C₂²×#59023 #72299764

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

Aliases

direct product

C₂²×#59023

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

106681 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)(9,10,13,16)(11,12,15,14)(18,21)(22,23);(5,6)(7,8)(9,11,13,15)(10,14,16,12)(17,18,20,21)(19,22,24,23);(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23);(9,11)(10,12)(13,15)(14,16)(17,18)(19,23)(20,21)(22,24);(9,12)(10,15)(11,16)(13,14);(3,4);(1,2)(3,4)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7849977, C₂²×#7670211, C₂²×#7685422
maximal normal subgroup of	C₂²×#7849977, C₂²×#7670211, C₂²×#7685422

GAP

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

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

C22×#59023 #72299764