Abstract group 1327104.dk: $A_4^2\wr C_2.C_2^2.C_2^3$

• Label: 1327104.dk
• Order: \( 2^{14} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{19} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: not computed
• Trans deg.: $24$
• Rank: $4$

Group information

Description:	$A_4^2\wr C_2.C_2^2.C_2^3$
Order:	\(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^8.C_3^4.C_2^6.C_2^4.C_2$, of order \(42467328\)\(\medspace = 2^{19} \cdot 3^{4} \)
Composition factors:	$C_2$ x 14, $C_3$ 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	12319	6560	237024	274016	373248	423936	1327104
Conjugacy classes	1	37	7	80	65	18	66	274
Divisions	1	37	7	78	65	16	62	266
Autjugacy classes	1	20	5	30	33	4	18	111

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$24$
Rank:	$4$
Inequivalent generating quadruples:	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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid b^{4}=f^{6}=g^{6}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(3,5,4,6)(7,9,8,10)(15,16)(17,18)(19,23,22)(20,24,21), (1,14,5,17,2,13,6,18) \!\cdots\! \rangle$
Transitive group:	24T21663	24T21672	36T39610		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:	$(C_2^8.C_3.S_3^3)$ . $D_4$	$C_2^8$ . $(C_3:S_3^3:D_4)$	$(A_4^2.S_4^2:C_2^3)$ . $C_2$	$(C_2^9.C_3:S_3^3)$ . $C_2^2$	all 31

Elements of the group are displayed as permutations of degree 24.

Homology

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

Subgroups

There are 85 normal subgroups (15 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4^2:\POPlus(4,3).C_2^3$
Commutator:	$G' \simeq$ $C_2^8.C_3^3.D_6$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $A_4^2:\POPlus(4,3).C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_3^4.Q_8:C_2^2$
Radical:	$R \simeq$ $A_4^2\wr C_2.C_2^2.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_3^4.Q_8:C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2.C_2^5$
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 64 (order at least 20736) are shown, as well as all normal subgroups of any index.

Order 1327104: $A_4^2\wr C_2.C_2^2.C_2^3$

Order 663552: $A_4^2\wr C_2.C_2^2.C_2^2$ x 4, $(C_2\times A_4^2\wr C_2).D_4$ x 4, $A_4^2.S_4^2:C_2^3$ x 2, $A_4^2\wr C_2.C_2^2.C_2^2$ x 2, $C_2.A_4^2\wr C_2.D_4$ x 2, $A_4^2.(C_2\times S_4^2:C_4)$

Order 331776: $A_4^2.S_4^2:C_2^2$ x 8, $A_4^2.S_4^2:C_2^2$ x 8, $C_2^9.C_3^4:D_4$ x 4, $C_2^9.C_3^4:D_4$ x 4, $(C_2\times A_4^2\wr C_2).C_4$ x 4, $A_4^2.(A_4^2:C_4\times C_2^2)$ x 3, $C_2^9.C_3:S_3^3$ x 2, $A_4^2\wr C_2.C_2^3$ x 2, $C_2^9.C_3^4:Q_8$ x 2, $C_2^9.C_3^4:D_4$ x 2, $C_2.A_4^2\wr C_2.C_4$ x 2, $C_2^8.C_3^4.C_2^4$, $C_2^9.C_3:S_3^3$

Order 165888: $A_4^2.S_4\wr C_2$ x 16, $C_2^9.C_3^4:C_4$ x 6, $A_4^2\wr C_2.C_2^2$ x 6, $C_2^8.C_3.S_3^3$ x 5, $A_4^2.(C_2\times A_4^2:C_4)$ x 5, $A_4^2\wr C_2.C_2^2$ x 4, $C_2^8.C_3^4.C_2^3$ x 3, $C_2^9.C_3\wr C_2^2$ x 2, $C_2^8.C_3^4:D_4$ x 2, $C_2^9.C_3^2:S_3^2$ x 2, $A_4^2\wr C_2.C_2^2$ x 2, $C_2^6.C_3^3.A_4.C_2^3$, $C_2^9.C_3\wr C_2^2$

Order 110592: $C_2^9.S_3^3$ x 2, $C_2^9.S_3^3$ x 2, $C_2^6.C_3^3.C_2.C_2^5$, $C_2^9.S_3^3$

