non-abelian, supersoluble, monomial, rational
Aliases: C32:D6, He3:C22, C3:S3:S3, C3.2S32, C32:C6:C2, He3:C2:C2, SmallGroup(108,17)
Series: Derived ►Chief ►Lower central ►Upper central
| He3 — C32:D6 |
Generators and relations for C32:D6
G = < a,b,c,d | a3=b3=c6=d2=1, ab=ba, cac-1=dad=a-1b-1, cbc-1=b-1, bd=db, dcd=c-1 >
Quotients: C1, C2, C22, S3, D6, S32, C32:D6
Character table of C32:D6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 6A | 6B | 6C | |
| size | 1 | 9 | 9 | 9 | 2 | 6 | 6 | 12 | 18 | 18 | 18 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | linear of order 2 |
| ρ5 | 2 | -2 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | 1 | 0 | orthogonal lifted from D6 |
| ρ6 | 2 | 0 | -2 | 0 | 2 | -1 | 2 | -1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | 0 | -1 | 0 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | 0 | 2 | -1 | 2 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ9 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | 1 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ10 | 6 | 0 | 0 | -2 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | orthogonal faithful |
| ρ11 | 6 | 0 | 0 | 2 | -3 | 0 | 0 | 0 | 0 | 0 | -1 | orthogonal faithful |
►On 9 points - transitive group 9T18
Generators in S9
(1 7 4)(2 5 6)(3 9 8)
(1 3 2)(4 8 6)(5 7 9)
(2 3)(4 5 6 7 8 9)
(4 9)(5 8)(6 7)
G:=sub<Sym(9)| (1,7,4)(2,5,6)(3,9,8), (1,3,2)(4,8,6)(5,7,9), (2,3)(4,5,6,7,8,9), (4,9)(5,8)(6,7)>;
G:=Group( (1,7,4)(2,5,6)(3,9,8), (1,3,2)(4,8,6)(5,7,9), (2,3)(4,5,6,7,8,9), (4,9)(5,8)(6,7) );
G=PermutationGroup([[(1,7,4),(2,5,6),(3,9,8)], [(1,3,2),(4,8,6),(5,7,9)], [(2,3),(4,5,6,7,8,9)], [(4,9),(5,8),(6,7)]])
G:=TransitiveGroup(9,18);
►On 18 points - transitive group 18T51
Generators in S18
(1 18 11)(2 8 15)(3 10 13)(4 16 7)(5 14 9)(6 12 17)
(1 5 4)(2 3 6)(7 11 9)(8 10 12)(13 17 15)(14 16 18)
(1 2)(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 2)(3 5)(4 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 18)
G:=sub<Sym(18)| (1,18,11)(2,8,15)(3,10,13)(4,16,7)(5,14,9)(6,12,17), (1,5,4)(2,3,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18)>;
G:=Group( (1,18,11)(2,8,15)(3,10,13)(4,16,7)(5,14,9)(6,12,17), (1,5,4)(2,3,6)(7,11,9)(8,10,12)(13,17,15)(14,16,18), (1,2)(3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18) );
G=PermutationGroup([[(1,18,11),(2,8,15),(3,10,13),(4,16,7),(5,14,9),(6,12,17)], [(1,5,4),(2,3,6),(7,11,9),(8,10,12),(13,17,15),(14,16,18)], [(1,2),(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,2),(3,5),(4,6),(7,8),(9,12),(10,11),(13,16),(14,15),(17,18)]])
G:=TransitiveGroup(18,51);
►On 18 points - transitive group 18T55
Generators in S18
(1 7 10)(2 16 13)(3 18 17)(4 14 15)(5 11 12)(6 9 8)
(1 6 5)(2 3 4)(7 9 11)(8 12 10)(13 17 15)(14 16 18)
(3 4)(5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 2)(3 6)(4 5)(7 15)(8 14)(9 13)(10 18)(11 17)(12 16)
G:=sub<Sym(18)| (1,7,10)(2,16,13)(3,18,17)(4,14,15)(5,11,12)(6,9,8), (1,6,5)(2,3,4)(7,9,11)(8,12,10)(13,17,15)(14,16,18), (3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,2)(3,6)(4,5)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16)>;
G:=Group( (1,7,10)(2,16,13)(3,18,17)(4,14,15)(5,11,12)(6,9,8), (1,6,5)(2,3,4)(7,9,11)(8,12,10)(13,17,15)(14,16,18), (3,4)(5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,2)(3,6)(4,5)(7,15)(8,14)(9,13)(10,18)(11,17)(12,16) );
G=PermutationGroup([[(1,7,10),(2,16,13),(3,18,17),(4,14,15),(5,11,12),(6,9,8)], [(1,6,5),(2,3,4),(7,9,11),(8,12,10),(13,17,15),(14,16,18)], [(3,4),(5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,2),(3,6),(4,5),(7,15),(8,14),(9,13),(10,18),(11,17),(12,16)]])
G:=TransitiveGroup(18,55);
►On 18 points - transitive group 18T56
Generators in S18
(1 18 16)(2 10 6)(3 7 5)(4 13 15)(8 17 12)(9 14 11)
(1 7 14)(2 15 8)(3 9 16)(4 17 10)(5 11 18)(6 13 12)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(4 6)(7 9)(10 12)(13 17)(14 16)
G:=sub<Sym(18)| (1,18,16)(2,10,6)(3,7,5)(4,13,15)(8,17,12)(9,14,11), (1,7,14)(2,15,8)(3,9,16)(4,17,10)(5,11,18)(6,13,12), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,3)(4,6)(7,9)(10,12)(13,17)(14,16)>;
