C₂²×#36146 #72320567

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

Aliases

direct product

C₂²×#36146

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

118285 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:

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

Conjugacy classes & character table

280 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂²×#7744685
maximal normal subgroup of	C₂²×#7744685
maximal quotient of	C₂²×#519339, C₂²×#612460, C₂²×#612810, …

GAP

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

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

C22×#36146 #72320567