#638⋊C₃ #6223

order	216
isoclinism	isoclinic class of #634⋊C₃
automorphism	Aut(G) = C₂²×((C₃⋊D₃)⋊C₃)⋊C₂ (order 432) · Inn(G) = C₂×(C₃⋊D₃)⋊C₃ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂×C₃

Aliases

Fitting	C₂²×H(3,𝔽₃) ⋊ C₂
Sylow	H(3,𝔽₃) ⋊ D₄

66 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

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

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

Polycyclic presentation:

2^2,3,2,3^2|b^a=bd,e^a=e^2,f^a=f^2,e^c=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 60);