D₃×D₅×A₅ #72247603

order	3600
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂²

Aliases

D₃×D₅×A₅

Fitting	C₃×C₅ ⋊ C₂²×A₅
socle, derived	C₃×C₅×A₅ ⋊ C₂²

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, socle, hypocenter	C₃×C₅×A₅	C₂²
Fitting, cosocle	C₃×C₅	C₂²×A₅
LCS	G ▷ C₃×C₅×A₅
UCS	C₁

318 conjugacy classes of subgroups; only the 20 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.

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

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 75

A minimal faithful permutation representation — 2 generators on 13 points:

(2,3)(4,6,8,7,5)(9,12,10,11,13);(1,2)(4,8)(5,7)(9,10)(11,12)

maximal subgroup of	D₃×A₅², D₃×A₅⋊AGL(1,𝔽₅)
maximal normal subgroup of	D₃×A₅⋊AGL(1,𝔽₅)

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

gap> G := Group((2,3)(4,6,8,7,5)(9,12,10,11,13), (1,2)(4,8)(5,7)(9,10)(11,12));