C₂²×#39885 #72291850

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

Aliases

direct product

C₂²×#39885

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

107309 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7837443, C₂²×#7850713, C₂²×#7687902
maximal normal subgroup of	C₂²×#7837443, C₂²×#7850713, C₂²×#7687902
maximal quotient of	C₂²×#845307

GAP

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

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

C22×#39885 #72291850