C₂²×C₁₃ #263

order	52
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₃×C₄×D₃ (order 72) · Inn(G) = C₁ · Out(G) = C₃×C₄×D₃
holomorph	G ⋊ Aut(G) = S₄×AGL(1,𝔽₁₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

C₂²×C₁₃

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(52, 5);