direct product, metabelian, supersoluble, monomial
Aliases: S3xDic6, D6.7D6, C12.20D6, Dic3.5D6, C4.5S32, (C3xS3):Q8, C3:2(S3xQ8), (C4xS3).1S3, C32:2(C2xQ8), C3:1(C2xDic6), (S3xDic3).C2, (S3xC12).2C2, (C3xDic6):4C2, C32:2Q8:2C2, (C3xC6).1C23, C6.1(C22xS3), C32:4Q8:4C2, (S3xC6).5C22, (C3xC12).16C22, C3:Dic3.4C22, (C3xDic3).1C22, C2.4(C2xS32), SmallGroup(144,137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3xDic6
G = < a,b,c,d | a3=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 228 in 82 conjugacy classes, 36 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2xC4, Q8, C32, Dic3, Dic3, Dic3, C12, C12, D6, C2xC6, C2xQ8, C3xS3, C3xC6, Dic6, Dic6, C4xS3, C4xS3, C2xDic3, C2xC12, C3xQ8, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xDic6, S3xQ8, S3xDic3, C32:2Q8, C3xDic6, S3xC12, C32:4Q8, S3xDic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2xQ8, Dic6, C22xS3, S32, C2xDic6, S3xQ8, C2xS32, S3xDic6
Character table of S3xDic6
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 12I | |
| size | 1 | 1 | 3 | 3 | 2 | 2 | 4 | 2 | 6 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | 12 | 12 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | 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 | -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 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | -1 | 1 | 1 | 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 |
| ρ6 | 1 | 1 | -1 | -1 | 1 | 1 | 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 |
| ρ7 | 1 | 1 | -1 | -1 | 1 | 1 | 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 |
| ρ8 | 1 | 1 | -1 | -1 | 1 | 1 | 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 |
| ρ9 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -2 | 0 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | 2 | -2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | 2 | 0 | 0 | -2 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | 2 | -2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | 1 | -1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 0 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | -2 | -2 | 2 | -1 | -1 | -2 | 0 | 0 | 2 | 0 | 0 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -1 | -1 | 0 | 0 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -2 | -2 | 2 | 0 | 0 | 0 | 2 | -1 | -1 | 0 | 0 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | -1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | -2 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -1 | √3 | -√3 | 0 | -√3 | √3 | -√3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ20 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | 1 | -1 | -√3 | √3 | 0 | √3 | -√3 | √3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ21 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 1 | √3 | -√3 | 0 | -√3 | √3 | √3 | -√3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ22 | 2 | -2 | 2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -2 | 1 | -1 | 1 | -√3 | √3 | 0 | √3 | -√3 | -√3 | √3 | 0 | 0 | symplectic lifted from Dic6, Schur index 2 |
| ρ23 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS32 |
| ρ24 | 4 | 4 | 0 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | -2 | -2 | -2 | 1 | 1 | 0 | 0 | 0 | 0 | orthogonal lifted from S32 |
| ρ25 | 4 | -4 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from S3xQ8, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | -2√3 | 2√3 | 0 | -√3 | √3 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 2√3 | -2√3 | 0 | √3 | -√3 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 34)(2 35)(3 36)(4 25)(5 26)(6 27)(7 28)(8 29)(9 30)(10 31)(11 32)(12 33)(13 45)(14 46)(15 47)(16 48)(17 37)(18 38)(19 39)(20 40)(21 41)(22 42)(23 43)(24 44)
(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 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 14 7 20)(2 13 8 19)(3 24 9 18)(4 23 10 17)(5 22 11 16)(6 21 12 15)(25 43 31 37)(26 42 32 48)(27 41 33 47)(28 40 34 46)(29 39 35 45)(30 38 36 44)
G:=sub<Sym(48)| (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44)>;
G:=Group( (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,34)(2,35)(3,36)(4,25)(5,26)(6,27)(7,28)(8,29)(9,30)(10,31)(11,32)(12,33)(13,45)(14,46)(15,47)(16,48)(17,37)(18,38)(19,39)(20,40)(21,41)(22,42)(23,43)(24,44), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,14,7,20)(2,13,8,19)(3,24,9,18)(4,23,10,17)(5,22,11,16)(6,21,12,15)(25,43,31,37)(26,42,32,48)(27,41,33,47)(28,40,34,46)(29,39,35,45)(30,38,36,44) );
G=PermutationGroup([[(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,34),(2,35),(3,36),(4,25),(5,26),(6,27),(7,28),(8,29),(9,30),(10,31),(11,32),(12,33),(13,45),(14,46),(15,47),(16,48),(17,37),(18,38),(19,39),(20,40),(21,41),(22,42),(23,43),(24,44)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,14,7,20),(2,13,8,19),(3,24,9,18),(4,23,10,17),(5,22,11,16),(6,21,12,15),(25,43,31,37),(26,42,32,48),(27,41,33,47),(28,40,34,46),(29,39,35,45),(30,38,36,44)]])
S3xDic6 is a maximal subgroup of
C24.3D6 Dic12:S3 Dic6.19D6 D12.11D6 D12.33D6 D12.34D6 Dic6.24D6 D12.25D6 S32xQ8 C3:S3:Dic6 C12.85S32 C33:5(C2xQ8) C3:S3:4Dic6
S3xDic6 is a maximal quotient of
Dic3:5Dic6 C62.9C23 C62.10C23 Dic3:6Dic6 Dic3.Dic6 C62.16C23 D6:Dic6 D6:6Dic6 D6:7Dic6 Dic3:Dic6 C62.37C23 D6:1Dic6 D6:2Dic6 D6:3Dic6 D6:4Dic6 C12:3Dic6 C3:S3:Dic6 C33:5(C2xQ8) C3:S3:4Dic6
Matrix representation of S3xDic6 ►in GL6(F13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
S3xDic6 in GAP, Magma, Sage, TeX
S_3\times {\rm Dic}_6 % in TeX
G:=Group("S3xDic6"); // GroupNames label
G:=SmallGroup(144,137);
// by ID
G=gap.SmallGroup(144,137);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,116,50,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^12=1,d^2=c^6,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
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