4:C_4)$ x 4, $(C_3^2\times A_4^2):(C_2\times C_4)$ x 4, $A_4^2:\SOPlus(4,2)$ x 4, $C_2\times (C_3\times A_4^2).D_6$ x 2, $A_4^3:S_3$ x 2, $A_4^3:S_3$ x 2, $A_4^3:C_6$ x 2, $C_2\times C_2^6.C_3^4$, $C_2\times (C_6^2\times A_4).D_6$, $(C_6\times A_4^2).D_6$, $C_2\times A_4^2:S_3^2$

Order 9216: $A_4^2.D_4^2$ x 20, $C_2^6.(S_3\times C_3:D_4)$ x 16, $A_4^2:C_2\wr C_4$ x 14, $C_2^4.A_4^2:C_4$ x 13, $A_4^2.C_2^4.C_2^2$ x 12, $C_2^4.A_4^2:C_4$ x 10, $C_2^5.\PSOPlus(4,3)$ x 8, $C_2^8.S_3^2$ x 8, $A_4^2:C_2\wr C_4$ x 8, $C_2^5.\PSOPlus(4,3)$ x 7, $C_2^8.S_3^2$ x 6, $(C_2^2:S_4)^2$ x 6, $A_4^2:\OD_{16}:C_2^2$ x 6, $A_4^2:C_2^4:C_4$ x 6, $(C_2^2\times S_4^2):C_4$ x 6, $C_2^7.\SOPlus(4,2)$ x 6, $C_2^7:\SOPlus(4,2)$ x 6, $C_2^7.\SOPlus(4,2)$ x 6, $C_2^8:S_3^2$ x 6, $A_4^2.C_2^2.C_2^3.C_2$ x 4, $C_2^6.C_6^2.C_2^2$ x 4, $C_2^5.\PSOPlus(4,3)$ x 4, $A_4^2.C_2^4.C_2^2$ x 3, $A_4^2.C_2^4.C_2^2$ x 3, $C_2^4.S_4^2$ x 3, $C_2^5.\PSOPlus(4,3)$ x 3, $C_2^6.C_6^2.C_2^2$ x 2, $C_2^6.C_6^2.C_2^2$ x 2, $(C_2\times A_4^2).C_2^5$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^6.C_6^2.C_2^2$ x 2, $A_4^2.C_2^5.C_2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4.C_2^5.C_6.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^3\times C_2^2.A_4^2.C_2$ x 2, $C_2\times C_2^8.C_3.S_3$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^6.C_6^2.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.S_4^2$ x 2, $C_2^7.(S_3\times A_4)$ x 2, $C_2^6.(A_4\times D_6)$ x 2, $C_2^6.(C_6\times S_4)$ x 2, $C_2^7.(C_3\times S_4)$ x 2, $C_2^9.(C_3\times S_3)$ x 2, $C_2^8.(C_6\times S_3)$ x 2, $A_4^2.C_2^5.C_2$, $C_2\times A_4^2.(C_2^3.C_4)$, $A_4^2.C_4.C_2^3.C_2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_2^4.C_2^2$, $C_2^4.C_6^2.C_2^3.C_2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_4.C_2^4$, $A_4^2.C_2^6$, $A_4^2.C_2^4.C_2^2$, $C_2\times A_4^2.C_4.C_2^3$, $C_2^2\times A_4^2.C_2^3.C_2$, $A_4^2.C_2^5.C_2$, $C_2^2\times A_4^2.C_2^3.C_2$, $A_4^2.C_2^5.C_2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_4.C_4^2$, $A_4^2.C_2^4.C_2^2$, $C_2\times A_4^2.C_2^3.C_2^2$, $C_2\times C_2^4.C_6^2.C_2^3$, $C_2\times A_4^2.Q_8:C_2^2$, $C_2\times A_4^2.C_4\wr C_2$, $C_2^6.C_6^2.C_2^2$, $A_4^2.C_2^4.C_2^2$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.(A_4\times S_4)$, $C_2^5.(A_4\times S_4)$, $C_2^6.A_4^2$, $C_2^6.(C_{12}\times A_4)$, $C_2\times S_4^2:D_4$, $(C_2\times S_4^2).D_4$

