Abstract group 660602880.c: $C_2^{10}.C_2^5.A_8$

• Label: 660602880.c
• Order: \( 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{20} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $64$
• Trans deg.: $64$
• Rank: $2$

Group information

Description:	$C_2^{10}.C_2^5.A_8$
Order:	\(660602880\)\(\medspace = 2^{21} \cdot 3^{2} \cdot 5 \cdot 7 \)
Exponent:	\(840\)\(\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Automorphism group:	Group of order \(330301440\)\(\medspace = 2^{20} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 15, $A_8$
Derived length:	$1$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	7	8	10	12	14	15	20	24	28	30	60
Elements	1	206975	530432	39393152	5505024	64153600	2949120	94617600	27525120	127074304	79626240	11010048	11010048	13762560	106168320	55050240	22020096	660602880
Conjugacy classes	1	30	2	108	1	23	2	24	3	24	18	2	1	1	12	6	2	260
Divisions	1	30	2	108	1	23	1	24	3	24	9	1	1	1	6	3	1	239
Autjugacy classes	1	30	2	108	1	23	2	24	3	24	18	2	1	1	12	6	2	260

Minimal presentations

Permutation degree:	$64$
Transitive degree:	$64$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $64$ $\langle(1,3)(5,7)(9,11)(13,15)(17,19)(21,23)(25,27)(29,31)(33,35)(37,39)(41,43) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^9.C_2^6)$ . $A_8$	$C_2^{10}$ . $(C_2^5.A_8)$	$C_2$ . $(C_2^9.C_2^5.A_8)$	$C_2^2$ . $(C_2^{12}.C_2.A_8)$	all 14
Aut. group:	$\Aut(D_4\times C_2^4)$

Elements of the group are displayed as permutations of degree 64.

Homology

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

Subgroups

There are 17 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	not computed
Frattini:	a subgroup isomorphic to $C_2^6$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^8.C_2^6.C_2^6.C_2$

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 840 (order at least 786432) are shown, as well as all normal subgroups of any index.

Order 660602880: $C_2^{10}.C_2^5.A_8$

Order 330301440: $C_2^6.C_2^4.C_2^4.A_8$

Order 82575360: $C_2^{10}.C_2^5.A_7$

Order 44040192: $C_2^9.C_2^4.C_2^5.\PSL(2,7)$, $C_2^7.C_2^6.C_2^5.\PSL(2,7)$

Order 41287680: $C_2^{10}.C_2.A_8$, $C_2^6.C_2^5.A_8$, $C_2^6.C_2^4.C_2^4.A_7$

Order 23592960: $C_2^6.C_2^4.C_2^5.A_6.C_2$

Order 22020096: $C_2^{10}.C_2^7:\GL(3,2)$ x 2, $C_2^{12}.C_2^5.\PSL(2,7)$, $C_2^9.C_2^6.C_2^2.\PSL(2,7)$, $C_2^9.C_2^4.C_2^4.\PSL(2,7)$, $C_2^{10}.C_2^6:(C_2\times \GL(3,2))$

Order 20643840: $C_2^5.C_2^5.A_8$ x 3, $C_2^9.C_2.A_8$ x 2, $C_2^{10}.A_8$, $C_2^5.C_2^5.A_8$

Order 18874368: $C_2^{12}.C_2^6.S_3^2.C_2$

Order 11796480: $C_2^6.C_2^4.C_2^4.A_6.C_2$ x 2, $C_2^{10}.C_2^4.C_6.S_5$, $C_2^6.C_2^4.C_2^5.A_6$

Order 11010048: $C_2^{12}.C_2^4.\PSL(2,7)$ x 2, $C_2^7.C_2^6.C_2^3.\PSL(2,7)$, $C_2^6.C_2^4.C_2^6.\PSL(2,7)$

Order 10321920: $C_2^5.C_2^4.A_8$ x 2, $C_2^9.A_8$, $C_2^8.C_2.A_8$, $C_2\times C_2^8.A_8$

