C₅×#91877 #72277292

order	2400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₂

non-solvable

Aliases

direct product

C₅×#91877

Subgroups

	subgroup	G/H
center, socle	C₂×C₅	C₂×S₅
commutator, cosocle, hypocenter	SL(2,𝔽₅)	C₂²×C₅
Frattini	C₂	C₂×C₅×S₅
Fitting, hypercenter	C₂²×C₅	S₅
LCS	G ▷ SL(2,𝔽₅)
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₅

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

normal subgroups only

This group has 137 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:

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

Conjugacy classes & character table

90 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277310
maximal normal subgroup of	#72277310

GAP

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

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