C₄⋊(C₃⋊D₃) #636

order	72
isoclinism	isoclinic class of #634
automorphism	Aut(G) = D₄×AGL(2,𝔽₃) (order 3456) · Inn(G) = C₂×C₃⋊D₃ · Out(G) = C₂×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃

Aliases

C₄⋊(C₃⋊D₃)

Fitting	C₃²×C₄ ⋊ C₂
Sylow	C₃² ⋊ D₄

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 10 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(72, 33);