order	170
isoclinism	isoclinic class of D₁₇
automorphism	Aut(G) = C₄×AGL(1,𝔽₁₇) (order 1088) · Inn(G) = D₁₇ · Out(G) = C₄×C₈
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₇
unambiguous semidirect product	C₁₇⋊(C₂×C₅)

Splittings

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

	subgroup	G/H
center, hypercenter	C₅	D₁₇
commutator, cosocle, hypocenter	C₁₇	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₅×C₁₇	C₂
LCS	G ▷ C₁₇
UCS	C₁ ◁ C₅

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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(170, 2);