Order 8192: $C_2^5.C_2^6.C_2^2$

Order 6912: $A_4^3:C_2^2$ x 21, $A_4:S_4^2$ x 20, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ x 16, $A_4^3:C_2^2$ x 12, $(C_2\times C_6):S_4^2$ x 8, $(C_2^2\times A_4^2):D_6$ x 8, $A_4.A_4^2.C_2^2$ x 7, $C_2\times C_2^6.C_3^3.C_2$ x 4, $A_4^3:C_2^2$ x 4, $C_2^2\times C_2^6.C_3^3$ x 3, $C_2^8.C_3^3$ x 3, $C_2\times C_2^6.C_3^3.C_2$ x 2, $(C_3\times A_4^2).C_2^3.C_2$ x 2, $A_4.A_4^2.C_4$ x 2, $A_4.(C_6\times A_4).C_2^3$ x 2, $C_2\times (C_3\times A_4).C_2^4.C_6$ x 2, $(C_3\times A_4^2).C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_4$ x 2, $C_2\times C_2^2.A_4^2.C_6$ x 2, $(C_3\times A_4^2).C_2^3.C_2$ x 2, $C_2^2\times C_3:S_4^2$ x 2, $C_2^2.A_4^3$ x 2, $C_2^2.A_4^3$ x 2, $C_2^3\times A_4.(S_3\times A_4)$, $C_2^3\times (C_3\times A_4^2).C_2$, $A_4^2.C_6.C_2^3$, $A_4^2.C_6.C_2^3$, $A_4^2.C_6.C_2^3$, $(C_2^4\times C_6).C_6^2.C_2$, $(C_3\times A_4^2).C_2^3.C_2$, $A_4^2.C_6.C_2^3$, $(C_3\times A_4^2).C_2^3.C_2$, $A_4^2.C_6.C_2^3$, $C_2^2\times A_4^2.D_6$, $A_4^3:C_4$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $A_4^3:C_4$

Order 6144: $C_2^4:C_3.C_2^4.C_2^3$, $A_4.C_2^6.C_2^3$, $C_2\times C_2^4.C_2^4:S_3.C_2$, $C_2^9.D_6$, $C_2^9.D_6$, $C_2^9.D_6$

Order 5184: $(C_3^2\times A_4^2):C_4$ x 7, $C_3^2:S_4^2$ x 6, $(A_4\times C_6^2):D_6$ x 4, $A_4^2:S_3^2$ x 3, $C_2\times C_2^4.C_3^4.C_2$ x 2, $A_4^2.C_6^2$, $C_2\times A_4^2.C_3:S_3$, $C_2\times (C_3\times A_4^2).C_6$, $A_4.(C_6\times S_3\times A_4)$, $C_2\times C_3^2.A_4^2.C_2$, $C_3\times A_4^3$, $C_6\times A_4^2:S_3$, $(C_3\times A_4^2):C_{12}$

Order 2592: $C_3^4:(C_4\times D_4)$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 5184) are shown, as well as all normal subgroups of any index.

Order 663552: $A_4^2.S_4\wr C_2.C_4$

Order 331776: $A_4^2.S_4^2:C_2^2$, $C_2^9.C_3^4:D_4$, $A_4^2.(A_4^2:C_4\times C_2^2)$, $C_2^9.C_3^4:Q_8$, $C_2^9.C_3^4:D_4$, $C_2^8.C_3^4:C_4^2$, $C_2\times A_4^2\wr C_2.C_4$

Order 165888: $C_2^9.C_3^4:C_4$ x 4, $C_2\times C_2^8.C_3^3.D_6$, $A_4^2\wr C_2.C_2^2$, $C_2^9.C_3^2:S_3^2$, $A_4^2\wr C_2.C_4$, $A_4^2.(C_2\times A_4^2:C_4)$, $A_4^2.S_4\wr C_2$

Order 82944: $A_4^2:(A_4^2:C_4)$ x 4, $C_2^8.C_3^3.D_6$ x 2, $A_4^2:\POPlus(4,3)$ x 2, $C_2\times A_4^2.A_4^2.C_2$, $C_2\times C_2^8.C_3^4.C_2$, $C_2\times A_4^2\wr C_2$, $C_2.A_4^2\wr C_2$

