C₂×#721616 #72337979

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#721616

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

41211 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232357, C₂×#72232305, C₂×#72232298
maximal normal subgroup of	C₂×#72232357, C₂×#72232305, C₂×#72232298

GAP

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

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

C2×#721616 #72337979