metabelian, supersoluble, monomial
Aliases: D12⋊5S3, D6.1D6, C12.21D6, Dic3.8D6, C4.6S32, (C4×S3)⋊1S3, (S3×C12)⋊2C2, (C3×D12)⋊4C2, D6⋊S3⋊2C2, C3⋊3(C4○D12), (S3×Dic3)⋊4C2, C32⋊1(C4○D4), (C3×C6).2C23, C6.2(C22×S3), C3⋊2(D4⋊2S3), C32⋊4Q8⋊5C2, (S3×C6).1C22, (C3×C12).17C22, C3⋊Dic3.5C22, (C3×Dic3).8C22, C2.5(C2×S32), SmallGroup(144,138)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊5S3
G = < a,b,c,d | a12=b2=c3=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd=a6b, dcd=c-1 >
Subgroups: 260 in 86 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C4○D4, C3×S3, C3×C6, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C4○D12, D4⋊2S3, S3×Dic3, D6⋊S3, S3×C12, C3×D12, C32⋊4Q8, D12⋊5S3
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C22×S3, S32, C4○D12, D4⋊2S3, C2×S32, D12⋊5S3
Character table of D12⋊5S3
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 12A | 12B | 12C | 12D | 12E | 12F | 12G | |
| size | 1 | 1 | 6 | 6 | 6 | 2 | 2 | 4 | 2 | 3 | 3 | 18 | 18 | 2 | 2 | 4 | 6 | 6 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 6 | 6 | |
| 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 | -2 | 2 | 0 | 2 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 1 | -1 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | 0 | 0 | -2 | -1 | 2 | -1 | -2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | -1 | -1 | orthogonal lifted from D6 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from S3 |
| ρ13 | 2 | 2 | 0 | 0 | -2 | -1 | 2 | -1 | 2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | 1 | 1 | 0 | 0 | -1 | -1 | -1 | 2 | -1 | 1 | 1 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | -2 | -2 | -2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | 1 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ15 | 2 | 2 | 2 | -2 | 0 | 2 | -1 | -1 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | -1 | 1 | -2 | -2 | 1 | 1 | 1 | 0 | 0 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 2 | -1 | -1 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 0 | 1 | 1 | 2 | 2 | -1 | -1 | -1 | 0 | 0 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | 2i | -2i | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2i | -2i | complex lifted from C4○D4 |
| ρ18 | 2 | -2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | -2i | 2i | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2i | 2i | complex lifted from C4○D4 |
| ρ19 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | √-3 | -√-3 | 0 | 0 | √3 | -√3 | -√3 | 0 | √3 | -i | i | complex lifted from C4○D12 |
| ρ20 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | √-3 | -√-3 | 0 | 0 | -√3 | √3 | √3 | 0 | -√3 | i | -i | complex lifted from C4○D12 |
| ρ21 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | 2i | -2i | 0 | 0 | -2 | 1 | 1 | -√-3 | √-3 | 0 | 0 | -√3 | √3 | √3 | 0 | -√3 | -i | i | complex lifted from C4○D12 |
| ρ22 | 2 | -2 | 0 | 0 | 0 | -1 | 2 | -1 | 0 | -2i | 2i | 0 | 0 | -2 | 1 | 1 | -√-3 | √-3 | 0 | 0 | √3 | -√3 | -√3 | 0 | √3 | i | -i | complex lifted from C4○D12 |
| ρ23 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | -2 | 1 | 0 | 0 | orthogonal lifted from S32 |
| ρ24 | 4 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | -4 | 0 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 2 | -1 | 0 | 0 | orthogonal lifted from C2×S32 |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from D4⋊2S3, Schur index 2 |
| ρ26 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | -2√3 | 2√3 | -√3 | 0 | √3 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -1 | 0 | 0 | 0 | 0 | 2√3 | -2√3 | √3 | 0 | -√3 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 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)(37 38 39 40 41 42 43 44 45 46 47 48)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 27)(8 26)(9 25)(10 36)(11 35)(12 34)(13 41)(14 40)(15 39)(16 38)(17 37)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 45)(26 46)(27 47)(28 48)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 43)(36 44)
G:=sub<Sym(48)| (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44)>;
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)(37,38,39,40,41,42,43,44,45,46,47,48), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,36)(11,35)(12,34)(13,41)(14,40)(15,39)(16,38)(17,37)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,45)(26,46)(27,47)(28,48)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,43)(36,44) );
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),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,27),(8,26),(9,25),(10,36),(11,35),(12,34),(13,41),(14,40),(15,39),(16,38),(17,37),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,45),(26,46),(27,47),(28,48),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,43),(36,44)]])
D12⋊5S3 is a maximal subgroup of
D24⋊S3 C24.3D6 D6.1D12 D24⋊7S3 D12⋊9D6 D12.22D6 D12.24D6 D12.12D6 D12.34D6 S3×C4○D12 D12⋊24D6 S3×D4⋊2S3 D12⋊12D6 D12.25D6 D12⋊15D6 D12⋊5D9 D36⋊5S3 C12.84S32 C12.S32 D6.S32 D6.4S32 (C3×D12)⋊S3 C12.57S32 C12⋊S3⋊12S3
D12⋊5S3 is a maximal quotient of
C62.11C23 Dic3⋊6Dic6 C62.17C23 D6⋊7Dic6 C62.28C23 C62.29C23 C12.27D12 C62.31C23 C62.39C23 C62.47C23 C62.49C23 C62.54C23 C62.55C23 D6.9D12 Dic3×D12 D6⋊3Dic6 C62.75C23 D6⋊2D12 C62.83C23 D12⋊5D9 D36⋊5S3 C12⋊S3⋊S3 D6.S32 D6.4S32 (C3×D12)⋊S3 C12.57S32 C12⋊S3⋊12S3
Matrix representation of D12⋊5S3 ►in GL6(𝔽13)
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 11 | 0 | 0 | 0 | 0 |
| 6 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 | 12 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [5,0,0,0,0,0,0,8,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,6,0,0,0,0,11,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
D12⋊5S3 in GAP, Magma, Sage, TeX
D_{12}\rtimes_5S_3 % in TeX
G:=Group("D12:5S3"); // GroupNames label
G:=SmallGroup(144,138);
// by ID
G=gap.SmallGroup(144,138);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,55,218,50,490,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^6*b,d*c*d=c^-1>;
// generators/relations
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