C₂²×#37536 #72325821

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷×C₄²

Aliases

direct product

C₂²×#37536

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

108077 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7726041, C₂²×#7653121, C₂²×#7645336
maximal normal subgroup of	C₂²×#7726041, C₂²×#7653121, C₂²×#7645336

GAP

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

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

C22×#37536 #72325821