direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×D9, C9⋊1D6, C3⋊1D18, 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
| C3×C9 — S3×D9 |
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 |
►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 12 25)(2 13 26)(3 14 27)(4 15 19)(5 16 20)(6 17 21)(7 18 22)(8 10 23)(9 11 24)
(10 23)(11 24)(12 25)(13 26)(14 27)(15 19)(16 20)(17 21)(18 22)
(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)(10 13)(11 12)(14 18)(15 17)(19 21)(22 27)(23 26)(24 25)
G:=sub<Sym(27)| (1,12,25)(2,13,26)(3,14,27)(4,15,19)(5,16,20)(6,17,21)(7,18,22)(8,10,23)(9,11,24), (10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22), (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)(10,13)(11,12)(14,18)(15,17)(19,21)(22,27)(23,26)(24,25)>;
G:=Group( (1,12,25)(2,13,26)(3,14,27)(4,15,19)(5,16,20)(6,17,21)(7,18,22)(8,10,23)(9,11,24), (10,23)(11,24)(12,25)(13,26)(14,27)(15,19)(16,20)(17,21)(18,22), (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)(10,13)(11,12)(14,18)(15,17)(19,21)(22,27)(23,26)(24,25) );
G=PermutationGroup([[(1,12,25),(2,13,26),(3,14,27),(4,15,19),(5,16,20),(6,17,21),(7,18,22),(8,10,23),(9,11,24)], [(10,23),(11,24),(12,25),(13,26),(14,27),(15,19),(16,20),(17,21),(18,22)], [(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),(10,13),(11,12),(14,18),(15,17),(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 C32⋊5D18 He3.6D6
S3×D9 is a maximal quotient of C9⋊Dic6 C18.D6 C3⋊D36 D6⋊D9 C9⋊D12 C32⋊D18 C32⋊5D18
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
Export
Subgroup lattice of S3×D9 in TeX
Character table of S3×D9 in TeX