Order 82944: $C_2^8.C_3^3.D_6$ x 9, $A_4^2.\POPlus(4,3)$ x 8, $A_4^2:\POPlus(4,3)$ x 8, $A_4^2:(A_4^2:C_4)$ x 6, $C_2^6.C_3^3.A_4.C_2^2$ x 4, $C_2\times A_4^2\wr C_2$ x 4, $A_4^2.A_4^2.C_2^2$ x 3, $C_2.A_4^2\wr C_2$ x 2, $C_2^6.C_3^4.C_2^4$, $C_2^8.C_3^3.D_6$

Order 73728: $C_2^6.S_4\wr C_2$ x 4, $A_4^2.C_2^2.C_2^6.C_2$ x 2, $C_2^4.S_4^2:D_4$ x 2, $A_4^2.C_2^3.C_2^5.C_2$

Order 55296: $C_2^8.S_3^3$ x 16, $C_2^6.C_3^3.C_2^4.C_2$ x 8, $C_2^6.C_3^3.C_2.C_2^4$ x 8, $C_2^9.(C_3\times S_3^2)$ x 4, $C_2^5.A_4^2:D_6$ x 4, $A_4^2.C_2^4:S_4$ x 4, $C_2^8.(C_6\times S_3^2)$ x 4, $C_2^6.C_3^3.C_2.C_2^4$ x 3, $C_2^8.C_3^3.C_2^3$ x 3, $A_4^2.(C_2^4\times A_4).C_2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^2.D_6.C_2^3$ x 2, $C_2^4:A_4^2.D_{12}$ x 2, $C_2^9.(C_3\times S_3^2)$ x 2, $C_2^5.A_4^2:D_6$ x 2, $C_2^3.A_4^3:C_2^2$ x 2, $A_4^2.C_2^2\wr S_3$ x 2, $C_2^9.(C_3\times S_3^2)$ x 2, $C_2^7.(A_4\times S_3^2)$ x 2, $A_4^2.(D_4\times A_4).C_2^2$, $C_2^6.C_3^3.C_2^5$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.C_2^5$, $A_4^2:C_2^4.S_4$, $C_2^7.(A_4\times S_3^2)$

Order 41472: $C_3.S_4^3$ x 8, $A_4^2.A_4^2.C_2$ x 6, $A_4^2\wr C_2$ x 4, $C_2^6.C_3^4.C_2^3$ x 3, $(C_3\times A_4^2).A_4.C_2^3$ x 2, $C_2^8.C_3^4.C_2$ x 2, $A_4^2:D_6^2:C_2$ x 2, $A_4^2.(S_3\times A_4).C_2^2$, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $A_4^2:(C_6^2.D_4)$

Order 36864: $(C_2^3\times S_4^2).D_4$ x 32, $A_4^2.C_2^2.C_2^5.C_2$ x 8, $A_4^2.C_2^2.C_2^5.C_2$ x 8, $C_2^9.\SOPlus(4,2)$ x 8, $A_4^2.C_2^3.C_2^4.C_2$ x 4, $A_4^2.C_2^3.C_2^4.C_2$ x 4, $C_2^6.S_4^2$ x 4, $C_2^6.A_4^2:C_4$ x 4, $C_2^6.\POPlus(4,3)$ x 4, $C_2^9.\SOPlus(4,2)$ x 4, $C_2^9.\SOPlus(4,2)$ x 4, $C_2^9.\SOPlus(4,2)$ x 4, $C_2^9.\SOPlus(4,2)$ x 4, $A_4^2.C_2^3.C_2^4.C_2$ x 3, $A_4^2.C_2^3.C_2^4.C_2$ x 3, $A_4^2.C_2^3.C_2^5$ x 2, $A_4^2.C_2^2.C_2^5.C_2$ x 2, $A_4^2.C_2^4.C_2^4$ x 2, $A_4^2.C_2^2.C_2^5.C_2$ x 2, $A_4^2.C_2^6.C_2^2$ x 2, $C_2^4.(C_4\times S_4^2)$ x 2, $C_2^9.\PSU(3,2)$ x 2, $C_2^4.(C_4\times A_4^2):C_4$ x 2, $C_2^9.(S_3\times D_6)$ x 2, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^4.C_2^4$, $A_4^2.D_4^2.C_2^2$, $A_4^2.C_2^2.C_2^5.C_2$, $A_4^2.C_4:D_4.C_2^3$

