C₂×#3867204 #72331817

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#3867204

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

36383 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)(5,6)(8,10)(11,12,14,16)(13,17,18,15)(20,23)(24,26);(3,4)(5,6)(7,8)(9,10)(11,12,14,16)(13,15,18,17)(21,25)(24,26);(3,4)(5,6)(19,20,22,23)(21,26,25,24);(1,2);(3,4,5,6)(11,12)(13,17)(14,16)(15,18)(19,20)(21,26)(22,23)(24,25);(3,4)(5,6)(11,13)(12,15)(14,18)(16,17)(19,21)(20,24)(22,25)(23,26)

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323324, C₂×#72323299, C₂×#72323295
maximal normal subgroup of	C₂×#72323324, C₂×#72323299, C₂×#72323295

GAP

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

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

C2×#3867204 #72331817