metabelian, supersoluble, monomial
Aliases: C12⋊1S3, C3⋊1D12, C32⋊5D4, C6.14D6, C4⋊(C3⋊S3), (C3×C12)⋊1C2, (C3×C6).13C22, (C2×C3⋊S3)⋊2C2, C2.4(C2×C3⋊S3), SmallGroup(72,33)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊S3
G = < a,b,c | a12=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >
Character table of C12⋊S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4 | 6A | 6B | 6C | 6D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 1 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | 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 | linear of order 2 |
| ρ5 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | -1 | -1 | -1 | 2 | -2 | 1 | 1 | -2 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ7 | 2 | 2 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -2 | -2 | orthogonal lifted from D6 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | -1 | 2 | -1 | -1 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 2 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | -1 | 1 | -2 | 1 | 1 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ13 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ14 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -2 | 1 | 1 | 1 | -√3 | 0 | √3 | √3 | -√3 | 0 | √3 | -√3 | orthogonal lifted from D12 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 1 | 1 | -2 | 1 | √3 | √3 | √3 | -√3 | -√3 | -√3 | 0 | 0 | orthogonal lifted from D12 |
| ρ16 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 1 | -2 | 1 | 1 | -√3 | √3 | 0 | √3 | 0 | -√3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 2 | -1 | -1 | 0 | 1 | 1 | -2 | 1 | -√3 | -√3 | -√3 | √3 | √3 | √3 | 0 | 0 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 1 | 1 | 1 | -2 | 0 | √3 | -√3 | 0 | √3 | -√3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | 0 | 0 | -1 | -1 | 2 | -1 | 0 | 1 | 1 | 1 | -2 | 0 | -√3 | √3 | 0 | -√3 | √3 | -√3 | √3 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | 0 | 1 | -2 | 1 | 1 | √3 | -√3 | 0 | -√3 | 0 | √3 | √3 | -√3 | orthogonal lifted from D12 |
| ρ21 | 2 | -2 | 0 | 0 | -1 | -1 | -1 | 2 | 0 | -2 | 1 | 1 | 1 | √3 | 0 | -√3 | -√3 | √3 | 0 | -√3 | √3 | orthogonal lifted from D12 |
►On 36 points
(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)
(1 35 13)(2 36 14)(3 25 15)(4 26 16)(5 27 17)(6 28 18)(7 29 19)(8 30 20)(9 31 21)(10 32 22)(11 33 23)(12 34 24)
(2 12)(3 11)(4 10)(5 9)(6 8)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)(21 27)(22 26)(23 25)(24 36)
G:=sub<Sym(36)| (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), (1,35,13)(2,36,14)(3,25,15)(4,26,16)(5,27,17)(6,28,18)(7,29,19)(8,30,20)(9,31,21)(10,32,22)(11,33,23)(12,34,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36)>;
G:=Group( (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), (1,35,13)(2,36,14)(3,25,15)(4,26,16)(5,27,17)(6,28,18)(7,29,19)(8,30,20)(9,31,21)(10,32,22)(11,33,23)(12,34,24), (2,12)(3,11)(4,10)(5,9)(6,8)(13,35)(14,34)(15,33)(16,32)(17,31)(18,30)(19,29)(20,28)(21,27)(22,26)(23,25)(24,36) );
G=PermutationGroup([[(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)], [(1,35,13),(2,36,14),(3,25,15),(4,26,16),(5,27,17),(6,28,18),(7,29,19),(8,30,20),(9,31,21),(10,32,22),(11,33,23),(12,34,24)], [(2,12),(3,11),(4,10),(5,9),(6,8),(13,35),(14,34),(15,33),(16,32),(17,31),(18,30),(19,29),(20,28),(21,27),(22,26),(23,25),(24,36)]])
C12⋊S3 is a maximal subgroup of
C3⋊D24 C32⋊5SD16 C24⋊2S3 C32⋊5D8 C32⋊7D8 C32⋊11SD16 D6.6D6 S3×D12 C12.59D6 D4×C3⋊S3 C12.26D6 He3⋊4D4 C36⋊S3 C33⋊8D4 C33⋊12D4 C12⋊S4 C12.7S4 C15⋊D12 C60⋊S3
C12⋊S3 is a maximal quotient of
C24⋊2S3 C32⋊5D8 C32⋊5Q16 C12⋊Dic3 C6.11D12 C36⋊S3 He3⋊5D4 C33⋊8D4 C33⋊12D4 C12⋊S4 C15⋊D12 C60⋊S3
Matrix representation of C12⋊S3 ►in GL4(𝔽13) generated by
| 0 | 1 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 6 | 3 |
| 0 | 0 | 10 | 3 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(13))| [0,12,0,0,1,12,0,0,0,0,6,10,0,0,3,3],[1,0,0,0,0,1,0,0,0,0,12,1,0,0,12,0],[1,12,0,0,0,12,0,0,0,0,12,0,0,0,12,1] >;
C12⋊S3 in GAP, Magma, Sage, TeX
C_{12}\rtimes S_3 % in TeX
G:=Group("C12:S3"); // GroupNames label
G:=SmallGroup(72,33);
// by ID
G=gap.SmallGroup(72,33);
# by ID
G:=PCGroup([5,-2,-2,-2,-3,-3,61,26,323,1204]);
// Polycyclic
G:=Group<a,b,c|a^12=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
Export
Subgroup lattice of C12⋊S3 in TeX
Character table of C12⋊S3 in TeX