Order 55296: $C_2^6.C_3^3.C_2^4.C_2$, $C_2\times C_2^8.C_3^2.D_6$, $C_2^5.A_4^2:D_6$, $C_2^5.A_4^2:D_6$, $A_4^2.C_2^4:S_4$, $C_2^3.A_4^3:C_2^2$

Order 41472: $C_2^8.C_3^4.C_2$ x 2, $C_2\times C_2^8.C_3^4$, $A_4^2.A_4^2.C_2$, $C_2\times C_2^6.C_3^3.D_6$, $A_4^2\wr C_2$

Order 36864: $A_4^2.C_2^3.C_2^4.C_2$, $C_2\times A_4^2.D_4^2.C_2$, $A_4^2.C_2^3.C_2^4.C_2$, $C_2^9.\PSU(3,2)$, $C_2^6.A_4^2:C_4$, $C_2^4.A_4^2:C_4^2$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$

Order 27648: $C_2^6.C_3^3.C_2^3.C_2$ x 4, $C_2\times C_2^8.C_3^3.C_2$ x 3, $C_2^8.C_3^2.D_6$ x 2, $C_2\times C_2^8.C_3^3.C_2$ x 2, $A_4^2.C_2^3:S_4$ x 2, $C_2^4.A_4^2:D_6$ x 2, $C_2^2\times C_2^6.C_3^2.D_6$, $A_4.C_2^5.C_6^2.C_2$, $C_2\times C_2^8.C_3^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^6.C_3^3.C_2^3.C_2$, $C_2^3\times A_4^2.A_4.C_2$, $C_2\times A_4^2.C_2^4.C_6$, $C_2\times C_2^4:C_3.A_4^2.C_2$, $C_2^4:A_4^2.C_{12}$, $C_2^4.A_4^2:D_6$, $(C_2^3\times A_4^2).S_4$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 20736: $A_4^3.D_6$ x 2, $C_2\times (C_3\times A_4).A_4^2.C_2$, $C_2\times A_4^2.A_4.C_6$, $C_2\times C_2^6.C_3^4.C_2$, $C_2^8.C_3^4$, $C_3^2:S_4^2:C_2^2$, $C_3^2:S_4^2:C_4$, $A_4^2:(C_6^2:C_4)$

Order 18432: $C_2^5.A_4^2:C_4$ x 8, $C_2^5.A_4^2:C_4$ x 4, $A_4^2.C_2^3.C_2^3.C_2$ x 3, $A_4^2.C_2^3.C_2^3.C_2$ x 2, $A_4^2.C_2^5.C_2^2$ x 2, $C_2^5.S_4^2$ x 2, $A_4^2.C_2^5:C_4$ x 2, $(C_2^2\times S_4^2).D_4$ x 2, $C_2^9.S_3^2$ x 2, $C_2^9.S_3^2$ x 2, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^2.C_2^4.C_2$, $C_2\times A_4^2.C_2\wr C_2^2$, $C_2\times A_4^2.D_4:D_4$, $C_2\times A_4^2.C_2\wr C_2^2$, $C_2^6.D_6^2.C_2$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2\wr C_4$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_4^2:C_4$, $A_4^2.C_2^2.C_2^4.C_2$, $A_4^2.C_2^5.C_2^2$, $C_2\times A_4^2.C_2\wr C_4$, $A_4^2.C_2^3.C_2^3.C_2$, $A_4^2.C_2^4.C_2^3$, $C_2^8.\SOPlus(4,2)$, $C_2^5.A_4^2:C_4$, $C_2^6.\PSOPlus(4,3)$, $C_2^6.C_{12}:S_4$, $C_2^9.S_3^2$, $C_2^9.S_3^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$

