C₂×#7676227 #72332268

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

Aliases

direct product

C₂×#7676227

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

124897 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323293
maximal normal subgroup of	C₂×#72323293
maximal quotient of	C₂×#72323404

GAP

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

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

C2×#7676227 #72332268