C₂×#7642247 #72255585

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

Aliases

direct product

C₂×#7642247

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

161821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 26 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72236365, C₂×#72218920, C₂×#72232275, …
maximal normal subgroup of	C₂×#72236365, C₂×#72218920, C₂×#72232275, …
maximal quotient of	C₂×#72232282, C₂×#72240189

GAP

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

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

C2×#7642247 #72255585