Order 9437184: $C_2^8.C_2^6.C_2^4.S_3^2$ x 3, $C_2^8.C_2^6.A_4^2.C_2^2$ x 2, $C_2^{12}.C_2^6.S_3^2$, $C_2^6.C_2^4.C_2^4.C_3.C_2^4.C_6.C_2$

Order 6291456: $C_2^{12}.C_2^4.C_6.C_2^4$, $C_2^{10}.C_2^6.C_6.C_2^4$, $C_2^6.C_2^6.A_4.C_2^5.C_2^2$

Order 5898240: $C_2^6.C_2^4.C_2^4.C_3.S_5$ x 2, $C_2^6.C_2^4.C_2^4.C_6.A_5$, $C_2^6.C_2^4.C_2^4.A_6$

Order 5505024: $C_2^9.C_2^6.\PSL(2,7)$ x 3, $C_2^{12}.F_8.C_6.C_2^2$, $C_2^{10}.C_2^5.\PSL(2,7)$, $C_2^7.C_2^6.F_8.C_6.C_2$, $C_2^7.C_2^6.C_2^2.\PSL(2,7)$, $C_2^6.C_2^6.C_2^3.\PSL(2,7)$, $C_2^6.C_2^5.C_2^4.\PSL(2,7)$, $C_2^6.C_2^4.C_2^5.\PSL(2,7)$

Order 5160960: $C_2^{10}.C_2.A_7$, $C_2^8.A_8$, $C_2^6.C_2^5.A_7$

Order 4718592: $C_2^8.C_2^6.A_4^2.C_2$ x 3, $C_2^{12}.C_2^6.C_3.S_3$, $C_2^{12}.A_4^2.C_2^3$, $C_2^{12}.A_4.C_2^4.C_6$, $C_2^{10}.C_2^6.S_3^2.C_2$, $C_2^8.C_2^4.C_2^4.(S_3\times A_4)$, $C_2^6.C_2^4.C_2^4.C_3.C_2^4.C_6$

Order 3932160: $C_2^{10}.C_2^5.S_5$ x 2

Order 3145728: $C_2^8.C_2^6.(D_4\times S_4)$ x 13, $C_2^6.C_2^6.A_4.C_2\wr C_2^2$ x 5, $C_2^8.C_2^6.A_4.D_4.C_2$ x 4, $C_2^8.C_2^6.A_4.C_2^4$ x 4, $C_2^{12}.A_4.C_2^4.C_2^2$ x 2, $C_2^{10}.C_2^6.C_6.C_2^3$ x 2, $C_2^8.C_2^6.A_4.C_2^3.C_2$ x 2, $C_2^8.C_2.C_2^6.A_4.C_2^3$ x 2, $C_2^6.C_2^6.A_4.C_2^4.C_2^2$ x 2, $C_2^{12}.C_2\wr D_6$ x 2, $C_2^{12}.C_2^4.D_{12}.C_2$, $C_2^{12}.A_4.C_2^5.C_2$, $C_2^{12}.A_4.C_2\wr C_2^2$, $C_2^{10}.C_2^5.A_4.C_2^3$, $C_2^8.C_2^6.C_6.C_2^2\wr C_2$, $C_2^8.C_2^6.(C_4\times S_4).C_2$, $C_2^6.C_2^6.A_4.C_2^5.C_2$, $C_2^6.C_2^6.A_4.C_2.C_2^5$

Order 2949120: $C_2^6.C_2^4.C_2^4.C_3.A_5$

Order 2752512: $C_2^9.C_2^5.\PSL(2,7)$ x 16, $C_2^6.C_2^4.C_2^4.\PSL(2,7)$ x 7, $C_2^8.C_2^6.\PSL(2,7)$ x 4, $C_2^6.C_2^5.C_2^3.\PSL(2,7)$ x 4, $C_2^{10}.C_2^4.\PSL(2,7)$ x 3, $C_2^7.C_2^6.F_8.C_6$ x 3, $C_2^6.C_2^6.F_8.C_6.C_2$ x 2, $C_2^6.C_2^6.C_2^2.\PSL(2,7)$ x 2, $C_2^7.C_2^6.C_2.\PSL(2,7)$, $C_2^7.C_2^4.C_2^3.\PSL(2,7)$, $C_2^6.C_2^6.F_8.C_{12}$, $C_2^5.C_2^4.C_2^5.\PSL(2,7)$

