C₂×#7630503 #72331667

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

Aliases

direct product

C₂×#7630503

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

125229 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)(11,12,16,15)(13,17,20,18)(14,22,23,19)(21,25,26,24);(4,6)(7,8)(9,10)(13,20)(14,23)(17,18)(19,22);(11,13)(12,17)(14,21)(15,18)(16,20)(19,24)(22,25)(23,26);(1,2);(3,4)(5,6)(11,13)(12,18)(14,21)(15,17)(16,20)(19,25)(22,24)(23,26);(11,14)(12,19)(13,21)(15,22)(16,23)(17,24)(18,25)(20,26);(11,15,16,12)(13,18,20,17)(14,22,23,19)(21,25,26,24)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327577
maximal normal subgroup of	C₂×#72327577
maximal quotient of	C₂×#72327602

GAP

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

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

C2×#7630503 #72331667