Order 13824: $A_4^3.C_2^3$ x 7, $C_2^8.C_3^3.C_2$ x 6, $C_2^6.C_3^3.D_4$ x 4, $C_2^2\times A_4^2.A_4.C_2$ x 4, $C_2^2.A_4^2:S_4$ x 4, $C_2\times C_2^8.C_3^3$ x 3, $C_2^6.C_3^3.D_4$ x 2, $C_2^6.C_3^2.D_6.C_2$ x 2, $C_2\times C_2^8.C_3^3$ x 2, $A_4.C_2^5.C_6^2$ x 2, $C_2^6.C_3^3.D_4$ x 2, $C_2^2:A_4^2.S_4$ x 2, $C_2^2\times A_4.A_4^2.C_2$, $A_4^2.C_2^4.C_6$, $C_2\times C_2^8.C_3^3$, $C_2^3\times C_2^6.C_3^3$, $C_2^6.C_3^3.C_2^2.C_2$, $C_2^2\times C_2^6.C_3^3.C_2$, $C_2^4:C_3.A_4^2.C_2$, $A_4^2.C_6.C_2^4$, $(C_3\times A_4^2).C_2^4.C_2$, $A_4.A_4^2.C_2^2.C_2$, $C_2\times C_2^6.C_3^2.D_6$, $A_4^2.C_6.C_2^4$, $C_2^4:A_4^2.C_6$, $C_2^8.(S_3\times C_3^2)$, $A_4^3.C_2^3$, $A_4^2:(C_4\times S_4)$, $(C_2^2\times A_4^2).S_4$, $C_2^8:(S_3\times C_3^2)$

Order 10368: $C_2^5:(C_3^4:C_4)$ x 4, $C_2\times (C_3\times A_4^2).D_6$ x 2, $A_4^3:S_3$ x 2, $C_2\times C_2^6.C_3^4$, $C_2\times (C_6^2\times A_4).D_6$, $A_4^3:S_3$, $A_4^3:C_6$, $(C_6\times A_4^2).D_6$, $C_2\times A_4^2:S_3^2$, $(C_3^2\times A_4^2):(C_2\times C_4)$, $A_4^2:\SOPlus(4,2)$

Order 9216: $A_4^2.D_4^2$ x 12, $C_2^4.A_4^2:C_4$ x 8, $A_4^2:C_2\wr C_4$ x 8, $C_2^5.\PSOPlus(4,3)$ x 7, $A_4^2.C_2^4.C_2^2$ x 6, $C_2^5.\PSOPlus(4,3)$ x 6, $C_2^6.(S_3\times C_3:D_4)$ x 4, $A_4^2.C_2^2.C_2^3.C_2$ x 4, $C_2^4.A_4^2:C_4$ x 4, $C_2^8.S_3^2$ x 4, $(C_2^2:S_4)^2$ x 4, $C_2^8:S_3^2$ x 4, $A_4^2.C_2^4.C_2^2$ x 3, $C_2^6.C_6^2.C_2^2$ x 3, $A_4^2.C_2^4.C_2^2$ x 3, $(C_2\times A_4^2).C_2^5$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^5.C_2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_2^2$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^5.\PSOPlus(4,3)$ x 2, $C_2^4.S_4^2$ x 2, $C_2^8.S_3^2$ x 2, $C_2^9.(C_3\times S_3)$ x 2, $C_2^8.(C_6\times S_3)$ x 2, $A_4^2:\OD_{16}:C_2^2$ x 2, $A_4^2:C_2\wr C_4$ x 2, $A_4^2:C_2^4:C_4$ x 2, $(C_2^2\times S_4^2):C_4$ x 2, $C_2^7.\SOPlus(4,2)$ x 2, $C_2^7:\SOPlus(4,2)$ x 2, $C_2^7.\SOPlus(4,2)$ x 2, $C_2^5.\PSOPlus(4,3)$ x 2, $A_4^2.C_2^5.C_2$, $C_2^6.C_6^2.C_2^2$, $C_2\times A_4^2.(C_2^3.C_4)$, $C_2^6.C_6^2.C_2^2$, $A_4^2.C_4.C_2^3.C_2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_2^4.C_2^2$, $C_2^4.C_6^2.C_2^3.C_2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_2^4.C_2^2$, $A_4^2.C_4.C_2^4$, $A_4^2.C_2^6$, $A_4^2.C_2^4.C_2^2$, $C_2\times A_4^2.C_4.C_2^3$, $C_2^2\times A_4^2.C_2^3.C_2$, $A_4^2.C_2^5.C_2$, $C_2^6.C_6^2.C_2^2$, $C_2^2\times A_4^2.C_2^3.C_2$, $A_4^2.C_2^5.C_2$, $C_2\times A_4^2.C_2^3.C_2^2$, $A_4^2.C_4.C_4^2$, $A_4^2.C_2^4.C_2^2$, $A_4.C_2^5.C_6.C_2^2$, $C_2\times A_4^2.C_2^3.C_2^2$, $C_2\times C_2^4.C_6^2.C_2^3$, $C_2\times A_4^2.Q_8:C_2^2$, $C_2^3\times C_2^2.A_4^2.C_2$, $C_2\times A_4^2.C_4\wr C_2$, $C_2\times C_2^8.C_3.S_3$, $C_2^6.C_6^2.C_2^2$, $A_4^2.C_2^4.C_2^2$, $C_2^6.C_6^2.C_2^2$, $C_2^4.S_4^2$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.(A_4\times S_4)$, $C_2^5.(A_4\times S_4)$, $C_2^7.(S_3\times A_4)$, $C_2^6.(A_4\times D_6)$, $C_2^6.(C_6\times S_4)$, $C_2^7.(C_3\times S_4)$, $C_2^6.A_4^2$, $C_2^6.(C_{12}\times A_4)$, $C_2\times S_4^2:D_4$, $(C_2\times S_4^2).D_4$

