C₂²×#59191 #72324902

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

Aliases

direct product

C₂²×#59191

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂²×#2403	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

107297 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7763449
maximal normal subgroup of	C₂²×#7763449

GAP

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

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

C22×#59191 #72324902