C₂²×#42424 #72299740

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

Aliases

direct product

C₂²×#42424

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

90353 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 7 generators on 24 points:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7849930, C₂²×#7758045
maximal normal subgroup of	C₂²×#7849930, C₂²×#7758045

GAP

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

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

C22×#42424 #72299740