C₂²×C₅×D₅ #6063

order	200
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₄×AGL(1,𝔽₅)×S₄ (order 1920) · Inn(G) = D₅ · Out(G) = C₂×C₄×S₄
homology	Abelianization (H₁) = C₂³×C₅ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×C₅×D₅
unambiguous semidirect product	C₅⋊(C₂³×C₅)
unambiguous semidirect product	(C₂×C₅)⋊(C₂²×C₅)

	subgroup	G/H
center, hypercenter	C₂²×C₅	D₅
commutator, cosocle, hypocenter	C₅	C₂³×C₅
Frattini	C₁	G
Fitting, socle	C₂²×C₅²	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₂²×C₅

74 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 3 generators on 14 points:

(1,2)(3,4)(6,7)(8,9)(10,11,12,13,14);(3,4)(5,6,8,9,7);(1,2)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 50);