D9 - GroupNames

Permutation representations of D₉
►On 9 points - transitive group 9T3

Generators in S₉

(1 2 3 4 5 6 7 8 9)

(1 9)(2 8)(3 7)(4 6)

G:=sub<Sym(9)| (1,2,3,4,5,6,7,8,9), (1,9)(2,8)(3,7)(4,6)>;

G:=Group( (1,2,3,4,5,6,7,8,9), (1,9)(2,8)(3,7)(4,6) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9)], [(1,9),(2,8),(3,7),(4,6)]])

G:=TransitiveGroup(9,3);

Generators in S₁₈

(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)

(1 13)(2 12)(3 11)(4 10)(5 18)(6 17)(7 16)(8 15)(9 14)

G:=sub<Sym(18)| (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14)>;

G:=Group( (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,13)(2,12)(3,11)(4,10)(5,18)(6,17)(7,16)(8,15)(9,14) );

G=PermutationGroup([[(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,13),(2,12),(3,11),(4,10),(5,18),(6,17),(7,16),(8,15),(9,14)]])

G:=TransitiveGroup(18,5);

Polynomial with Galois group D₉ over ℚ

action	f(x)	Disc(f)
9T3	x⁹+3x⁸-67x⁷-226x⁶+699x⁵+1211x⁴-3137x³+940x²+904x-392	2²²·7⁴·13²·19⁶·29²·229⁴·547²

16	10
7	14

14	6
10	3

G:=sub<GL(2,GF(17))| [16,7,10,14],[14,10,6,3] >;

D₉ in GAP, Magma, Sage, TeX

D_9

% in TeX

G:=Group("D9");

// GroupNames label

G:=SmallGroup(18,1);

// by ID

G=gap.SmallGroup(18,1);

# by ID

G:=PCGroup([3,-2,-3,-3,97,22,110]);

// Polycyclic

G:=Group<a,b|a^9=b^2=1,b*a*b=a^-1>;

// generators/relations

Export

Subgroup lattice of D₉ in TeX
Character table of D₉ in TeX

G = D9 order 18 = 2·32