C₂×#3860926 #72332756

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

Aliases

direct product

C₂×#3860926

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

48551 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72323407, C₂×#72323415, C₂×#72323404
maximal normal subgroup of	C₂×#72323407, C₂×#72323415, C₂×#72323404

GAP

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

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

C2×#3860926 #72332756