direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×Dic3, D6.S3, C6.1D6, (C3×S3)⋊C4, C2.1S32, C3⋊3(C4×S3), (S3×C6).C2, C32⋊2(C2×C4), C3⋊Dic3⋊1C2, C3⋊1(C2×Dic3), (C3×Dic3)⋊2C2, (C3×C6).1C22, SmallGroup(72,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — S3×Dic3 |
Generators and relations for S3×Dic3
G = < a,b,c,d | a3=b2=c6=1, d2=c3, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Character table of S3×Dic3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 12A | 12B | |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 3 | 3 | 9 | 9 | 2 | 2 | 4 | 6 | 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 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -i | i | -i | i | -1 | -1 | -1 | 1 | -1 | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | i | -i | -i | i | -1 | -1 | -1 | -1 | 1 | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -i | i | i | -i | -1 | -1 | -1 | -1 | 1 | i | -i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | i | -i | i | -i | -1 | -1 | -1 | 1 | -1 | -i | i | linear of order 4 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | -2 | -2 | 2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 1 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ14 | 2 | -2 | 2 | -2 | -1 | 2 | -1 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -1 | 0 | 0 | symplectic lifted from Dic3, Schur index 2 |
| ρ15 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | 0 | 0 | i | -i | complex lifted from C4×S3 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | 0 | 0 | -i | i | complex lifted from C4×S3 |
| ρ17 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ18 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 24 points - transitive group 24T60
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 15 17)(14 16 18)(19 23 21)(20 24 22)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)
(1 8 4 11)(2 7 5 10)(3 12 6 9)(13 20 16 23)(14 19 17 22)(15 24 18 21)
G:=sub<Sym(24)| (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,20,16,23)(14,19,17,22)(15,24,18,21)>;
G:=Group( (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,15,17)(14,16,18)(19,23,21)(20,24,22), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24), (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,8,4,11)(2,7,5,10)(3,12,6,9)(13,20,16,23)(14,19,17,22)(15,24,18,21) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,15,17),(14,16,18),(19,23,21),(20,24,22)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24)], [(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24)], [(1,8,4,11),(2,7,5,10),(3,12,6,9),(13,20,16,23),(14,19,17,22),(15,24,18,21)]])
G:=TransitiveGroup(24,60);
S3×Dic3 is a maximal subgroup of
D12⋊5S3 D12⋊S3 C4×S32 D6.3D6 D6.4D6 C6.S32 He3⋊(C2×C4) C33⋊9(C2×C4) Dic3.4S4 D30.S3
S3×Dic3 is a maximal quotient of
D6.Dic3 D6⋊Dic3 Dic3⋊Dic3 C6.S32 C33⋊9(C2×C4) D30.S3
Matrix representation of S3×Dic3 ►in GL4(𝔽5) generated by
| 0 | 0 | 0 | 4 |
| 0 | 4 | 3 | 0 |
| 0 | 3 | 0 | 0 |
| 1 | 0 | 0 | 4 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 3 | 0 |
| 0 | 3 | 1 | 0 |
| 4 | 0 | 0 | 1 |
| 0 | 3 | 1 | 0 |
| 3 | 0 | 0 | 2 |
| 0 | 0 | 0 | 4 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [0,0,0,1,0,4,3,0,0,3,0,0,4,0,0,4],[0,1,0,0,1,0,0,0,0,0,0,3,0,0,2,0],[0,0,0,4,0,0,3,0,0,3,1,0,1,0,0,1],[0,3,0,0,3,0,0,0,1,0,0,1,0,2,4,0] >;
S3×Dic3 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_3 % in TeX
G:=Group("S3xDic3"); // GroupNames label
G:=SmallGroup(72,20);
// by ID
G=gap.SmallGroup(72,20);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,26,168,1204]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^6=1,d^2=c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations
Export
Subgroup lattice of S3×Dic3 in TeX
Character table of S3×Dic3 in TeX