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G = S3×D9order 108 = 22·33

Direct product of S3 and D9

direct product, metabelian, supersoluble, monomial, A-group

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

C1C3×C9 — S3×D9
C1C3C32C3×C9S3×C9 — S3×D9
C3×C9 — S3×D9
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 >

3C2
9C2
27C2
2C3
27C22
3C6
3S3
9S3
9S3
9C6
18S3
2C9
9D6
9D6
3D9
3C18
3C3×S3
3C3⋊S3
6D9
3D18
3S32

Character table of S3×D9

 class 12A2B2C3A3B3C6A6B9A9B9C9D9E9F18A18B18C
 size 13927224618222444666
ρ1111111111111111111    trivial
ρ21-1-11111-1-1111111-1-1-1    linear of order 2
ρ31-11-1111-11111111-1-1-1    linear of order 2
ρ411-1-11111-1111111111    linear of order 2
ρ520-202-1-101222-1-1-1000    orthogonal lifted from D6
ρ6220022220-1-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ720202-1-10-1222-1-1-1000    orthogonal lifted from S3
ρ82-200222-20-1-1-1-1-1-1111    orthogonal lifted from D6
ρ92-200-12-110ζ9792ζ989ζ9594ζ9792ζ9594ζ98998997929594    orthogonal lifted from D18
ρ102200-12-1-10ζ989ζ9594ζ9792ζ989ζ9792ζ9594ζ9594ζ989ζ9792    orthogonal lifted from D9
ρ112-200-12-110ζ989ζ9594ζ9792ζ989ζ9792ζ959495949899792    orthogonal lifted from D18
ρ122-200-12-110ζ9594ζ9792ζ989ζ9594ζ989ζ979297929594989    orthogonal lifted from D18
ρ132200-12-1-10ζ9594ζ9792ζ989ζ9594ζ989ζ9792ζ9792ζ9594ζ989    orthogonal lifted from D9
ρ142200-12-1-10ζ9792ζ989ζ9594ζ9792ζ9594ζ989ζ989ζ9792ζ9594    orthogonal lifted from D9
ρ1540004-2-200-2-2-2111000    orthogonal lifted from S32
ρ164000-2-210097+2ζ9298+2ζ995+2ζ9497929594989000    orthogonal faithful
ρ174000-2-210098+2ζ995+2ζ9497+2ζ9298997929594000    orthogonal faithful
ρ184000-2-210095+2ζ9497+2ζ9298+2ζ995949899792000    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

18100
18000
0010
0001
,
01800
18000
00180
00018
,
1000
0100
00512
00717
,
18000
01800
00717
00512
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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Subgroup lattice of S3×D9 in TeX
Character table of S3×D9 in TeX

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