D₈₅ #4292

order	170
isoclinism	isoclinic class of D₈₅
automorphism	Aut(G) = AGL(1,𝔽₅)×AGL(1,𝔽₁₇) (order 5440) · Inn(G) = G · Out(G) = C₂×C₁₆
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₅
unambiguous semidirect product	C₅⋊D₁₇
unambiguous semidirect product	C₁₇⋊D₅

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

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

Polycyclic presentation:

2,5,17|b^a=b^4,c^a=c^16

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(170, 3);