C₂²×#44001 #72301090

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

Aliases

direct product

C₂²×#44001

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

106105 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)(9,10,13,11)(12,14,16,15)(17,18)(19,23)(20,21)(22,24);(5,6)(7,8)(17,18)(19,22)(20,21)(23,24);(18,21)(19,24);(9,11,13,10)(12,14,16,15)(17,18)(19,23)(20,21)(22,24);(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,24)(21,23);(3,4);(1,2)(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7819605, C₂²×#7763450, C₂²×#7738214
maximal normal subgroup of	C₂²×#7819605, C₂²×#7763450, C₂²×#7738214

GAP

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

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

C22×#44001 #72301090