D₃₉ #669

order	78
isoclinism	isoclinic class of D₃₉
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₃) (order 936) · Inn(G) = G · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₃₉
unambiguous semidirect product	C₃⋊D₁₃
unambiguous semidirect product	C₁₃⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₃	C₂
LCS	G ▷ C₃×C₁₃
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 39

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

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

Polycyclic presentation:

2,3,13|b^a=b^2,c^a=c^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(78, 5);