Order 27648: $A_4^2.(C_2\times D_4\times A_4)$ x 20, $C_2^6.C_3:(S_3\times D_{12})$ x 16, $C_2^8.(C_3\times S_3^2)$ x 16, $A_4^2.C_2^3:S_4$ x 16, $C_2\times S_4^3$ x 16, $C_2^8.C_3^2.D_6$ x 12, $C_2^6.C_3^3.C_2^4$ x 12, $C_2^6.C_3^3.C_2^3.C_2$ x 12, $C_2^6.C_3^3.D_4.C_2$ x 10, $(C_2^4\times A_4).D_6:D_6$ x 8, $C_2^6.C_3^3.C_2^3.C_2$ x 8, $C_2^6.C_3^3.C_2^3.C_2$ x 8, $C_2^6.C_3^2.D_6.C_2^2$ x 8, $C_2^6.(A_4\times S_3^2)$ x 8, $A_4^2.C_2^3:S_4$ x 8, $C_2^4.A_4^2:D_6$ x 8, $C_2^6.C_3^2.D_6.C_2^2$ x 6, $A_4^2.(D_4\times A_4).C_2$ x 4, $A_4^2.(C_2\times D_4\times A_4)$ x 4, $C_2^6.C_3^3.C_2.C_2^3$ x 4, $C_2^4.A_4^2:D_6$ x 4, $(C_2^3\times A_4^2).S_4$ x 4, $C_2^8.(C_3^2\times D_6)$ x 4, $C_2^8.(C_3^2\times D_6)$ x 4, $C_2^6.C_3^3.C_2^4$ x 3, $A_4.C_2^5.C_6^2.C_2$ x 3, $(C_3\times A_4^2).C_2.C_2^5$ x 3, $C_2^6.C_3^3.D_4.C_2$ x 3, $C_2^6.C_3^3.D_4.C_2$ x 3, $A_4^2.A_4.C_2^4$ x 3, $A_4^2.A_4.C_2^4$ x 3, $A_4^2.C_2^4.C_6.C_2$ x 3, $C_2^8.C_3^2.D_6$ x 3, $C_2^4:C_3.A_4^2.C_2^2$ x 3, $C_2^6.C_3^2.D_6.C_2^2$ x 2, $C_2^8.C_3^2.D_6$ x 2, $A_4^2.A_4.C_2^3.C_2$ x 2, $A_4^2.(D_4\times A_4).C_2$ x 2, $A_4^3.C_2^4$ x 2, $C_2^4:A_4^2.C_{12}$ x 2, $(C_2^3\times A_4^2).S_4$ x 2, $C_2^9.(S_3\times C_3^2)$ x 2, $C_2^9.(S_3\times C_3^2)$ x 2, $C_2^8.C_3^2.D_6$, $C_2^6.C_3^3.C_2.C_2^3$, $A_4^2.A_4.C_4.C_2^2$, $C_2^6.C_3^3.D_4.C_2$, $C_2^7.S_3^3$

Order 20736: $C_3^2:S_4^2:C_2^2$ x 16, $A_4^3.D_6$ x 12, $A_4^3:D_6$ x 12, $C_3^2:S_4^2:C_2^2$ x 8, $C_3\times A_4:S_4^2$ x 4, $C_3^2:S_4^2:C_4$ x 4, $(C_3\times A_4).A_4^2.C_2^2$ x 3, $A_4^2.C_2^2.C_6^2$ x 3, $C_2^4.C_3^4.C_2^4$ x 3, $A_4^2:(C_6^2:C_4)$ x 3, $A_4^2:D_6^2$ x 2, $C_2^6.C_3^3.D_6$, $C_2^8.C_3^4$, $(C_2^4\times C_6).S_3^3$

Order 16384: $C_2^8.C_2.C_2^5$

Order 5184: $C_3:S_3^3:D_4$

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 64 (order at least 20736) are shown, as well as all normal subgroups of any index.

