C₂²×#37658 #72324554

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

direct product

C₂²×#37658

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

133701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 20 points:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7708684, C₂²×#7916101
maximal normal subgroup of	C₂²×#7708684, C₂²×#7916101

GAP

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

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

C22×#37658 #72324554