D₉⋊C₂₇ #92185

order	486
isoclinism	isoclinic class of AGL(1,ℤ/9)
automorphism	Aut(G) = C₉×AGL(1,ℤ/9) (order 486) · Inn(G) = AGL(1,ℤ/9) · Out(G) = C₉
homology	Abelianization (H₁) = C₂×C₂₇ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	D₉⋊C₂₇
unambiguous semidirect product	(C₉⋊C₂₇)⋊C₂

Fitting, Sylow	C₉⋊C₂₇ ⋊ C₂
derived	C₉ ⋊ C₂×C₂₇

31 conjugacy classes of subgroups.

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 54

A minimal faithful permutation representation — 2 generators on 36 points:

(2,6,9,4,5,8)(3,7)(10,15,27,18,30,29,32,11,20,13,23,33,26,35,14,17,16,28,19,31,36,12,22,21,25,24,34);(1,9,4,7,5,6,3,2,8)

Polycyclic presentation:

2,3^5|b^3=c,c^3=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(486, 30);