Order 1327104: $A_4^2\wr C_2.C_2^2.C_2^3$

Order 663552: $A_4^2.(C_2\times S_4^2:C_4)$, $A_4^2.S_4^2:C_2^3$, $A_4^2\wr C_2.C_2^2.C_2^2$, $A_4^2\wr C_2.C_2^2.C_2^2$, $(C_2\times A_4^2\wr C_2).D_4$, $C_2.A_4^2\wr C_2.D_4$

Order 331776: $A_4^2.S_4^2:C_2^2$ x 2, $A_4^2.(A_4^2:C_4\times C_2^2)$ x 2, $C_2^8.C_3^4.C_2^4$, $C_2^9.C_3:S_3^3$, $C_2^9.C_3^4:D_4$, $C_2^9.C_3:S_3^3$, $A_4^2\wr C_2.C_2^3$, $A_4^2.S_4^2:C_2^2$, $C_2^9.C_3^4:Q_8$, $C_2^9.C_3^4:D_4$, $C_2^9.C_3^4:D_4$, $C_2.A_4^2\wr C_2.C_4$, $(C_2\times A_4^2\wr C_2).C_4$

Order 165888: $C_2^8.C_3.S_3^3$ x 2, $C_2^8.C_3^4.C_2^3$ x 2, $C_2^9.C_3^4:C_4$ x 2, $A_4^2.(C_2\times A_4^2:C_4)$ x 2, $A_4^2\wr C_2.C_2^2$ x 2, $A_4^2.S_4\wr C_2$ x 2, $C_2^6.C_3^3.A_4.C_2^3$, $C_2^9.C_3\wr C_2^2$, $C_2^8.C_3^4:D_4$, $A_4^2\wr C_2.C_2^2$, $C_2^9.C_3^2:S_3^2$, $C_2^9.C_3\wr C_2^2$, $A_4^2\wr C_2.C_2^2$

Order 110592: $C_2^6.C_3^3.C_2.C_2^5$, $C_2^9.S_3^3$, $C_2^9.S_3^3$, $C_2^9.S_3^3$

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

Order 73728: $A_4^2.C_2^2.C_2^6.C_2$, $A_4^2.C_2^3.C_2^5.C_2$, $C_2^6.S_4\wr C_2$, $C_2^4.S_4^2:D_4$

Order 55296: $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^8.C_3^3.C_2^3$ x 2, $C_2^8.S_3^3$ x 2, $A_4^2.(D_4\times A_4).C_2^2$, $A_4^2.(C_2^4\times A_4).C_2$, $C_2^6.C_3^3.C_2^5$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^3.C_2.C_2^4$, $C_2^6.C_3^2.D_6.C_2^3$, $C_2^6.C_3^3.C_2^5$, $C_2^4:A_4^2.D_{12}$, $C_2^9.(C_3\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$, $C_2^5.A_4^2:D_6$, $C_2^5.A_4^2:D_6$, $A_4^2.C_2^4:S_4$, $C_2^3.A_4^3:C_2^2$, $A_4^2:C_2^4.S_4$, $A_4^2.C_2^2\wr S_3$, $C_2^8.(C_6\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$, $C_2^7.(A_4\times S_3^2)$, $C_2^7.(A_4\times S_3^2)$

Order 41472: $A_4^2.A_4^2.C_2$ x 2, $C_2^8.C_3^4.C_2$ x 2, $C_2^6.C_3^4.C_2^3$ x 2, $C_3.S_4^3$ x 2, $A_4^2.(S_3\times A_4).C_2^2$, $(C_3\times A_4^2).A_4.C_2^3$, $C_2^8.C_3^4.C_2$, $C_2^6.C_3^4.C_2^3$, $A_4^2:D_6^2:C_2$, $A_4^2:(C_6^2.D_4)$, $A_4^2\wr C_2$

Order 36864: $A_4^2.C_2^3.C_2^4.C_2$ x 2, $A_4^2.C_2^2.C_2^5.C_2$ x 2, $A_4^2.C_2^2.C_2^5.C_2$ x 2, $A_4^2.C_2^3.C_2^4.C_2$ x 2, $(C_2^3\times S_4^2).D_4$ x 2, $C_2^9.\SOPlus(4,2)$ x 2, $A_4^2.C_2^3.C_2^4.C_2$, $A_4^2.C_2^5.C_2^3$, $A_4^2.C_2^3.C_2^4.C_2$, $A_4^2.C_2^3.C_2^5$, $A_4^2.C_2^4.C_2^4$, $A_4^2.C_2^2.C_2^5.C_2$, $A_4^2.D_4^2.C_2^2$, $A_4^2.C_2^4.C_2^4$, $A_4^2.C_2^2.C_2^5.C_2$, $A_4^2.C_4:D_4.C_2^3$, $A_4^2.C_2^2.C_2^5.C_2$, $A_4^2.C_2^6.C_2^2$, $C_2^6.S_4^2$, $C_2^4.(C_4\times S_4^2)$, $C_2^9.\PSU(3,2)$, $C_2^6.A_4^2:C_4$, $C_2^4.(C_4\times A_4^2):C_4$, $C_2^6.\POPlus(4,3)$, $C_2^9.(S_3\times D_6)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$, $C_2^9.\SOPlus(4,2)$

