Abstract group 2654208.kw: $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$

• Label: 2654208.kw
• Order: \( 2^{15} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{18} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $24$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$
Order:	\(2654208\)\(\medspace = 2^{15} \cdot 3^{4} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_2^6.C_3^3.C_2^6.C_6.C_2^5$, of order \(21233664\)\(\medspace = 2^{18} \cdot 3^{4} \)
Composition factors:	$C_2$ x 15, $C_3$ 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	23295	19160	423168	404520	110592	147456	862464	442368	221184	2654208
Conjugacy classes	1	78	4	153	112	16	2	100	6	16	488
Divisions	1	78	4	147	109	14	2	88	4	7	454
Autjugacy classes	1	70	4	99	104	8	1	58	2	4	351

Minimal presentations

Permutation degree:	$24$
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:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n \mid c^{2}=e^{6}=f^{12}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,2,3,5,8,11,6,7)(13,14,15,17)(16,18)(19,21,23,24)(20,22), (2,4,7,10)(3,6,8)(5,9)(11,12)(14,16,19,22)(17,20,23,18)(21,24)\rangle$
Transitive group:	36T46557	36T46638			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:	$C_2^9$ . $(C_6^3:S_4)$	$C_2^4$ . $(A_4:S_4^2.S_4)$	$(C_2^5.A_4^3:S_4)$ . $C_2$	$(C_2^5.A_4^3:S_4)$ . $C_2$	all 33

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

Homology

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

Subgroups

There are 43 normal subgroups (39 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $A_4^3.C_2^5:S_4$
Commutator:	$G' \simeq$ $C_2^5.A_4^3:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $A_4^3.C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.C_2^6$	$G/\operatorname{Fit} \simeq$ $C_3^3:S_4$
Radical:	$R \simeq$ $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_6^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2^6.C_2$
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 32 (order at least 82944) are shown, as well as all normal subgroups of any index.

Order 2654208: $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$

Order 1327104: $C_2^5.A_4^3:S_4$, $C_2^5.A_4^3:S_4$, $A_4^3.C_2^6:A_4$

Order 884736: $A_4^2.A_4.C_2^6.C_2^3$

Order 663552: $C_2^5.A_4^3:A_4$, $A_4^3.C_2^4:S_4$, $A_4^3.C_2^4:S_4$, $A_4^3.C_2\wr S_3$, $A_4^3.C_2^5:A_4$, $A_4^3.C_2^4:S_4$

Order 442368: $A_4^2.A_4.C_2^5.C_2^3$, $A_4^2.A_4.C_2^5.C_2^3$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.C_2^6.C_6.C_2^3$, $A_4^2.A_4.C_2^4.C_2^4$, $A_4^2.A_4.C_2^4.C_2^4$, $C_2\times C_2^6.C_3^3.C_2^5.C_2^2$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^5.C_2^3$, $A_4^2.A_4.C_2^4.C_2^4$, $C_2^6.C_6^3.C_2.C_2^4$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^5.C_2^3$

Order 331776: $A_4^3.C_2^3:S_4$ x 4, $C_2^3.A_4^3:S_4$ x 4, $C_2^4.A_4^3:A_4$ x 2, $C_2^4.A_4^3:A_4$, $A_4^3.C_2^2\wr C_3$, $C_2^5.A_4^3:S_3$, $A_4^3.C_2^3:S_4$, $C_2^9.C_6\wr C_3$

