C₂²×#59018 #72324723

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

Aliases

direct product

C₂²×#59018

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

94429 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:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7916101, C₂²×#7694192
maximal normal subgroup of	C₂²×#7916101, C₂²×#7694192

GAP

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

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

C22×#59018 #72324723