non-abelian, supersoluble, monomial
Aliases: C32⋊2D9, C33.2S3, (C3×C9)⋊4S3, C32⋊C9⋊3C2, C3.3(C9⋊S3), C32.8(C3⋊S3), C3.1(He3⋊C2), SmallGroup(162,17)
Series: Derived ►Chief ►Lower central ►Upper central
| C32⋊C9 — C32⋊2D9 |
Generators and relations for C32⋊2D9
G = < a,b,c,d | a3=b3=c9=d2=1, ab=ba, cac-1=ab-1, dad=a-1b, bc=cb, bd=db, dcd=c-1 >
Character table of C32⋊2D9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | |
| size | 1 | 27 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 27 | 27 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 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 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ7 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ8 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ12 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ13 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ14 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ15 | 2 | 0 | 2 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ16 | 3 | 1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 0 | 0 | 0 | ζ32 | ζ3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 0 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ18 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -3+3√-3/2 | -3-3√-3/2 | 3 | 0 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ19 | 3 | 1 | -3+3√-3/2 | -3-3√-3/2 | -3-3√-3/2 | -3+3√-3/2 | 3 | 0 | 0 | 0 | ζ3 | ζ32 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ20 | 6 | 0 | -3-3√-3 | -3+3√-3 | 3-3√-3/2 | 3+3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ21 | 6 | 0 | -3+3√-3 | -3-3√-3 | 3+3√-3/2 | 3-3√-3/2 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 18 points - transitive group 18T87
Generators in S18
(2 8 5)(3 6 9)(11 14 17)(12 18 15)
(1 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)
(1 2 3 4 5 6 7 8 9)(10 11 12 13 14 15 16 17 18)
(1 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)
G:=sub<Sym(18)| (2,8,5)(3,6,9)(11,14,17)(12,18,15), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)>;
G:=Group( (2,8,5)(3,6,9)(11,14,17)(12,18,15), (1,4,7)(2,5,8)(3,6,9)(10,16,13)(11,17,14)(12,18,15), (1,2,3,4,5,6,7,8,9)(10,11,12,13,14,15,16,17,18), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10) );
G=PermutationGroup([[(2,8,5),(3,6,9),(11,14,17),(12,18,15)], [(1,4,7),(2,5,8),(3,6,9),(10,16,13),(11,17,14),(12,18,15)], [(1,2,3,4,5,6,7,8,9),(10,11,12,13,14,15,16,17,18)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10)]])
G:=TransitiveGroup(18,87);
►On 27 points - transitive group 27T65
Generators in S27
(2 12 19)(3 20 13)(5 15 22)(6 23 16)(8 18 25)(9 26 10)
(1 27 11)(2 19 12)(3 20 13)(4 21 14)(5 22 15)(6 23 16)(7 24 17)(8 25 18)(9 26 10)
(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 11)(12 18)(13 17)(14 16)(19 25)(20 24)(21 23)(26 27)
G:=sub<Sym(27)| (2,12,19)(3,20,13)(5,15,22)(6,23,16)(8,18,25)(9,26,10), (1,27,11)(2,19,12)(3,20,13)(4,21,14)(5,22,15)(6,23,16)(7,24,17)(8,25,18)(9,26,10), (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)(10,11)(12,18)(13,17)(14,16)(19,25)(20,24)(21,23)(26,27)>;
G:=Group( (2,12,19)(3,20,13)(5,15,22)(6,23,16)(8,18,25)(9,26,10), (1,27,11)(2,19,12)(3,20,13)(4,21,14)(5,22,15)(6,23,16)(7,24,17)(8,25,18)(9,26,10), (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)(10,11)(12,18)(13,17)(14,16)(19,25)(20,24)(21,23)(26,27) );
G=PermutationGroup([[(2,12,19),(3,20,13),(5,15,22),(6,23,16),(8,18,25),(9,26,10)], [(1,27,11),(2,19,12),(3,20,13),(4,21,14),(5,22,15),(6,23,16),(7,24,17),(8,25,18),(9,26,10)], [(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),(10,11),(12,18),(13,17),(14,16),(19,25),(20,24),(21,23),(26,27)]])
G:=TransitiveGroup(27,65);
C32⋊2D9 is a maximal subgroup of
C32⋊D18 C32⋊C9.S3 C32⋊C9⋊C6 C3.3C3≀S3 C32⋊C9.C6 C33.(C3×S3) C32⋊2D9.C3 C33⋊1D9 (C3×C9)⋊D9 (C3×C9)⋊3D9 C92⋊4S3 C34.7S3 (C32×C9)⋊S3 C9⋊C9⋊2S3 C92⋊6S3 C92⋊5S3 C33⋊6D9 He3⋊4D9
C32⋊2D9 is a maximal quotient of
C32⋊2Dic9 C3.2(C9⋊D9) C32⋊2D27 C33⋊2D9 (C3×C9)⋊5D9 (C3×C9)⋊6D9 C33.D9 He3⋊2D9 3- 1+2⋊D9 He3.3D9 He3.4D9 C33⋊6D9
Matrix representation of C32⋊2D9 ►in GL5(𝔽19)
| 16 | 11 | 0 | 0 | 0 |
| 8 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 7 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 7 |
| 0 | 18 | 0 | 0 | 0 |
| 1 | 3 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 7 | 0 |
| 0 | 0 | 0 | 0 | 11 |
| 8 | 2 | 0 | 0 | 0 |
| 16 | 11 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 0 | 0 | 0 | 12 | 0 |
G:=sub<GL(5,GF(19))| [16,8,0,0,0,11,2,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,7,0,0,0,0,0,7],[0,1,0,0,0,18,3,0,0,0,0,0,1,0,0,0,0,0,7,0,0,0,0,0,11],[8,16,0,0,0,2,11,0,0,0,0,0,18,0,0,0,0,0,0,12,0,0,0,8,0] >;
C32⋊2D9 in GAP, Magma, Sage, TeX
C_3^2\rtimes_2D_9
% in TeX
G:=Group("C3^2:2D9"); // GroupNames label
G:=SmallGroup(162,17);
// by ID
G=gap.SmallGroup(162,17);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,221,186,182,457,723]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^9=d^2=1,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d=a^-1*b,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C32⋊2D9 in TeX
Character table of C32⋊2D9 in TeX