C₂×C₃²×C₇ #1267

order	126
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×GL(2,𝔽₃) (order 288) · Inn(G) = C₁ · Out(G) = C₂×C₃×GL(2,𝔽₃)
holomorph	G ⋊ Aut(G) = C₂×AGL(1,𝔽₇)×AGL(2,𝔽₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₂×C₃²×C₇

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 126

A minimal faithful permutation representation — 2 generators on 15 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(126, 16);