Order 294912: $A_4^2.C_2^6.C_2^5$

Order 221184: $A_4^2.A_4.C_2^6.C_2$ x 12, $A_4^2.A_4.C_2^6.C_2$ x 9, $C_2^6.C_3^3.C_2^5.C_2^2$ x 6, $A_4^2.A_4.C_2^5.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^3$ x 4, $A_4^2.A_4.C_2^6.C_2$ x 4, $A_4^2.A_4.C_2^5.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^3$ x 4, $A_4^2.A_4.C_2^5.C_2^2$ x 4, $A_4^2.A_4.C_2^5.C_2^2$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^5.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^3$ x 2, $A_4^2.A_4.C_2^3.C_2^4$ x 2, $A_4^2.A_4.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2^4$ x 2, $A_4^2.(C_2^4\times A_4).C_2^3$ x 2, $A_4^2.A_4.C_2^4.C_2^3$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2^2$ x 2, $C_2^4:A_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6.(C_2\times C_6^3).C_2^3$, $C_2\times A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^4$, $C_2^6.C_6^3.D_4.C_2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^4$, $A_4^2.C_6.C_2^5.C_2^3$, $C_2\times C_2^6.C_3^3.C_2^6$, $C_2^6.C_6^3.C_2^4$, $C_2\times C_2^6.C_3^3.C_2^5.C_2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.D_4.C_2$, $C_2\times C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.C_2^4.C_6.C_2^4$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.C_2^4.C_6.C_2^4$, $C_2^6.C_3^3.C_2^6.C_2$, $C_2^6.C_6^3.C_2.C_2^3$, $C_2^6.C_6^3.D_4.C_2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^4$, $C_2\times C_2^6.C_3^2.D_6.C_2^4$, $C_2^4:A_4^2.\GL(2,\mathbb{Z}/4)$

Order 165888: $A_4^3.C_2^3:A_4$ x 10, $A_4:S_4^2.S_4$ x 6, $A_4^3.C_2^3:A_4$ x 3, $A_4^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.A_4\wr C_3$, $A_4^3.C_2^2:S_4$, $A_4^3.\GL(2,\mathbb{Z}/4)$, $C_2^8.C_6\wr C_3$

Order 147456: $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $C_2\times A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $C_2\times A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $C_2^6.C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$

Order 110592: $A_4^2.A_4.C_2^5.C_2$ x 61, $A_4^2.A_4.C_2^5.C_2$ x 59, $C_2^6.C_3^3.C_2^4.C_2^2$ x 26, $C_2^6.C_6^3.C_2^3$ x 15, $A_4^2.A_4.C_2^5.C_2$ x 12, $C_2^6.C_3^3.C_2^6$ x 11, $C_2^6.C_3^3.C_2^4.C_2^2$ x 10, $C_2^6.C_3^3.C_2^5.C_2$ x 10, $C_2\times C_2^6.C_3^3.C_2^5$ x 9, $C_2^6.C_3^3.C_2^4.C_2^2$ x 9, $A_4^2.C_2^4.C_6.C_2^3$ x 8, $C_2^6.C_3^3.C_2.C_2^5$ x 8, $A_4^2.C_2^4.C_6.C_2^3$ x 8, $A_4^2.C_2^3.C_6.C_2^4$ x 8, $A_4^2.A_4.C_2^3.C_2^3$ x 6, $C_2^6.C_3^3.C_2^5.C_2$ x 6, $C_2^6.C_3^2.D_6.C_2^4$ x 6, $C_2^6.C_3^3.C_2^5.C_2$ x 6, $A_4^2.A_4.C_2^4.C_2^2$ x 6, $C_2\times A_4^2.A_4.C_2^4.C_2$ x 5, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.C_2^4.C_6.C_2^3$ x 4, $C_2^6.C_6^3.C_2^3$ x 4, $A_4^2.C_2^4.C_3.C_4^2$ x 4, $C_2\times C_2^6.C_3^3.C_2^4.C_2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $C_2^6.C_3^3.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $A_4^2.A_4.C_2^5.C_2$ x 3, $C_2^6.C_6^3.C_2^3$ x 3, $C_2\times C_2^6.C_3^3.C_2^4.C_2$ x 3, $A_4^2.C_2^4.C_6.C_2^3$ x 3, $C_2^6.C_6^3.C_2^3$ x 3, $C_2^6.C_3^3.C_2^4.C_2^2$ x 3, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2^3$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $A_4^2.A_4.C_2^3.C_2^3$ x 2, $C_2\times C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $A_4^2.C_2^4.C_6.C_2^3$ x 2, $C_2\times C_2^6.C_3:S_3:D_{12}.C_2$ x 2, $C_2\times A_4^2.(C_2^4\times A_4).C_2$ x 2, $C_2\times C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2\times C_2^6.C_3^2.D_6.C_2^3$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2^6.C_3^3.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2\times A_4^2.C_2^3.C_6.C_2^3$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2^{10}.C_{18}:C_6$ x 2, $C_2^{10}.C_9:C_{12}$ x 2, $C_2^{10}.C_{18}:C_6$ x 2, $C_2^2\times C_2^6.C_3^3.C_2^4$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.A_4.C_2^4.C_2^2$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^8.C_6^3.C_2$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_6.C_2^5.C_2^2$, $A_4^2.C_6.C_2^5.C_2^2$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.A_4.C_2^4.C_2^2$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_6.C_2^6.C_2$, $C_2^6.C_6^3.C_4.C_2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2^2$, $C_2\times C_2^6.C_3^2.D_6.C_2^3$, $A_4^2.C_6.C_2^6.C_2$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_2^4.C_6.C_2^3$, $C_2^6.C_6^3.C_2^3$, $A_4^2.A_4.C_2^4.C_2^2$, $C_2^6.C_3^3.C_2^5.C_2$, $C_2\times C_2^6.C_3^3.C_2^4.C_2$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2^2$, $C_2\times A_4^2.A_4.C_2^5$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^5.C_2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2\times (C_3\times A_4^2).C_2^5.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_4.C_2$, $A_4^2.C_6.C_2^5.C_2^2$, $A_4^2.C_6.C_2^4.C_2^3$, $A_4^2.A_4.C_2^4.C_2^2$, $C_2^7.C_6^2:S_4$, $C_2^7.C_6^2:S_4$, $C_2^{10}.(C_2^2\times \He_3)$

