non-abelian, supersoluble, monomial
Aliases: C32:2D9, C33.2S3, (C3xC9):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 |
(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);
(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 C3.3C3wrS3 C32:C9.C6 C33.(C3xS3) C32:2D9.C3 C33:1D9 (C3xC9):D9 (C3xC9):3D9 C92:4S3 C34.7S3 (C32xC9):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 (C3xC9):5D9 (C3xC9):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