G:=Group( (1,18,16)(2,10,6)(3,7,5)(4,13,15)(8,17,12)(9,14,11), (1,7,14)(2,15,8)(3,9,16)(4,17,10)(5,11,18)(6,13,12), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,3)(4,6)(7,9)(10,12)(13,17)(14,16) );
G=PermutationGroup([[(1,18,16),(2,10,6),(3,7,5),(4,13,15),(8,17,12),(9,14,11)], [(1,7,14),(2,15,8),(3,9,16),(4,17,10),(5,11,18),(6,13,12)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,3),(4,6),(7,9),(10,12),(13,17),(14,16)]])
G:=TransitiveGroup(18,56);
►On 18 points - transitive group 18T57
Generators in S18
(2 11 16)(3 12 17)(5 13 8)(6 14 9)
(1 15 10)(2 11 16)(3 17 12)(4 7 18)(5 13 8)(6 9 14)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)
(1 3)(4 6)(7 9)(10 12)(14 18)(15 17)
G:=sub<Sym(18)| (2,11,16)(3,12,17)(5,13,8)(6,14,9), (1,15,10)(2,11,16)(3,17,12)(4,7,18)(5,13,8)(6,9,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,3)(4,6)(7,9)(10,12)(14,18)(15,17)>;
G:=Group( (2,11,16)(3,12,17)(5,13,8)(6,14,9), (1,15,10)(2,11,16)(3,17,12)(4,7,18)(5,13,8)(6,9,14), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18), (1,3)(4,6)(7,9)(10,12)(14,18)(15,17) );
G=PermutationGroup([[(2,11,16),(3,12,17),(5,13,8),(6,14,9)], [(1,15,10),(2,11,16),(3,17,12),(4,7,18),(5,13,8),(6,9,14)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18)], [(1,3),(4,6),(7,9),(10,12),(14,18),(15,17)]])
G:=TransitiveGroup(18,57);
►On 27 points - transitive group 27T29
Generators in S27
(1 21 18)(2 8 5)(3 11 14)(4 12 24)(6 20 23)(7 27 15)(9 26 17)(10 25 19)(13 16 22)
(1 24 27)(2 22 25)(3 26 23)(4 15 21)(5 16 10)(6 11 17)(7 18 12)(8 13 19)(9 20 14)
(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)
(2 3)(4 18)(5 17)(6 16)(7 21)(8 20)(9 19)(10 11)(12 15)(13 14)(22 26)(23 25)
G:=sub<Sym(27)| (1,21,18)(2,8,5)(3,11,14)(4,12,24)(6,20,23)(7,27,15)(9,26,17)(10,25,19)(13,16,22), (1,24,27)(2,22,25)(3,26,23)(4,15,21)(5,16,10)(6,11,17)(7,18,12)(8,13,19)(9,20,14), (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), (2,3)(4,18)(5,17)(6,16)(7,21)(8,20)(9,19)(10,11)(12,15)(13,14)(22,26)(23,25)>;
G:=Group( (1,21,18)(2,8,5)(3,11,14)(4,12,24)(6,20,23)(7,27,15)(9,26,17)(10,25,19)(13,16,22), (1,24,27)(2,22,25)(3,26,23)(4,15,21)(5,16,10)(6,11,17)(7,18,12)(8,13,19)(9,20,14), (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), (2,3)(4,18)(5,17)(6,16)(7,21)(8,20)(9,19)(10,11)(12,15)(13,14)(22,26)(23,25) );
G=PermutationGroup([[(1,21,18),(2,8,5),(3,11,14),(4,12,24),(6,20,23),(7,27,15),(9,26,17),(10,25,19),(13,16,22)], [(1,24,27),(2,22,25),(3,26,23),(4,15,21),(5,16,10),(6,11,17),(7,18,12),(8,13,19),(9,20,14)], [(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)], [(2,3),(4,18),(5,17),(6,16),(7,21),(8,20),(9,19),(10,11),(12,15),(13,14),(22,26),(23,25)]])
G:=TransitiveGroup(27,29);
C32:D6 is a maximal subgroup of
He3:D4 He3:D6 He3.D6 He3.2D6 He3:5D6 He3:6D6 He3.6D6 C62:5D6 AGL2(F3)
C32:D6 is a maximal quotient of
He3:2Q8 C6.S32 He3:2D4 He3:(C2xC4) He3:3D4 C32:D18 He3:D6 He3.D6 He3.2D6 He3:5D6 He3:6D6 C62:5D6
| action | f(x) | Disc(f) |
|---|---|---|
| 9T18 | x9-3x8-39x7+167x6-24x5-480x4+136x3+384x2+144x+16 | 231·312·76·373 |
Matrix representation of C32:D6 ►in GL6(Z)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | -1 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | -1 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| -1 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(6,Integers())| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,1,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0],[0,0,0,0,1,-1,0,0,0,0,0,-1,1,-1,0,0,0,0,0,-1,0,0,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0],[0,0,-1,0,0,0,0,0,0,-1,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;
C32:D6 in GAP, Magma, Sage, TeX
C_3^2\rtimes D_6
% in TeX
G:=Group("C3^2:D6"); // GroupNames label
G:=SmallGroup(108,17);
// by ID
G=gap.SmallGroup(108,17);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,67,483,253,1804,909]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^6=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C32:D6 in TeX
Character table of C32:D6 in TeX