Order 98304: $C_2^5.C_2^6.C_6.C_2^3$, $(C_2^4\times A_4).C_2^6.C_2^3$

Order 82944: $A_4:S_4^2.A_4$ x 9, $C_2\times A_4^3:S_4$ x 3, $A_4^3.A_4:C_4$ x 3, $C_2^2\times A_4^3:A_4$ x 3, $C_2\times A_4^3.S_4$ x 2, $(C_2\times A_4^3).S_4$ x 2, $C_2^{10}:C_3\wr C_3$, $C_2^2\times A_4^3.A_4$

Order 55296: $C_2^2\times C_2^6.C_6^3$, $C_2^2\times C_2^6.C_3^3.C_2^3$, $C_2\times C_2^6.C_3^3.C_2^4$

Order 41472: $(C_2\times C_6^3).\GL(2,\mathbb{Z}/4)$

Order 32768: $C_2^8.C_2^6.C_2$

Order 27648: $C_2^6.C_3^3.C_2^4$, $C_2\times C_2^6.C_3^3.C_2^3$, $C_2^3\times C_2^6.C_3^3.C_2$, $C_2\times C_2^6.C_6^3$

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

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

Order 4096: $C_2^6.C_2^6$

Order 3456: $C_2\times A_4^3$

Order 2048: $C_2^{11}$

Order 1728: $A_4^3$

Order 1024: $C_2^{10}$ x 2

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 128: $C_2^7$

Order 81: $C_3\wr C_3$

Order 64: $C_2^6$, $C_2^3:D_4$

Order 32: $C_2^5$

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 32 (order at least 82944) are shown, as well as all normal subgroups of any index.

Order 2654208: $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$

Order 1327104: $C_2^5.A_4^3:S_4$, $C_2^5.A_4^3:S_4$, $A_4^3.C_2^6:A_4$

Order 884736: $A_4^2.A_4.C_2^6.C_2^3$

Order 663552: $C_2^5.A_4^3:A_4$, $A_4^3.C_2^4:S_4$, $A_4^3.C_2^4:S_4$, $A_4^3.C_2\wr S_3$, $A_4^3.C_2^5:A_4$, $A_4^3.C_2^4:S_4$

Order 442368: $A_4^2.A_4.C_2^5.C_2^3$, $A_4^2.A_4.C_2^5.C_2^3$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.C_2^6.C_6.C_2^3$, $A_4^2.A_4.C_2^4.C_2^4$, $A_4^2.A_4.C_2^4.C_2^4$, $C_2\times C_2^6.C_3^3.C_2^5.C_2^2$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^5.C_2^3$, $A_4^2.A_4.C_2^4.C_2^4$, $C_2^6.C_6^3.C_2.C_2^4$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^6.C_2^2$, $A_4^2.A_4.C_2^5.C_2^3$

