C₂²×#35262 #72337187

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁸×C₄

Aliases

direct product

C₂²×#35262

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

114325 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7641850, C₂²×#7646027, C₂²×#7631310
maximal normal subgroup of	C₂²×#7641850, C₂²×#7646027, C₂²×#7631310

GAP

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

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

C22×#35262 #72337187