C₂×#5955305 #72339398

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

direct product

C₂×#5955305

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#534	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

72157 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)(7,8,9,10)(15,16)(17,18)(19,22)(20,21);(15,17)(16,20)(18,22)(19,21);(4,6)(8,10)(11,12,13,14);(1,2);(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,17)(16,18)(19,21)(20,22);(15,18)(16,21)(17,22)(19,20)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72327582, C₂×#72327612
maximal normal subgroup of	C₂×#72327582, C₂×#72327612

GAP

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

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

C2×#5955305 #72339398