order	75
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₄×C₅ (order 40) · Inn(G) = C₁ · Out(G) = C₂×C₄×C₅
holomorph	G ⋊ Aut(G) = D₃×AGL(1,ℤ/25)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₇₅
abelian	C₃×C₂₅

Subgroups

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 75

A minimal faithful permutation representation — 1 generator on 28 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(75, 1);