order	124
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₃×C₅ (order 60) · Inn(G) = C₁ · Out(G) = C₂²×C₃×C₅
holomorph	G ⋊ Aut(G) = D₄×AGL(1,𝔽₃₁)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₁₂₄
abelian	C₄×C₃₁

Subgroups

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 124

A minimal faithful permutation representation — 1 generator on 35 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(124, 2);