Order 8192: $C_2^5.C_2^6.C_2^2$

Order 6912: $A_4^3:C_2^2$ x 15, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ x 8, $A_4^3:C_2^2$ x 8, $A_4:S_4^2$ x 8, $A_4.A_4^2.C_2^2$ x 5, $(C_2^2\times A_4^2):D_6$ x 4, $C_2\times C_2^6.C_3^3.C_2$ x 3, $C_2^2\times C_2^6.C_3^3$ x 3, $C_2^8.C_3^3$ x 3, $(C_3\times A_4^2).C_2^3.C_2$ x 2, $A_4.(C_6\times A_4).C_2^3$ x 2, $(C_3\times A_4^2).C_2^3.C_2$ x 2, $C_2^6.C_3^3.C_4$ x 2, $(C_3\times A_4^2).C_2^3.C_2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $C_2^2\times C_3:S_4^2$ x 2, $C_2^2.A_4^3$ x 2, $A_4^3:C_2^2$ x 2, $C_2^3\times A_4.(S_3\times A_4)$, $C_2^3\times (C_3\times A_4^2).C_2$, $C_2\times C_2^6.C_3^3.C_2$, $A_4^2.C_6.C_2^3$, $A_4^2.C_6.C_2^3$, $A_4.A_4^2.C_4$, $A_4^2.C_6.C_2^3$, $(C_2^4\times C_6).C_6^2.C_2$, $C_2\times (C_3\times A_4).C_2^4.C_6$, $(C_3\times A_4^2).C_2^3.C_2$, $A_4^2.C_6.C_2^3$, $(C_3\times A_4^2).C_2^3.C_2$, $C_2\times C_2^2.A_4^2.C_6$, $A_4^2.C_6.C_2^3$, $C_2^2\times A_4^2.D_6$, $C_2^2.A_4^3$, $A_4^3:C_4$, $A_4^3:C_2^2$, $A_4^3:C_2^2$, $A_4^3:C_4$

Order 6144: $C_2^4:C_3.C_2^4.C_2^3$, $A_4.C_2^6.C_2^3$, $C_2\times C_2^4.C_2^4:S_3.C_2$, $C_2^9.D_6$, $C_2^9.D_6$, $C_2^9.D_6$

Order 5184: $C_3^2:S_4^2$ x 4, $(C_3^2\times A_4^2):C_4$ x 4, $C_2\times C_2^4.C_3^4.C_2$ x 2, $(A_4\times C_6^2):D_6$ x 2, $A_4^2:S_3^2$ x 2, $A_4^2.C_6^2$, $C_2\times A_4^2.C_3:S_3$, $C_2\times (C_3\times A_4^2).C_6$, $A_4.(C_6\times S_3\times A_4)$, $C_2\times C_3^2.A_4^2.C_2$, $C_3\times A_4^3$, $C_6\times A_4^2:S_3$, $(C_3\times A_4^2):C_{12}$

Order 2592: $C_3^4:(C_4\times D_4)$

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 663552: $A_4^2.S_4\wr C_2.C_4$ ($C_1$)