Order 2580480: $C_2^6.C_2^4.A_7$ x 3, $C_2^6.C_2.A_8$ x 2, $C_2^{10}.A_7$, $C_2^9.C_2.A_7$, $C_2^8.C_2^2.A_7$, $C_2^5.C_2^5.A_7$

Order 2359296: $C_2^{12}.C_2^4.S_3^2$ x 3, $C_2^{10}.C_2^6.S_3^2$ x 2, $C_2^6.C_2^6.A_4^2.C_2^2$ x 2, $C_2^{12}.C_2^4.C_3.D_6$, $C_2^{12}.C_2^4.(C_6\times S_3)$, $C_2^{12}.A_4^2.C_2^2$, $C_2^{10}.C_3.C_2^6.C_6.C_2$, $C_2^9.(C_2^3\times A_4).A_4.C_2^2$, $C_2^8.C_2^6.A_4^2$, $C_2^8.C_2^6.A_4.D_6$, $C_2^8.C_2^6.(S_3\times S_4)$, $C_2^8.C_2^4.A_4\wr C_2.C_2$, $C_2^6.C_2^6.A_4\wr C_2.C_2$, $C_2^6.C_2^4.A_4^2.D_4.C_2$, $C_2^{12}.C_2^4:S_3^2$

Order 2097152: $C_2^8.C_2^6.C_2^6.C_2$

Order 1966080: $C_2^6.C_2^4.C_2^4.S_5$ x 3, $C_2^{10}.F_{16}.C_4.C_2$, $C_2^{10}.C_2^5.A_5$, $C_2^{10}.C_2^4.S_5$, $C_2^6.C_2^5.C_2^4.A_5$

Order 1835008: $C_2^7.C_2^6.F_8.C_2^2$, $C_2^6.C_2.C_2^6.F_8.C_2^2$

Order 1572864: $C_2^8.C_2^6.A_4.D_4$ x 23, $C_2^8.C_2^6.A_4.C_2^3$ x 21, $C_2^6.C_2^6.A_4.C_2^4.C_2$ x 12, $C_2^6.C_2^6.(D_4\times S_4).C_2$ x 9, $C_2^{10}.C_2^6.D_6.C_2$ x 8, $C_2^8.C_2^5.C_2^4.C_6.C_2$ x 7, $C_2^{12}.A_4.C_2^4.C_2$ x 5, $C_2^8.C_2^6.(C_4\times S_4)$ x 5, $C_2^6.C_2^6.A_4.C_2^3.C_2^2$ x 4, $C_2^{12}.C_2^4:S_4$ x 4, $C_2^{12}.C_2^4.D_6.C_2$ x 3, $C_2^{12}.(C_2^5.S_3).C_2$ x 3, $C_2^8.C_2^6.S_4.C_2^2$ x 3, $C_2^8.C_2^6.A_4.C_4.C_2$ x 3, $C_2^6.C_2^6.A_4.C_2^2\wr C_2$ x 3, $C_2^6.C_2^6.A_4.C_2.C_2^4$ x 3, $C_2^{12}.C_2^4:S_4$ x 3, $C_2^{12}.C_2^4.D_{12}$ x 2, $C_2^{12}.C_2^4.C_{12}.C_2$ x 2, $C_2^{12}.A_4.C_2^3.C_2^2$ x 2, $C_2^{12}.A_4.C_2^2\wr C_2$ x 2, $C_2^{10}.C_2^6.D_{12}$ x 2, $C_2^{10}.C_2^6.C_6.C_2^2$ x 2, $C_2^8.(C_2^9.D_6)$ x 2, $C_2^{12}.(D_4\times S_4).C_2$, $C_2^9.C_6.C_2^6.C_2^3$, $C_2^9.C_2^4.C_6.C_2^5$, $C_2^9.A_4.C_2^6.C_2^2$, $C_2^8.C_2^6.C_6.C_2^3.C_2$, $C_2^8.C_2^6.A_4.C_2^2.C_2$, $C_2^8.C_2^6.(D_4\times A_4)$, $C_2^8.C_2^6.(C_6.D_4).C_2$, $C_2^8.C_2^4.C_6.C_2^5.C_2$, $C_2^7.C_2^6.C_6.C_2^5$, $C_2^7.C_2^4.C_6.C_2^4.C_2^3$, $C_2^7.C_2^4.C_6.C_2^3.C_2^4$, $C_2^6.C_2^6.C_2^4.C_{12}.C_2$, $C_2^6.C_2^6.C_2^4.C_6.C_2^2$, $C_2^6.C_2^6.C_2^4.C_3.D_4$, $C_2^6.C_2^6.A_4.C_2^5$, $C_2^6.C_2^5.C_2^4.C_6.C_2^3$, $C_2^4.C_2^6.C_2^4.C_6.C_2^4$

