C₂×#7678613 #72331661

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

Aliases

direct product

C₂×#7678613

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

124437 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:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327577
maximal normal subgroup of	C₂×#72327577
maximal quotient of	C₂×#72327589

GAP

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

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

C2×#7678613 #72331661