C₂×#4306666 #72337939

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#4306666

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

43299 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

220 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232339, C₂×#72232305, C₂×#72232287
maximal normal subgroup of	C₂×#72232339, C₂×#72232305, C₂×#72232287

GAP

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

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

C2×#4306666 #72337939