order	100
isoclinism	isoclinic class of C₅⋊D₅
automorphism	Aut(G) = C₂×AGL(2,𝔽₅) (order 24000) · Inn(G) = C₅⋊D₅ · Out(G) = C₂×A₅⋊C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₅

Aliases

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

Splittings

Fitting, socle	C₂×C₅² ⋊ C₂
derived, Sylow	C₅² ⋊ C₂²

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

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 50

A minimal faithful permutation representation — 3 generators on 12 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(100, 15);