C₄⋊(C₃⋊D₉) #6228

order	216
isoclinism	isoclinic class of #6226
automorphism	Aut(G) = D₄×#72210336 (order 23328) · Inn(G) = C₂×C₃⋊D₉ · Out(G) = C₂²×#272
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

C₄⋊(C₃⋊D₉)

Fitting	C₃×C₄×C₉ ⋊ C₂
Sylow	C₃×C₉ ⋊ D₄

80 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 108

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 65);