Order 331776: $A_4^2.S_4^2:C_2^2$ ($C_2$), $C_2^9.C_3^4:D_4$ ($C_2$), $A_4^2.(A_4^2:C_4\times C_2^2)$ ($C_2$), $C_2^9.C_3^4:Q_8$ ($C_2$), $C_2^9.C_3^4:D_4$ ($C_2$), $C_2^8.C_3^4:C_4^2$ ($C_2$), $C_2\times A_4^2\wr C_2.C_4$ ($C_2$)

Order 165888: $C_2^9.C_3^4:C_4$ ($C_2^2$) x 4, $A_4^2.S_4\wr C_2$ ($C_4$) x 4, $C_2\times C_2^8.C_3^3.D_6$ ($C_2^2$), $A_4^2\wr C_2.C_2^2$ ($C_2^2$), $C_2^9.C_3^2:S_3^2$ ($C_2^2$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_2\times C_4$) x 2, $A_4^2:(A_4^2:C_4)$ ($D_4$) x 2, $A_4^2:(A_4^2:C_4)$ ($C_2\times C_4$) x 2, $A_4^2:\POPlus(4,3)$ ($C_2\times C_4$) x 2, $C_2\times C_2^8.C_3^4.C_2$ ($C_2^3$)

Order 41472: $C_2\times C_2^8.C_3^4$ ($D_4:C_2$), $C_2^8.C_3^4.C_2$ ($C_2\times D_4$), $C_2^8.C_3^4.C_2$ ($C_2^2\times C_4$)

Order 20736: $C_2^8.C_3^4$ ($C_4\times D_4$)

Order 512: $C_2^9$ ($C_3^4:D_4:C_2$)

Order 256: $C_2^8$ ($C_3^4:(C_4\times D_4)$)

Order 2: $C_2$ ($A_4^2:\POPlus(4,3).C_2^2$)

Order 1: $C_1$ ($A_4^2.S_4\wr C_2.C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 663552: $A_4^2.S_4\wr C_2.C_4$ ($C_1$)

Order 331776: $A_4^2.S_4^2:C_2^2$ ($C_2$), $C_2^9.C_3^4:D_4$ ($C_2$), $A_4^2.(A_4^2:C_4\times C_2^2)$ ($C_2$), $C_2^9.C_3^4:Q_8$ ($C_2$), $C_2^9.C_3^4:D_4$ ($C_2$), $C_2^8.C_3^4:C_4^2$ ($C_2$), $C_2\times A_4^2\wr C_2.C_4$ ($C_2$)

Order 165888: $C_2^9.C_3^4:C_4$ ($C_2^2$) x 4, $C_2\times C_2^8.C_3^3.D_6$ ($C_2^2$), $A_4^2\wr C_2.C_2^2$ ($C_2^2$), $C_2^9.C_3^2:S_3^2$ ($C_2^2$), $A_4^2.S_4\wr C_2$ ($C_4$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_2\times C_4$), $C_2\times C_2^8.C_3^4.C_2$ ($C_2^3$), $A_4^2:(A_4^2:C_4)$ ($D_4$), $A_4^2:(A_4^2:C_4)$ ($C_2\times C_4$), $A_4^2:\POPlus(4,3)$ ($C_2\times C_4$)

Order 41472: $C_2\times C_2^8.C_3^4$ ($D_4:C_2$), $C_2^8.C_3^4.C_2$ ($C_2\times D_4$), $C_2^8.C_3^4.C_2$ ($C_2^2\times C_4$)

Order 20736: $C_2^8.C_3^4$ ($C_4\times D_4$)

Order 512: $C_2^9$ ($C_3^4:D_4:C_2$)

Order 256: $C_2^8$ ($C_3^4:(C_4\times D_4)$)

Order 2: $C_2$ ($A_4^2:\POPlus(4,3).C_2^2$)

Order 1: $C_1$ ($A_4^2.S_4\wr C_2.C_4$)

Series

Derived series

$A_4^2.S_4\wr C_2.C_4$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$A_4^2.S_4\wr C_2.C_4$

$\rhd$

$A_4^2.(A_4^2:C_4\times C_2^2)$

$\rhd$

$C_2\times C_2^8.C_3^3.D_6$

$\rhd$

$C_2\times C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

$A_4^2.S_4\wr C_2.C_4$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

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

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

Character theory

Complex character table

The $176 \times 176$ character table is not available for this group.

Rational character table

The $157 \times 157$ rational character table is not available for this group.