Order 27648: $C_2\times S_4^3$ x 6, $C_2^6.C_3^3.C_2^4$ x 5, $C_2^8.C_3^2.D_6$ x 4, $C_2^6.C_3^3.D_4.C_2$ x 3, $C_2^6.C_3^3.C_2^3.C_2$ x 3, $A_4^2.(C_2\times D_4\times A_4)$ x 3, $C_2^6.C_3^3.C_2^4$ x 2, $A_4.C_2^5.C_6^2.C_2$ x 2, $C_2^6.C_3:(S_3\times D_{12})$ x 2, $(C_2^4\times A_4).D_6:D_6$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $(C_3\times A_4^2).C_2.C_2^5$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $C_2^6.C_3^3.D_4.C_2$ x 2, $A_4^2.A_4.C_2^4$ x 2, $A_4^2.A_4.C_2^4$ x 2, $C_2^6.C_3^2.D_6.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2$ x 2, $C_2^8.C_3^2.D_6$ x 2, $A_4^2.(C_2\times D_4\times A_4)$ x 2, $C_2^6.C_3^3.C_2^3.C_2$ x 2, $C_2^4:C_3.A_4^2.C_2^2$ x 2, $A_4^2.C_2^3:S_4$ x 2, $C_2^4.A_4^2:D_6$ x 2, $C_2^8.C_3^2.D_6$, $C_2^6.C_3^3.C_2.C_2^3$, $C_2^6.C_3^2.D_6.C_2^2$, $C_2^8.C_3^2.D_6$, $A_4^2.A_4.C_4.C_2^2$, $A_4^2.A_4.C_2^3.C_2$, $C_2^6.C_3^3.D_4.C_2$, $A_4^2.(D_4\times A_4).C_2$, $A_4^2.(D_4\times A_4).C_2$, $C_2^6.C_3^3.C_2.C_2^3$, $C_2^6.C_3^2.D_6.C_2^2$, $C_2^8.(C_3\times S_3^2)$, $C_2^6.(A_4\times S_3^2)$, $A_4^2.C_2^3:S_4$, $A_4^3.C_2^4$, $C_2^7.S_3^3$, $C_2^4:A_4^2.C_{12}$, $C_2^4.A_4^2:D_6$, $(C_2^3\times A_4^2).S_4$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 20736: $A_4^3.D_6$ x 4, $(C_3\times A_4).A_4^2.C_2^2$ x 2, $A_4^2.C_2^2.C_6^2$ x 2, $C_2^4.C_3^4.C_2^4$ x 2, $A_4^3:D_6$ x 2, $C_3^2:S_4^2:C_2^2$ x 2, $A_4^2:(C_6^2:C_4)$ x 2, $C_2^6.C_3^3.D_6$, $C_2^8.C_3^4$, $C_3\times A_4:S_4^2$, $C_3^2:S_4^2:C_4$, $(C_2^4\times C_6).S_3^3$, $A_4^2:D_6^2$, $C_3^2:S_4^2:C_2^2$

Order 16384: $C_2^8.C_2.C_2^5$

Order 5184: $C_3:S_3^3:D_4$

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 1327104: $A_4^2\wr C_2.C_2^2.C_2^3$ ($C_1$)

Order 663552: $A_4^2\wr C_2.C_2^2.C_2^2$ ($C_2$) x 4, $(C_2\times A_4^2\wr C_2).D_4$ ($C_2$) x 4, $A_4^2.S_4^2:C_2^3$ ($C_2$) x 2, $A_4^2\wr C_2.C_2^2.C_2^2$ ($C_2$) x 2, $C_2.A_4^2\wr C_2.D_4$ ($C_2$) x 2, $A_4^2.(C_2\times S_4^2:C_4)$ ($C_2$)

