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## G = S3×D9order 108 = 22·33

### Direct product of S3 and D9

Aliases: S3×D9, C91D6, C31D18, C32.2D6, C9⋊S3⋊C2, (S3×C9)⋊C2, C3.1S32, (C3×D9)⋊C2, (C3×C9)⋊C22, (C3×S3).S3, SmallGroup(108,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C9 — S3×D9
 Chief series C1 — C3 — C32 — C3×C9 — S3×C9 — S3×D9
 Lower central C3×C9 — S3×D9
 Upper central C1

Generators and relations for S3×D9
G = < a,b,c,d | a3=b2=c9=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Character table of S3×D9

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 9A 9B 9C 9D 9E 9F 18A 18B 18C size 1 3 9 27 2 2 4 6 18 2 2 2 4 4 4 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 1 -1 -1 1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ3 1 -1 1 -1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 1 1 -1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 2 0 -2 0 2 -1 -1 0 1 2 2 2 -1 -1 -1 0 0 0 orthogonal lifted from D6 ρ6 2 2 0 0 2 2 2 2 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 0 2 0 2 -1 -1 0 -1 2 2 2 -1 -1 -1 0 0 0 orthogonal lifted from S3 ρ8 2 -2 0 0 2 2 2 -2 0 -1 -1 -1 -1 -1 -1 1 1 1 orthogonal lifted from D6 ρ9 2 -2 0 0 -1 2 -1 1 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 orthogonal lifted from D18 ρ10 2 2 0 0 -1 2 -1 -1 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 orthogonal lifted from D9 ρ11 2 -2 0 0 -1 2 -1 1 0 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 orthogonal lifted from D18 ρ12 2 -2 0 0 -1 2 -1 1 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 orthogonal lifted from D18 ρ13 2 2 0 0 -1 2 -1 -1 0 ζ95+ζ94 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ98+ζ9 ζ97+ζ92 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 orthogonal lifted from D9 ρ14 2 2 0 0 -1 2 -1 -1 0 ζ97+ζ92 ζ98+ζ9 ζ95+ζ94 ζ97+ζ92 ζ95+ζ94 ζ98+ζ9 ζ98+ζ9 ζ97+ζ92 ζ95+ζ94 orthogonal lifted from D9 ρ15 4 0 0 0 4 -2 -2 0 0 -2 -2 -2 1 1 1 0 0 0 orthogonal lifted from S32 ρ16 4 0 0 0 -2 -2 1 0 0 2ζ97+2ζ92 2ζ98+2ζ9 2ζ95+2ζ94 -ζ97-ζ92 -ζ95-ζ94 -ζ98-ζ9 0 0 0 orthogonal faithful ρ17 4 0 0 0 -2 -2 1 0 0 2ζ98+2ζ9 2ζ95+2ζ94 2ζ97+2ζ92 -ζ98-ζ9 -ζ97-ζ92 -ζ95-ζ94 0 0 0 orthogonal faithful ρ18 4 0 0 0 -2 -2 1 0 0 2ζ95+2ζ94 2ζ97+2ζ92 2ζ98+2ζ9 -ζ95-ζ94 -ζ98-ζ9 -ζ97-ζ92 0 0 0 orthogonal faithful

Permutation representations of S3×D9
On 18 points - transitive group 18T50
Generators in S18
(1 7 4)(2 8 5)(3 9 6)(10 13 16)(11 14 17)(12 15 18)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 10)(8 11)(9 12)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 12)(2 11)(3 10)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)

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

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

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

G:=TransitiveGroup(18,50);

On 27 points - transitive group 27T30
Generators in S27
(1 15 25)(2 16 26)(3 17 27)(4 18 19)(5 10 20)(6 11 21)(7 12 22)(8 13 23)(9 14 24)
(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(18 19)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)(19 20 21 22 23 24 25 26 27)
(1 9)(2 8)(3 7)(4 6)(11 18)(12 17)(13 16)(14 15)(19 21)(22 27)(23 26)(24 25)

G:=sub<Sym(27)| (1,15,25)(2,16,26)(3,17,27)(4,18,19)(5,10,20)(6,11,21)(7,12,22)(8,13,23)(9,14,24), (10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9)(2,8)(3,7)(4,6)(11,18)(12,17)(13,16)(14,15)(19,21)(22,27)(23,26)(24,25)>;

G:=Group( (1,15,25)(2,16,26)(3,17,27)(4,18,19)(5,10,20)(6,11,21)(7,12,22)(8,13,23)(9,14,24), (10,20)(11,21)(12,22)(13,23)(14,24)(15,25)(16,26)(17,27)(18,19), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18)(19,20,21,22,23,24,25,26,27), (1,9)(2,8)(3,7)(4,6)(11,18)(12,17)(13,16)(14,15)(19,21)(22,27)(23,26)(24,25) );

G=PermutationGroup([(1,15,25),(2,16,26),(3,17,27),(4,18,19),(5,10,20),(6,11,21),(7,12,22),(8,13,23),(9,14,24)], [(10,20),(11,21),(12,22),(13,23),(14,24),(15,25),(16,26),(17,27),(18,19)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18),(19,20,21,22,23,24,25,26,27)], [(1,9),(2,8),(3,7),(4,6),(11,18),(12,17),(13,16),(14,15),(19,21),(22,27),(23,26),(24,25)])

G:=TransitiveGroup(27,30);

S3×D9 is a maximal subgroup of   C32⋊D18  He3.D6  He3.2D6  C325D18  He3.6D6
S3×D9 is a maximal quotient of   C9⋊Dic6  C18.D6  C3⋊D36  D6⋊D9  C9⋊D12  C32⋊D18  C325D18

Matrix representation of S3×D9 in GL4(𝔽19) generated by

 18 1 0 0 18 0 0 0 0 0 1 0 0 0 0 1
,
 0 18 0 0 18 0 0 0 0 0 18 0 0 0 0 18
,
 1 0 0 0 0 1 0 0 0 0 5 12 0 0 7 17
,
 18 0 0 0 0 18 0 0 0 0 7 17 0 0 5 12
G:=sub<GL(4,GF(19))| [18,18,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,18,0,0,18,0,0,0,0,0,18,0,0,0,0,18],[1,0,0,0,0,1,0,0,0,0,5,7,0,0,12,17],[18,0,0,0,0,18,0,0,0,0,7,5,0,0,17,12] >;

S3×D9 in GAP, Magma, Sage, TeX

S_3\times D_9
% in TeX

G:=Group("S3xD9");
// GroupNames label

G:=SmallGroup(108,16);
// by ID

G=gap.SmallGroup(108,16);
# by ID

G:=PCGroup([5,-2,-2,-3,-3,-3,337,282,483,909]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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