C₄⋊AGL(1,ℤ/9) #6217

order	216
isoclinism	isoclinic class of 3₋¹⁺²⋊Q₈
automorphism	Aut(G) = D₄×AGL(1,ℤ/9) (order 432) · Inn(G) = C₂×AGL(1,ℤ/9) · Out(G) = C₂²
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₄⋊AGL(1,ℤ/9)
unambiguous semidirect product	D₃₆⋊C₃

Fitting	C₄×3₋¹⁺² ⋊ C₂
Sylow	3₋¹⁺² ⋊ D₄

51 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 54);