Order 1474560: $C_2^6.C_2^5.A_6.C_2$ x 4, $C_2^{10}.C_2.A_6.C_2$, $C_2^{10}.(C_2\times S_6)$

Order 1376256: $C_2^6.C_2^7:\GL(3,2)$ x 13, $C_2^8.C_2^5.\PSL(2,7)$ x 12, $C_2^9.C_2^4.\PSL(2,7)$ x 10, $C_2^6.C_2^6.C_2.\PSL(2,7)$ x 6, $C_2^6.C_2^7:\GL(3,2)$ x 5, $C_2^{12}.C_2.\PSL(2,7)$ x 4, $C_2^7.C_2^6.\PSL(2,7)$ x 4, $C_2^6.C_2^4.C_2^3.\PSL(2,7)$ x 4, $C_2^6.C_2.C_2^6.\PSL(2,7)$ x 4, $C_2^6.C_2^6.F_8.C_6$ x 2, $C_2^5.C_2^6.C_2^2.\PSL(2,7)$ x 2, $C_2^5.C_2^5.C_2^3.\PSL(2,7)$ x 2, $C_2^5.C_2^4.C_2^4.\PSL(2,7)$ x 2, $C_2^9.C_2^6.C_7.C_6$, $C_2^7.C_2^6.F_8.C_3$, $C_2^3.C_2^4.C_2^6.\PSL(2,7)$

Order 1290240: $C_2^6.A_8$ x 5, $C_2^9.A_7$ x 2, $C_2^5.C_2^4.A_7$ x 2, $(C_2^4\times C_4).A_8$ x 2, $C_2^8.C_2.A_7$

Order 1179648: $C_2^8.C_2^6.A_4.S_3$ x 4, $C_2^8.C_2^6.A_4.C_6$ x 4, $C_2^6.C_2^4.C_2^4.(S_3\times A_4)$ x 4, $C_2^{10}.C_3.C_2^6.C_6$ x 2, $C_2^{10}.A_4^2.D_4$ x 2, $C_2^{10}.A_4.C_2^4.C_6$ x 2, $C_2^9.C_2^4.D_6^2$ x 2, $C_2^8.C_2^6.C_3:S_4$ x 2, $C_2^6.C_2^6.A_4^2.C_2$ x 2, $C_2^6.C_2^4.A_4^2.D_4$ x 2, $C_2^{12}.A_4^2.C_2$, $C_2^9.A_4^2.C_2^4$, $C_2^8.C_2^6.(S_3\times A_4)$, $C_2^8.A_4^2.C_2^4.C_2$, $C_2^6.C_2^6.C_3.C_2^4.C_6$, $C_2^6.C_2^6.A_4.D_6.C_2$, $C_2^6.C_2^6.(S_3\times S_4).C_2$, $C_2^6.C_2^4.A_4^2.C_4.C_2$, $C_2^6.C_2^4.A_4^2.C_2^3$, $C_2.C_2^6.C_2^6.D_6^2$