Order 331776: $A_4^3.C_2^3:S_4$ x 3, $C_2^3.A_4^3:S_4$ x 3, $C_2^4.A_4^3:A_4$ x 2, $C_2^4.A_4^3:A_4$, $A_4^3.C_2^2\wr C_3$, $C_2^5.A_4^3:S_3$, $A_4^3.C_2^3:S_4$, $C_2^9.C_6\wr C_3$

Order 294912: $A_4^2.C_2^6.C_2^5$

Order 221184: $A_4^2.A_4.C_2^6.C_2$ x 9, $A_4^2.A_4.C_2^6.C_2$ x 6, $C_2^6.C_3^3.C_2^5.C_2^2$ x 5, $A_4^2.A_4.C_2^6.C_2$ x 3, $A_4^2.A_4.C_2^5.C_2^2$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^3$ x 2, $A_4^2.A_4.C_2^5.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^3$ x 2, $A_4^2.A_4.C_2^3.C_2^4$ x 2, $A_4^2.A_4.C_2^5.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2^4$ x 2, $A_4^2.A_4.C_2^4.C_2^3$ x 2, $A_4^2.(C_2^4\times A_4).C_2^3$ x 2, $A_4^2.A_4.C_2^4.C_2^3$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.(C_2\times C_6^3).C_2^3$, $C_2\times A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^4$, $C_2^6.C_6^3.D_4.C_2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^4$, $A_4^2.C_6.C_2^5.C_2^3$, $C_2\times C_2^6.C_3^3.C_2^6$, $C_2^6.C_6^3.C_2^4$, $C_2\times C_2^6.C_3^3.C_2^5.C_2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.D_4.C_2$, $C_2\times C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.C_2^4.C_6.C_2^4$, $A_4^2.A_4.C_2^5.C_2^2$, $A_4^2.C_2^4.C_6.C_2^4$, $C_2^6.C_3^3.C_2^6.C_2$, $C_2^6.C_6^3.C_2.C_2^3$, $C_2^6.C_6^3.D_4.C_2$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^4$, $C_2\times C_2^6.C_3^2.D_6.C_2^4$, $A_4^2.A_4.C_2^5.C_2^2$, $C_2^4:A_4^2.\GL(2,\mathbb{Z}/4)$, $C_2^4:A_4^2.\GL(2,\mathbb{Z}/4)$

Order 165888: $A_4^3.C_2^3:A_4$ x 8, $A_4:S_4^2.S_4$ x 3, $A_4^3.C_2^3:A_4$ x 2, $A_4^3.\GL(2,\mathbb{Z}/4)$, $C_2^5.A_4\wr C_3$, $A_4^3.C_2^2:S_4$, $A_4^3.\GL(2,\mathbb{Z}/4)$, $C_2^8.C_6\wr C_3$

Order 147456: $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $C_2\times A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $C_2\times A_4^2.C_2^6.C_2^3$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^5.C_2^5$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $C_2^6.C_6^2.C_2^4.C_2^2$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$, $A_4^2.C_2^6.C_2^4$

