C₃×#634 #6303

order	216
isoclinism	isoclinic class of #634
automorphism	Aut(G) = C₂×D₄×AGL(2,𝔽₃) (order 6912) · Inn(G) = C₂×C₃⋊D₃ · Out(G) = C₂²×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₃

Aliases

C₃³ ⋊ Q₈

96 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 3 generators on 17 points:

(1,8,4,3)(2,7,6,5)(9,10,11)(15,16,17);(1,6,4,2)(3,5,8,7)(13,14)(16,17);(12,13,14)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 140);