Order 1048576: $C_2^7.C_2^6.C_2^5.C_2^2$ x 22, $C_2^9.C_2^6.C_2^5$ x 16, $C_2^8.C_2^6.C_2^5.C_2$ x 11, $C_2^8.C_2^6.C_2^6$ x 5, $C_2^7.C_2^6.C_2^6.C_2$ x 5, $C_2^7.C_2^6.C_2^4.C_2^3$ x 4

Order 983040: $C_2^6.C_2^4.F_{16}.C_4$ x 2, $C_2^6.C_2^4.C_2^4.A_5$ x 2, $C_2^6.C_2^4.F_{16}.C_2^2$

Order 917504: $C_2^{10}.C_2^6.C_{14}$ x 2, $C_2^6.C_2^6.F_8.C_2^2$ x 2, $C_2^{12}.F_8.C_4$, $C_2^7.C_2^6.F_8.C_2$

Order 786432: $C_2^7.C_2^6.C_6.C_2^4$ x 35, $C_2^9.A_4.C_2^3.C_2^4$ x 33, $C_2^9.C_2^4.C_6.C_2^4$ x 26, $C_2^5.C_2^6.C_6.C_2^4.C_2^2$ x 24, $C_2^8.C_2^6.A_4.C_2^2$ x 21, $C_2^6.C_2^6.A_4.C_2^3.C_2$ x 21, $C_2^{12}.(D_4\times S_4)$ x 19, $C_2^8.C_2^6.C_6.C_2^3$ x 18, $C_2^{12}.C_2^4.D_6$ x 16, $C_2^{10}.C_2^6.D_6$ x 14, $C_2^9.C_6.C_2^4.C_2^4$ x 14, $C_2^6.C_2^6.(D_4\times S_4)$ x 13, $C_2^8.C_2^6.S_4.C_2$ x 12, $C_2^8.C_2^4.C_3.D_4^2$ x 11, $C_2^9.A_4.C_2^5.C_2^2$ x 9, $C_2^9.A_4.C_2^4.C_2^3$ x 9, $C_2^8.C_2^6.D_6.C_2^2$ x 9, $C_2^{12}.A_4.D_4.C_2$ x 8, $C_2^6.C_2.C_2^6.A_4.C_2^3$ x 8, $C_2^9.C_6.C_2^5.C_2^3$ x 7, $C_2^8.C_2^4.C_6.C_2^4.C_2$ x 7, $C_2^{12}.A_4.C_2^3.C_2$ x 6, $C_2^8.C_6.C_2^5.C_2^4$ x 6, $C_2^8.C_2^6.A_4.C_4$ x 6, $C_2^7.C_2^4.C_6.C_2^4.C_2^2$ x 6, $C_2^6.C_2^6.A_4.D_4.C_2$ x 6, $C_2^8.C_2^5.A_4.C_2^3$ x 5, $C_2^{12}.C_2^4.C_6.C_2$ x 4, $C_2^{10}.C_2^6.C_6.C_2$ x 4, $C_2^6.C_2^6.C_3.D_4^2$ x 4, $C_2^6.C_2^6.A_4.C_2^4$ x 4, $C_2^{12}.A_4.C_2^4$ x 3, $C_2^9.C_6.C_2^6.C_2^2$ x 3, $C_2^9.A_4.C_2.C_2^6$ x 3, $C_2^8.C_2^6.D_{12}:C_2$ x 3, $C_2^8.C_2^5.A_4.C_2^2.C_2$ x 3, $C_2^8.C_2^4.C_6.C_2^5$ x 3, $C_2^6.C_2^6.C_6.C_2^4.C_2$ x 3, $C_2^5.C_2^6.C_6.C_2^5.C_2$ x 3, $C_2^{12}.S_4.C_2^3$ x 2, $C_2^{12}.Q_8:S_4$ x 2, $C_2^{12}.A_4.D_4:C_2$ x 2, $C_2^{12}.(C_2^5.S_3)$ x 2, $C_2^{10}.C_2^6.C_{12}$ x 2, $C_2^9.A_4.C_2^6.C_2$ x 2, $C_2^8.C_2^6.D_4:S_3$ x 2, $C_2^7.C_2^5.A_4.C_2^4$ x 2, $C_2^7.C_2^4.C_6.C_2^5.C_2$ x 2, $C_2^6.C_2^6.C_6.C_2^5$ x 2, $C_2^6.C_2^6.C_2^4.C_6.C_2$ x 2, $C_2^6.C_2^6.(C_2^5.S_3)$ x 2, $C_2^{12}.C_2^3:S_4$ x 2, $C_2^{12}.C_2^4.C_{12}$, $C_2^{12}.C_2.A_4.C_2^3$, $C_2^{12}.(C_4\times S_4).C_2$, $C_2^{10}.A_4.C_2\wr C_2^2$, $C_2^9.C_2^4.D_6.C_2^3$, $C_2^8.C_2^6.D_{12}.C_2$, $C_2^8.C_2^6.D_6:C_4$, $C_2^8.C_2^6.(C_6.D_4)$, $C_2^8.C_2^4.C_6.C_2.C_2^4$, $C_2^8.C_2^4.C_2^4:S_3.C_2$, $C_2^7.C_2^6.D_6.C_2^3$, $C_2^6.C_2^6.Q_8:S_4$, $C_2^6.C_2^6.A_4.D_4:C_2$, $C_2^6.C_2^6.A_4.C_2^2:C_4$, $C_2^6.C_2^6.(Q_8\times S_4)$, $C_2^6.C_2^6.(C_4\times S_4).C_2$, $C_2^6.C_2^5.C_2^4.C_6.C_2^2$, $C_2^6.C_2.C_2^6.A_4.C_4.C_2$, $C_2^5.C_2^5.C_2^4.C_6.C_2^3$, $C_2^{12}.(D_4\times S_4)$

