C₂²×#37800 #72292873

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

Aliases

direct product

C₂²×#37800

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

97421 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);(9,10,13,15)(11,12,16,14)(17,18,21,24)(19,20,23,22);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23);(6,8)(9,11,13,16)(10,14,15,12)(17,19)(18,20)(21,23)(22,24);(9,12,13,14)(10,11,15,16)(17,20)(18,23)(19,24)(21,22);(5,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,23)(19,24)(21,22);(1,2)(3,4)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7918389, C₂²×#7915509, C₂³×#37800
maximal normal subgroup of	C₂²×#7918389, C₂²×#7915509, C₂³×#37800
maximal quotient of	C₂²×#850465, C₂³×#37800

GAP

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

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

C22×#37800 #72292873