C₂×#3860269 #72332770

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

Aliases

direct product

C₂×#3860269

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

45763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 22 points:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323391, C₂×#72297536, C₂×#72323388
maximal normal subgroup of	C₂×#72323391, C₂×#72297536, C₂×#72323388

GAP

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

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

C2×#3860269 #72332770