C₂×#763813 #72331159

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#763813

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

75245 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72261369, C₂×#72323283, C₂×#72261284
maximal normal subgroup of	C₂×#72261369, C₂×#72323283, C₂×#72261284

GAP

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

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

C2×#763813 #72331159