C₂×#7670636 #72341896

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸

Aliases

direct product

C₂×#7670636

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

71181 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72241249
maximal normal subgroup of	C₂×#72241249

GAP

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

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

C2×#7670636 #72341896