C₂×#8105981 #72331686

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#8105981

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

68345 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 7 generators on 26 points:

(4,8)(5,9)(11,12,15,13)(14,16,18,17)(19,20,22,24)(21,25,26,23);(3,4)(5,6)(7,8)(9,10)(12,13)(14,18)(19,20)(21,23)(22,24)(25,26);(4,8)(5,9)(11,12,15,13)(14,16,18,17)(19,20)(21,23)(22,24)(25,26);(1,2);(12,13)(16,17)(21,26)(23,25);(3,5,7,9)(4,6,8,10)(11,13,15,12)(14,17,18,16)(19,20)(21,23)(22,24)(25,26);(3,6)(4,5)(7,10)(8,9)(11,14)(12,16)(13,17)(15,18)(19,21,22,26)(20,23,24,25)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241227
maximal normal subgroup of	C₂×#72241227
maximal quotient of	C₂×#72327592

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((4,8)(5,9)(11,12,15,13)(14,16,18,17)(19,20,22,24)(21,25,26,23), (3,4)(5,6)(7,8)(9,10)(12,13)(14,18)(19,20)(21,23)(22,24)(25,26), (4,8)(5,9)(11,12,15,13)(14,16,18,17)(19,20)(21,23)(22,24)(25,26), (1,2), (12,13)(16,17)(21,26)(23,25), (3,5,7,9)(4,6,8,10)(11,13,15,12)(14,17,18,16)(19,20)(21,23)(22,24)(25,26), (3,6)(4,5)(7,10)(8,9)(11,14)(12,16)(13,17)(15,18)(19,21,22,26)(20,23,24,25));

C2×#8105981 #72331686