order	110
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₂×C₅×AGL(1,𝔽₅) (order 200) · Inn(G) = D₅ · Out(G) = C₂²×C₅
holomorph	G ⋊ Aut(G) = AGL(1,𝔽₁₁)×D₅⋊AGL(1,𝔽₅)
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₁
unambiguous semidirect product	C₅⋊(C₂×C₁₁)

Splittings

Fitting, socle, Hall	C₅×C₁₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₁

	subgroup	G/H
center, hypercenter	C₁₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 55

A minimal faithful permutation representation — 2 generators on 16 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16);(1,2,4,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(110, 3);