C₂²×#44899 #72324641

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

Aliases

direct product

C₂²×#44899

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

98389 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)(9,10,12,13)(11,14,15,16)(17,18,22,21)(19,23,20,24);(6,8)(11,15)(14,16)(17,18,22,21)(19,23,20,24);(6,8)(10,13)(14,16)(17,19,22,20)(18,23,21,24);(3,4);(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,23,21,24);(1,2)(3,4);(5,6)(7,8)(9,10)(11,16)(12,13)(14,15)(17,21,22,18)(19,23,20,24)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

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

GAP

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

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

C22×#44899 #72324641