order	170
isoclinism	isoclinic class of D₅
automorphism	Aut(G) = C₁₆×AGL(1,𝔽₅) (order 320) · Inn(G) = D₅ · Out(G) = C₂×C₁₆
holomorph	G ⋊ Aut(G) = D₅⋊AGL(1,𝔽₅)×AGL(1,𝔽₁₇)
homology	Abelianization (H₁) = C₂×C₁₇ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₇
unambiguous semidirect product	C₅⋊(C₂×C₁₇)

Splittings

Fitting, socle, Hall	C₅×C₁₇ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₇

	subgroup	G/H
center, hypercenter	C₁₇	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₇
Frattini	C₁	G
Fitting, socle	C₅×C₁₇	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₇

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 85

A minimal faithful permutation representation — 2 generators on 22 points:

(2,3)(4,5)(6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22);(1,2,4,5,3)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(170, 1);