C₃²×C₇ #319

order	63
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) = AGL(1,𝔽₇)×AGL(2,𝔽₃)
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₃

Aliases

C₃²×C₇

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 63

A minimal faithful permutation representation — 2 generators on 13 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(63, 4);