((C₃⋊D₃)⋊C₃)⋊C₄ #6197

order	216
isoclinism	isoclinic class of ((C₃⋊D₃)⋊C₃)⋊C₂
automorphism	Aut(G) = C₂²×((C₃⋊D₃)⋊C₃)⋊C₂ (order 432) · Inn(G) = ((C₃⋊D₃)⋊C₃)⋊C₂ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

62 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 34);