#72277320

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₅×A₅
commutator, hypocenter	C₅×SL(2,𝔽₅)	C₂²
Fitting	C₂²×C₅	C₂×A₅
socle, cosocle	C₂×C₅	C₂²×A₅
hypercenter	C₂²	D₅×A₅
LCS	G ▷ C₅×SL(2,𝔽₅)
UCS	C₁ ◁ C₂ ◁ C₂²

182 conjugacy classes of subgroups; only the 16 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 240

A minimal faithful permutation representation — 2 generators on 53 points:

(1,2)(4,5)(6,40,33,53,10,13)(7,9,27,48,32,34)(8,29,28,46,14,17)(11,20,12,52,35,30)(15,37,39,42,24,50)(16,44,22,25,23,19)(18,21,31,36,47,26)(38,43,51,49,41,45);(1,3)(2,4)(6,31,28,47,26,37,21,40,24,53,10,17)(7,42,39,52,15,46,11,29,8,48,14,50)(9,27,44,32,30,23,20,12,51,35,34,45)(13,43,19,33,25,22,18,16,38,36,41,49)

Conjugacy classes & character table

60 conjugacy classes — view the full character table.

GAP

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

gap> G := Group((1,2)(4,5)(6,40,33,53,10,13)(7,9,27,48,32,34)(8,29,28,46,14,17)(11,20,12,52,35,30)(15,37,39,42,24,50)(16,44,22,25,23,19)(18,21,31,36,47,26)(38,43,51,49,41,45), (1,3)(2,4)(6,31,28,47,26,37,21,40,24,53,10,17)(7,42,39,52,15,46,11,29,8,48,14,50)(9,27,44,32,30,23,20,12,51,35,34,45)(13,43,19,33,25,22,18,16,38,36,41,49));