Order 110592: $A_4^2.A_4.C_2^5.C_2$ x 33, $A_4^2.A_4.C_2^5.C_2$ x 33, $C_2^6.C_3^3.C_2^4.C_2^2$ x 14, $C_2^6.C_3^3.C_2^6$ x 9, $C_2^6.C_6^3.C_2^3$ x 9, $A_4^2.A_4.C_2^5.C_2$ x 7, $C_2\times C_2^6.C_3^3.C_2^5$ x 7, $A_4^2.C_2^4.C_6.C_2^3$ x 7, $C_2^6.C_3^3.C_2^4.C_2^2$ x 6, $C_2^6.C_3^3.C_2.C_2^5$ x 6, $C_2^6.C_3^3.C_2^5.C_2$ x 6, $A_4^2.C_2^4.C_6.C_2^3$ x 6, $A_4^2.C_2^3.C_6.C_2^4$ x 6, $C_2^6.C_3^3.C_2^5.C_2$ x 6, $C_2^6.C_3^3.C_2^4.C_2^2$ x 5, $C_2^6.C_3^3.C_2^5.C_2$ x 5, $C_2^6.C_3^2.D_6.C_2^4$ x 4, $A_4^2.A_4.C_2^4.C_2^2$ x 4, $C_2^6.C_3^3.C_2^4.C_2^2$ x 4, $C_2\times A_4^2.A_4.C_2^4.C_2$ x 3, $A_4^2.A_4.C_2^3.C_2^3$ x 3, $C_2\times C_2^6.C_3^3.C_2^4.C_2$ x 3, $C_2^6.C_6^3.C_2^3$ x 3, $C_2^6.C_3^3.C_2^4.C_2^2$ x 3, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2^3$ x 2, $A_4^2.C_2^4.C_6.C_2^3$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $A_4^2.C_2^4.C_3.C_4^2$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $A_4^2.A_4.C_2^5.C_2$ x 2, $A_4^2.A_4.C_2^3.C_2^3$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2$ x 2, $C_2\times C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $C_2^6.C_6^3.C_2^3$ x 2, $A_4^2.C_2^4.C_6.C_2^3$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.C_2^4.C_6.C_2^3$ x 2, $C_2\times C_2^6.C_3:S_3:D_{12}.C_2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2\times A_4^2.(C_2^4\times A_4).C_2$ x 2, $C_2\times C_2^6.C_3^3.C_2.C_2^4$ x 2, $C_2\times C_2^6.C_3^2.D_6.C_2^3$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2^6.C_3^3.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2\times A_4^2.C_2^3.C_6.C_2^3$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2\times C_2^6.C_3^3.C_2^4.C_2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $A_4^2.A_4.C_2^4.C_2^2$ x 2, $C_2^2\times C_2^6.C_3^3.C_2^4$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.A_4.C_2^4.C_2^2$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.A_4.C_2^4.C_2^2$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^8.C_6^3.C_2$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_6.C_2^5.C_2^2$, $A_4^2.C_6.C_2^5.C_2^2$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.A_4.C_2^4.C_2^2$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_6.C_2^6.C_2$, $C_2^6.C_6^3.C_4.C_2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2^2$, $C_2\times C_2^6.C_3^2.D_6.C_2^3$, $A_4^2.C_6.C_2^6.C_2$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_2^4.C_6.C_2^3$, $C_2^6.C_6^3.C_2^3$, $A_4^2.A_4.C_2^4.C_2^2$, $C_2^6.C_3^3.C_2^5.C_2$, $C_2\times C_2^6.C_3^3.C_2^4.C_2$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^4.C_2^2$, $C_2\times A_4^2.A_4.C_2^5$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_3^3.C_2^5.C_2$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_6^3.C_2^3$, $C_2^6.C_6^3.C_2^3$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2\times (C_3\times A_4^2).C_2^5.C_2^2$, $C_2^6.C_3^3.C_2^4.C_2^2$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.C_2^4.C_6.C_2^3$, $A_4^2.C_6.C_2^5.C_2^2$, $C_2^6.C_6^3.C_4.C_2$, $A_4^2.C_6.C_2^5.C_2^2$, $A_4^2.A_4.C_2^4.C_2^2$, $A_4^2.A_4.C_2^4.C_2^2$, $A_4^2.C_6.C_2^4.C_2^3$, $A_4^2.A_4.C_2^4.C_2^2$, $C_2^7.C_6^2:S_4$, $C_2^7.C_6^2:S_4$, $C_2^{10}.C_{18}:C_6$, $C_2^{10}.C_9:C_{12}$, $C_2^{10}.(C_2^2\times \He_3)$, $C_2^{10}.C_{18}:C_6$

Order 98304: $C_2^5.C_2^6.C_6.C_2^3$, $(C_2^4\times A_4).C_2^6.C_2^3$

Order 82944: $A_4:S_4^2.A_4$ x 7, $C_2\times A_4^3:S_4$ x 2, $C_2\times A_4^3.S_4$ x 2, $A_4^3.A_4:C_4$ x 2, $C_2^2\times A_4^3:A_4$ x 2, $(C_2\times A_4^3).S_4$ x 2, $C_2^{10}:C_3\wr C_3$, $C_2^2\times A_4^3.A_4$

