C₂×#7672627 #72302267

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂×#7672627

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

122481 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72268516
maximal normal subgroup of	C₂×#72268516
maximal quotient of	C₂×#72280169, C₂×#72280294, #72315102, …

GAP

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

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

C2×#7672627 #72302267