3₋¹⁺²⋊C₈ #6178

order	216
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	Aut(G) = C₂²×AGL(1,ℤ/9) (order 216) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	3₋¹⁺²⋊C₈
unambiguous semidirect product	(C₉⋊C₈)⋊C₃

derived	C₉ ⋊ C₃×C₈
Sylow	3₋¹⁺² ⋊ C₈

27 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 72

A minimal faithful permutation representation — 2 generators on 17 points:

(2,6,9,4,5,8)(3,7)(10,11,12,14,13,15,16,17);(1,9,4,7,5,6,3,2,8)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 15);