Order 55296: $C_2^2\times C_2^6.C_6^3$, $C_2^2\times C_2^6.C_3^3.C_2^3$, $C_2\times C_2^6.C_3^3.C_2^4$

Order 41472: $(C_2\times C_6^3).\GL(2,\mathbb{Z}/4)$

Order 32768: $C_2^8.C_2^6.C_2$

Order 27648: $C_2^6.C_3^3.C_2^4$, $C_2\times C_2^6.C_3^3.C_2^3$, $C_2^3\times C_2^6.C_3^3.C_2$, $C_2\times C_2^6.C_6^3$

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

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

Order 4096: $C_2^6.C_2^6$

Order 3456: $C_2\times A_4^3$

Order 2048: $C_2^{11}$

Order 1728: $A_4^3$

Order 1024: $C_2^{10}$ x 2

Order 512: $C_2^9$

Order 256: $C_2^8$

Order 128: $C_2^7$

Order 81: $C_3\wr C_3$

Order 64: $C_2^6$, $C_2^3:D_4$

Order 32: $C_2^5$

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2654208: $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 1327104: $C_2^5.A_4^3:S_4$ ($C_2$), $C_2^5.A_4^3:S_4$ ($C_2$), $A_4^3.C_2^6:A_4$ ($C_2$)

Order 663552: $C_2^5.A_4^3:A_4$ ($C_2^2$)

Order 442368: $C_2\times C_2^6.C_3^3.C_2^5.C_2^2$ ($S_3$)

Order 331776: $C_2^4.A_4^3:A_4$ ($D_4$)

Order 221184: $C_2\times C_2^6.C_3^3.C_2^6$ ($D_6$)

Order 110592: $C_2^6.C_3^3.C_2^6$ ($C_3:D_4$), $C_2^2\times C_2^6.C_3^3.C_2^4$ ($S_4$), $C_2^8.C_6^3.C_2$ ($S_4$), $C_2^6.C_6^3.C_2^3$ ($S_4$)

Order 55296: $C_2^2\times C_2^6.C_6^3$ ($C_2\times S_4$), $C_2^2\times C_2^6.C_3^3.C_2^3$ ($C_2\times S_4$), $C_2\times C_2^6.C_3^3.C_2^4$ ($C_2\times S_4$)

Order 27648: $C_2^6.C_3^3.C_2^4$ ($\GL(2,\mathbb{Z}/4)$), $C_2\times C_2^6.C_3^3.C_2^3$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3\times C_2^6.C_3^3.C_2$ ($C_2^2:S_4$), $C_2\times C_2^6.C_6^3$ ($\GL(2,\mathbb{Z}/4)$)

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

Order 6912: $A_4:S_4^2$ ($C_2^4:S_4$) x 2, $C_2\times C_2^6.C_3^3.C_2$ ($C_2\wr S_3$)

Order 4096: $C_2^6.C_2^6$ ($C_3^3:S_4$)

Order 3456: $C_2\times A_4^3$ ($C_2^5:S_4$)

Order 2048: $C_2^{11}$ ($C_2\times C_3^3:S_4$)

Order 1728: $A_4^3$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 1024: $C_2^{10}$ ($C_3^3:\GL(2,\mathbb{Z}/4)$), $C_2^{10}$ ($(C_3\times C_6^2):S_4$)

Order 512: $C_2^9$ ($C_6^3:S_4$)

Order 256: $C_2^8$ ($(C_2\times C_6^3):S_4$)

Order 128: $C_2^7$ ($C_6:D_6^2:S_4$)

Order 64: $C_2^6$ ($(C_2\times C_6^3).\GL(2,\mathbb{Z}/4)$), $C_2^3:D_4$ ($A_4^3:S_4$)

Order 32: $C_2^5$ ($C_2\times A_4^3:S_4$)

Order 16: $C_2^4$ ($A_4:S_4^2.S_4$), $C_2^4$ ($A_4^3.\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($A_4^3.C_2^3:S_4$)

