C₂²×#37778 #72324866

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂²×#37778

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

101805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 24 points:

(11,15)(13,16)(17,18,21,23)(19,20,24,22);(6,8)(9,10,12,14)(11,13,15,16);(6,8)(17,19,21,24)(18,22,23,20);(3,4);(6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,4);(5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,19)(21,22)(23,24)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7768137
maximal normal subgroup of	C₂²×#7768137

GAP

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

gap> G := Group((11,15)(13,16)(17,18,21,23)(19,20,24,22), (6,8)(9,10,12,14)(11,13,15,16), (6,8)(17,19,21,24)(18,22,23,20), (3,4), (6,8)(9,11,12,15)(10,13,14,16)(17,19)(18,20)(21,24)(22,23), (1,2)(3,4), (5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,20)(18,19)(21,22)(23,24));

C22×#37778 #72324866