C₂²×#46836 #72295930

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

Aliases

direct product

C₂²×#46836

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

91289 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7763450, C₂²×#7837953
maximal normal subgroup of	C₂²×#7763450, C₂²×#7837953

GAP

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

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

C22×#46836 #72295930