D₅₂ #1060

order	104
isoclinism	isoclinic class of Dic₂₆
automorphism	Aut(G) = D₄×AGL(1,𝔽₁₃) (order 1248) · Inn(G) = C₂×D₁₃ · Out(G) = C₂×C₃×C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral	D₅₂
unambiguous semidirect product	C₄⋊D₁₃

Fitting	C₄×C₁₃ ⋊ C₂
Sylow	C₁₃ ⋊ D₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(104, 6);