C₂²×#38364 #72236634

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

Aliases

direct product

C₂²×#38364

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

112293 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);(1,2)(3,4);(9,10,13,14)(11,12,15,16)(17,18,22,24)(19,23,21,20);(6,8)(10,14)(11,15)(17,19,22,21)(18,20,24,23);(6,8)(17,20)(18,19)(21,24)(22,23);(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23);(5,6,7,8)(9,12)(10,15)(11,14)(13,16)(17,21,22,19)(18,23,24,20)

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂²×#7792761, C₂³×#38364
maximal normal subgroup of	C₂²×#7792761, C₂³×#38364
maximal quotient of	C₂²×#850465, C₂³×#38364, C₂×#72232284, …

GAP

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

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

C22×#38364 #72236634