D₅₁ #1052

order	102
isoclinism	isoclinic class of D₅₁
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₇) (order 1632) · Inn(G) = G · Out(G) = C₁₆
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₁
unambiguous semidirect product	C₃⋊D₁₇
unambiguous semidirect product	C₁₇⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₇	C₂
LCS	G ▷ C₃×C₁₇
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 51

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2,3,17|b^a=b^2,c^a=c^16

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(102, 3);