C₂²×#35939 #72325836

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

Aliases

direct product

C₂²×#35939

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁷
prime radical	C₂³×#546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

105517 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)(11,16)(12,15)(13,14)(17,18,22,24)(19,21,20,23);(10,14)(11,15)(17,18,22,24)(19,21,20,23);(5,6)(7,8)(17,19,22,20)(18,23,24,21);(3,4);(5,6,7,8)(9,11)(10,12)(13,15)(14,16)(17,20,22,19)(18,23,24,21);(1,2)(3,4);(5,6)(7,8)(9,12)(10,15)(11,14)(13,16)(17,21,22,23)(18,19,24,20)

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7653121, C₂²×#7685422
maximal normal subgroup of	C₂²×#7653121, C₂²×#7685422

GAP

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

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

C22×#35939 #72325836