C₂²×C₃₁ #1246

order	124
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₅×D₃ (order 180) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₅×D₃
holomorph	G ⋊ Aut(G) = S₄×AGL(1,𝔽₃₁)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂

Aliases

C₂²×C₃₁

10 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 35 points:

(1,2)(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);(3,4)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(124, 4);