Order 32768: $C_2^9.C_2^6$

Order 16384: $C_2^6.C_2^6.C_2^2$

Order 2048: $C_2^6.C_2^5$, $C_2^{10}.C_2$

Order 1024: $C_2^6.C_2^4$, $C_2^{10}$

Order 128: $D_4\times C_2^4$

Order 64: $C_2^6$ x 3, $C_2^4\times C_4$

Order 32: $C_2^5$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 660602880: $C_2^{10}.C_2^5.A_8$ ($C_1$)

Order 330301440: $C_2^6.C_2^4.C_2^4.A_8$ ($C_2$)

Order 32768: $C_2^9.C_2^6$ ($A_8$)

Order 16384: $C_2^6.C_2^6.C_2^2$ ($C_2\times A_8$)

Order 2048: $C_2^6.C_2^5$ ($C_2^4.A_8$), $C_2^{10}.C_2$ ($C_2^4.A_8$)

Order 1024: $C_2^6.C_2^4$ ($C_2^5.A_8$), $C_2^{10}$ ($C_2^5.A_8$)

Order 128: $D_4\times C_2^4$ ($C_2^8.A_8$)

Order 64: $C_2^6$ ($C_2^5.C_2^4.A_8$), $C_2^6$ ($C_2\times C_2^8.A_8$), $C_2^6$ ($C_2^5.C_2^4.A_8$), $C_2^4\times C_4$ ($C_2^5.C_2^4.A_8$)

Order 32: $C_2^5$ ($C_2^5.C_2^5.A_8$)

Order 4: $C_2^2$ ($C_2^{12}.C_2.A_8$)

Order 2: $C_2$ ($C_2^9.C_2^5.A_8$)

Order 1: $C_1$ ($C_2^{10}.C_2^5.A_8$)

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 2 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $260 \times 260$ character table is not available for this group.

Rational character table

The $239 \times 239$ rational character table is not available for this group.