direct product, metabelian, supersoluble, monomial
Aliases: S3×Dic6, D6.7D6, C12.20D6, Dic3.5D6, C4.5S32, (C3×S3)⋊Q8, C3⋊2(S3×Q8), (C4×S3).1S3, C32⋊2(C2×Q8), C3⋊1(C2×Dic6), (S3×Dic3).C2, (S3×C12).2C2, (C3×Dic6)⋊4C2, C32⋊2Q8⋊2C2, (C3×C6).1C23, C6.1(C22×S3), C32⋊4Q8⋊4C2, (S3×C6).5C22, (C3×C12).16C22, C3⋊Dic3.4C22, (C3×Dic3).1C22, C2.4(C2×S32), SmallGroup(144,137)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×Dic6
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, C2×C4, Q8, C32, Dic3, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, C3×S3, C3×C6, Dic6, Dic6, C4×S3, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, S3×Dic3, C32⋊2Q8, C3×Dic6, S3×C12, C32⋊4Q8, S3×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, Dic6, C22×S3, S32, C2×Dic6, S3×Q8, C2×S32, S3×Dic6
Character table of S3×Dic6
| 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 C2×S32 |
| 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 S3×Q8, 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)]])
S3×Dic6 is a maximal subgroup of
C24.3D6 Dic12⋊S3 Dic6.19D6 D12.11D6 D12.33D6 D12.34D6 Dic6.24D6 D12.25D6 S32×Q8 C3⋊S3⋊Dic6 C12.85S32 C33⋊5(C2×Q8) C3⋊S3⋊4Dic6
S3×Dic6 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(C2×Q8) C3⋊S3⋊4Dic6
Matrix representation of S3×Dic6 ►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] >;
S3×Dic6 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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