C₂²×#36721 #72324635

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

Aliases

direct product

C₂²×#36721

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

113757 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)(10,13)(11,15)(18,21)(19,23);(6,8)(10,13)(14,16)(17,18,22,21)(19,24,23,20);(17,19)(18,20)(21,24)(22,23);(3,4);(9,10,12,13)(11,16,15,14)(17,20,22,24)(18,23,21,19);(1,2)(3,4);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,21,22,18)(19,24,23,20)

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7708684, C₂²×#7721725, C₂²×#7642299
maximal normal subgroup of	C₂²×#7708684, C₂²×#7721725, C₂²×#7642299
maximal quotient of	C₂²×#625760

GAP

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

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

C22×#36721 #72324635