metabelian, supersoluble, monomial
Aliases: C32:1D9, C33.1S3, C9:S3:1C3, C3.(C9:C6), (C3xC9):1C6, C32:C9:2C2, C3.1(C3xD9), C3.1(C32:C6), C32.13(C3xS3), SmallGroup(162,5)
Series: Derived ►Chief ►Lower central ►Upper central
C3xC9 — C32:D9 |
Generators and relations for C32:D9
G = < a,b,c,d | a3=b3=c9=d2=1, cac-1=dad=ab=ba, bc=cb, dbd=b-1, dcd=c-1 >
Character table of C32:D9
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 | 2 | 2 | 2 | 2 | 3 | 3 | 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 | 1 | -1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ6 | ζ65 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 6 |
ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | 1 | 1 | ζ32 | ζ32 | ζ32 | 1 | ζ3 | ζ3 | linear of order 3 |
ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ65 | ζ6 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 6 |
ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | 1 | 1 | ζ3 | ζ3 | ζ3 | 1 | ζ32 | ζ32 | linear of order 3 |
ρ7 | 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 |
ρ8 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
ρ9 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
ρ10 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 0 | 0 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
ρ11 | 2 | 0 | 2 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | -1+√-3 | -1-√-3 | 0 | 0 | ζ6 | -1 | -1 | ζ65 | ζ65 | ζ65 | -1 | ζ6 | ζ6 | complex lifted from C3xS3 |
ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | ζ95+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ98+ζ9 | ζ98+ζ97 | ζ94+ζ92 | complex lifted from C3xD9 |
ρ13 | 2 | 0 | 2 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | -1-√-3 | -1+√-3 | 0 | 0 | ζ65 | -1 | -1 | ζ6 | ζ6 | ζ6 | -1 | ζ65 | ζ65 | complex lifted from C3xS3 |
ρ14 | 2 | 0 | 2 | -1 | -1 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | ζ98+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ9 | ζ92+ζ9 | ζ97+ζ95 | complex lifted from C3xD9 |
ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | ζ97+ζ95 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ94 | ζ98+ζ94 | ζ92+ζ9 | complex lifted from C3xD9 |
ρ16 | 2 | 0 | 2 | -1 | -1 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | ζ98+ζ97 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ92 | ζ94+ζ92 | ζ95+ζ9 | complex lifted from C3xD9 |
ρ17 | 2 | 0 | 2 | -1 | -1 | -1 | -1-√-3 | -1+√-3 | ζ6 | ζ65 | 0 | 0 | ζ94+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ95+ζ94 | ζ95+ζ9 | ζ98+ζ97 | complex lifted from C3xD9 |
ρ18 | 2 | 0 | 2 | -1 | -1 | -1 | -1+√-3 | -1-√-3 | ζ65 | ζ6 | 0 | 0 | ζ92+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ97+ζ92 | ζ97+ζ95 | ζ98+ζ94 | complex lifted from C3xD9 |
ρ19 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9:C6 |
ρ20 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32:C6 |
ρ21 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9:C6 |
(2 26 10)(3 11 27)(5 20 13)(6 14 21)(8 23 16)(9 17 24)
(1 25 18)(2 26 10)(3 27 11)(4 19 12)(5 20 13)(6 21 14)(7 22 15)(8 23 16)(9 24 17)
(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 23)(11 22)(12 21)(13 20)(14 19)(15 27)(16 26)(17 25)(18 24)
G:=sub<Sym(27)| (2,26,10)(3,11,27)(5,20,13)(6,14,21)(8,23,16)(9,17,24), (1,25,18)(2,26,10)(3,27,11)(4,19,12)(5,20,13)(6,21,14)(7,22,15)(8,23,16)(9,24,17), (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)(10,23)(11,22)(12,21)(13,20)(14,19)(15,27)(16,26)(17,25)(18,24)>;
G:=Group( (2,26,10)(3,11,27)(5,20,13)(6,14,21)(8,23,16)(9,17,24), (1,25,18)(2,26,10)(3,27,11)(4,19,12)(5,20,13)(6,21,14)(7,22,15)(8,23,16)(9,24,17), (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)(10,23)(11,22)(12,21)(13,20)(14,19)(15,27)(16,26)(17,25)(18,24) );
G=PermutationGroup([[(2,26,10),(3,11,27),(5,20,13),(6,14,21),(8,23,16),(9,17,24)], [(1,25,18),(2,26,10),(3,27,11),(4,19,12),(5,20,13),(6,21,14),(7,22,15),(8,23,16),(9,24,17)], [(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),(10,23),(11,22),(12,21),(13,20),(14,19),(15,27),(16,26),(17,25),(18,24)]])
G:=TransitiveGroup(27,59);
C32:D9 is a maximal subgroup of
C32:D18 (C3xHe3):S3 (C3xHe3).S3 C33.(C3:S3) C32:C9:6S3 C3.(C33:S3) C3.(He3:S3) C32:C9.10S3 C33:2D9 (C3xC9):5D9 (C3xC9):6D9 He3:2D9 3- 1+2:D9 C34.S3 C33:D9 C92:3C6 He3:3D9 C92:9C6 C9:He3:2C2 (C32xC9):C6 C92:10C6 C92:4C6 C92:5C6 C92:11C6
C32:D9 is a maximal quotient of
C32:Dic9 C9:S3:C9 C32:D27 C33:1D9 (C3xC9):D9 (C3xC9):3D9 He3:D9 He3.D9 He3.2D9 C33:D9
Matrix representation of C32:D9 ►in GL8(F19)
7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 18 | 18 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 18 | 18 |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 18 | 18 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 18 | 18 |
2 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
5 | 7 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
2 | 14 | 0 | 0 | 0 | 0 | 0 | 0 |
12 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 18 | 18 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 18 | 18 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(19))| [7,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[2,5,0,0,0,0,0,0,14,7,0,0,0,0,0,0,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[2,12,0,0,0,0,0,0,14,17,0,0,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,1,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
C32:D9 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_9
% in TeX
G:=Group("C3^2:D9");
// GroupNames label
G:=SmallGroup(162,5);
// by ID
G=gap.SmallGroup(162,5);
# by ID
G:=PCGroup([5,-2,-3,-3,-3,-3,992,187,282,723,2704]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^9=d^2=1,c*a*c^-1=d*a*d=a*b=b*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C32:D9 in TeX
Character table of C32:D9 in TeX