Order 331776: $A_4^2.S_4^2:C_2^2$ ($C_2^2$) x 8, $C_2^9.C_3^4:D_4$ ($C_2^2$) x 4, $C_2^9.C_3^4:D_4$ ($C_2^2$) x 4, $(C_2\times A_4^2\wr C_2).C_4$ ($C_2^2$) x 4, $A_4^2.(A_4^2:C_4\times C_2^2)$ ($C_2^2$) x 3, $C_2^9.C_3:S_3^3$ ($C_2^2$) x 2, $A_4^2\wr C_2.C_2^3$ ($C_2^2$) x 2, $C_2^9.C_3^4:Q_8$ ($C_2^2$) x 2, $C_2^9.C_3^4:D_4$ ($C_2^2$) x 2, $C_2.A_4^2\wr C_2.C_4$ ($C_2^2$) x 2, $C_2^8.C_3^4.C_2^4$ ($C_2^2$), $C_2^9.C_3:S_3^3$ ($C_2^2$)

Order 165888: $C_2^9.C_3^4:C_4$ ($C_2^3$) x 6, $C_2^8.C_3.S_3^3$ ($D_4$) x 4, $A_4^2\wr C_2.C_2^2$ ($C_2^3$) x 4, $C_2^8.C_3^4.C_2^3$ ($C_2^3$) x 3, $C_2^9.C_3^2:S_3^2$ ($C_2^3$) x 2

Order 82944: $C_2^8.C_3^3.D_6$ ($C_2\times D_4$) x 6, $C_2^8.C_3^3.D_6$ ($C_2^4$)

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

Order 20736: $C_2^8.C_3^4$ ($C_2^3:D_4$)

Order 512: $C_2^9$ ($C_3^4.Q_8:C_2^2$)

Order 256: $C_2^8$ ($C_3:S_3^3:D_4$)

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

Order 1: $C_1$ ($A_4^2\wr C_2.C_2^2.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1327104: $A_4^2\wr C_2.C_2^2.C_2^3$ ($C_1$)

Order 663552: $A_4^2.(C_2\times S_4^2:C_4)$ ($C_2$), $A_4^2.S_4^2:C_2^3$ ($C_2$), $A_4^2\wr C_2.C_2^2.C_2^2$ ($C_2$), $A_4^2\wr C_2.C_2^2.C_2^2$ ($C_2$), $(C_2\times A_4^2\wr C_2).D_4$ ($C_2$), $C_2.A_4^2\wr C_2.D_4$ ($C_2$)

Order 331776: $A_4^2.S_4^2:C_2^2$ ($C_2^2$) x 2, $A_4^2.(A_4^2:C_4\times C_2^2)$ ($C_2^2$) x 2, $C_2^8.C_3^4.C_2^4$ ($C_2^2$), $C_2^9.C_3:S_3^3$ ($C_2^2$), $C_2^9.C_3^4:D_4$ ($C_2^2$), $C_2^9.C_3:S_3^3$ ($C_2^2$), $A_4^2\wr C_2.C_2^3$ ($C_2^2$), $C_2^9.C_3^4:Q_8$ ($C_2^2$), $C_2^9.C_3^4:D_4$ ($C_2^2$), $C_2^9.C_3^4:D_4$ ($C_2^2$), $C_2.A_4^2\wr C_2.C_4$ ($C_2^2$), $(C_2\times A_4^2\wr C_2).C_4$ ($C_2^2$)

Order 165888: $C_2^8.C_3^4.C_2^3$ ($C_2^3$) x 2, $C_2^9.C_3^4:C_4$ ($C_2^3$) x 2, $C_2^8.C_3.S_3^3$ ($D_4$), $A_4^2\wr C_2.C_2^2$ ($C_2^3$), $C_2^9.C_3^2:S_3^2$ ($C_2^3$)

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

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

Order 20736: $C_2^8.C_3^4$ ($C_2^3:D_4$)

Order 512: $C_2^9$ ($C_3^4.Q_8:C_2^2$)

Order 256: $C_2^8$ ($C_3:S_3^3:D_4$)

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

Order 1: $C_1$ ($A_4^2\wr C_2.C_2^2.C_2^3$)

Series

Derived series

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

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$A_4^2.S_4^2:C_2^3$

$\rhd$

$C_2^8.C_3^4.C_2^4$

$\rhd$

$C_2^8.C_3^4.C_2^3$

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2^8.C_3^3.D_6$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

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

Character theory

Complex character table

The $274 \times 274$ character table is not available for this group.

Rational character table

The $266 \times 266$ rational character table is not available for this group.