Order 4: $C_2^2$ ($A_4^3.C_2\wr S_3$)

Order 2: $C_2$ ($A_4^3.C_2^5:S_4$)

Order 1: $C_1$ ($A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2654208: $A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 1327104: $C_2^5.A_4^3:S_4$ ($C_2$), $C_2^5.A_4^3:S_4$ ($C_2$), $A_4^3.C_2^6:A_4$ ($C_2$)

Order 663552: $C_2^5.A_4^3:A_4$ ($C_2^2$)

Order 442368: $C_2\times C_2^6.C_3^3.C_2^5.C_2^2$ ($S_3$)

Order 331776: $C_2^4.A_4^3:A_4$ ($D_4$)

Order 221184: $C_2\times C_2^6.C_3^3.C_2^6$ ($D_6$)

Order 110592: $C_2^6.C_3^3.C_2^6$ ($C_3:D_4$), $C_2^2\times C_2^6.C_3^3.C_2^4$ ($S_4$), $C_2^8.C_6^3.C_2$ ($S_4$), $C_2^6.C_6^3.C_2^3$ ($S_4$)

Order 55296: $C_2^2\times C_2^6.C_6^3$ ($C_2\times S_4$), $C_2^2\times C_2^6.C_3^3.C_2^3$ ($C_2\times S_4$), $C_2\times C_2^6.C_3^3.C_2^4$ ($C_2\times S_4$)

Order 27648: $C_2^6.C_3^3.C_2^4$ ($\GL(2,\mathbb{Z}/4)$), $C_2\times C_2^6.C_3^3.C_2^3$ ($\GL(2,\mathbb{Z}/4)$), $C_2^3\times C_2^6.C_3^3.C_2$ ($C_2^2:S_4$), $C_2\times C_2^6.C_6^3$ ($\GL(2,\mathbb{Z}/4)$)

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

Order 6912: $C_2\times C_2^6.C_3^3.C_2$ ($C_2\wr S_3$), $A_4:S_4^2$ ($C_2^4:S_4$)

Order 4096: $C_2^6.C_2^6$ ($C_3^3:S_4$)

Order 3456: $C_2\times A_4^3$ ($C_2^5:S_4$)

Order 2048: $C_2^{11}$ ($C_2\times C_3^3:S_4$)

Order 1728: $A_4^3$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 1024: $C_2^{10}$ ($C_3^3:\GL(2,\mathbb{Z}/4)$), $C_2^{10}$ ($(C_3\times C_6^2):S_4$)

Order 512: $C_2^9$ ($C_6^3:S_4$)

Order 256: $C_2^8$ ($(C_2\times C_6^3):S_4$)

Order 128: $C_2^7$ ($C_6:D_6^2:S_4$)

Order 64: $C_2^6$ ($(C_2\times C_6^3).\GL(2,\mathbb{Z}/4)$), $C_2^3:D_4$ ($A_4^3:S_4$)

Order 32: $C_2^5$ ($C_2\times A_4^3:S_4$)

Order 16: $C_2^4$ ($A_4:S_4^2.S_4$), $C_2^4$ ($A_4^3.\GL(2,\mathbb{Z}/4)$)

Order 8: $C_2^3$ ($A_4^3.C_2^3:S_4$)

Order 4: $C_2^2$ ($A_4^3.C_2\wr S_3$)

Order 2: $C_2$ ($A_4^3.C_2^5:S_4$)

Order 1: $C_1$ ($A_4^3.C_2^4:\GL(2,\mathbb{Z}/4)$)

Series

Derived series

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

$\rhd$

$C_2^5.A_4^3:A_4$

$\rhd$

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

$\rhd$

$A_4^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$A_4^3.C_2^6:A_4$

$\rhd$

$C_2^5.A_4^3:A_4$

$\rhd$

$C_2^4.A_4^3:A_4$

$\rhd$

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

$\rhd$

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

$\rhd$

$A_4:S_4^2$

$\rhd$

$A_4^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_2^5.A_4^3:A_4$

$\rhd$

$C_2^4.A_4^3:A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

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

The $488 \times 488$ character table is not available for this group.

Rational character table

The $454 \times 454$ rational character table is not available for this group.