direct product, metabelian, supersoluble, monomial, rational
Aliases: D4×C3⋊S3, C12⋊3D6, C62⋊4C22, C3⋊4(S3×D4), (C2×C6)⋊6D6, (C3×D4)⋊2S3, C12⋊S3⋊6C2, C32⋊11(C2×D4), (C3×C12)⋊4C22, (D4×C32)⋊5C2, C32⋊7D4⋊3C2, C6.34(C22×S3), (C3×C6).33C23, C3⋊Dic3⋊6C22, C4⋊1(C2×C3⋊S3), (C4×C3⋊S3)⋊3C2, C22⋊2(C2×C3⋊S3), (C2×C3⋊S3)⋊6C22, (C22×C3⋊S3)⋊4C2, C2.6(C22×C3⋊S3), SmallGroup(144,172)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C3⋊S3
G = < a,b,c,d,e | a4=b2=c3=d3=e2=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c-1, ede=d-1 >
Subgroups: 610 in 162 conjugacy classes, 49 normal (13 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, D4, D4, C23, C32, Dic3, C12, D6, C2×C6, C2×D4, C3⋊S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C3⋊D4, C3×D4, C22×S3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C2×C3⋊S3, C62, S3×D4, C4×C3⋊S3, C12⋊S3, C32⋊7D4, D4×C32, C22×C3⋊S3, D4×C3⋊S3
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊S3, C22×S3, C2×C3⋊S3, S3×D4, C22×C3⋊S3, D4×C3⋊S3
Character table of D4×C3⋊S3
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 12A | 12B | 12C | 12D | |
| size | 1 | 1 | 2 | 2 | 9 | 9 | 18 | 18 | 2 | 2 | 2 | 2 | 2 | 18 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | -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 | 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 | -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 | 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 | -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 | 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 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | 1 | -1 | -1 | -1 | 2 | 1 | 1 | -2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 1 | -1 | -1 | -1 | 2 | orthogonal lifted from D6 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | -1 | 1 | -2 | 1 | 1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | -2 | 0 | 2 | -1 | -1 | -1 | 1 | -1 | 2 | -1 | -1 | -2 | 1 | 1 | 1 | 1 | 1 | -2 | orthogonal lifted from D6 |
| ρ14 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | orthogonal lifted from S3 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ16 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | 2 | 0 | -1 | -1 | -1 | 2 | -2 | -2 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 2 | -1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | 0 | 0 | -2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | -2 | 2 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ19 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 2 | -1 | -1 | -1 | -2 | 0 | -1 | -1 | -1 | 2 | 2 | -2 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | orthogonal lifted from D6 |
| ρ20 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | -2 | 0 | -1 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | -1 | -1 | 2 | 1 | -2 | 1 | 1 | orthogonal lifted from D6 |
| ρ21 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | -1 | 2 | -1 | 2 | 0 | -1 | -1 | 2 | -1 | 1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | -1 | 2 | -1 | -1 | orthogonal lifted from D6 |
| ρ22 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ23 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ24 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | 2 | 0 | -1 | 2 | -1 | -1 | 1 | 1 | 1 | -2 | 1 | 1 | -2 | 1 | 2 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ25 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | 1 | -1 | -1 | 2 | -1 | 1 | -2 | 1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ26 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | -1 | 2 | -1 | -1 | -2 | 0 | -1 | 2 | -1 | -1 | -1 | 1 | 1 | -2 | 1 | -1 | 2 | -1 | -2 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ27 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 4 | -2 | -2 | 0 | 0 | 2 | -4 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ28 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -2 | -2 | -2 | 0 | 0 | 2 | 2 | 2 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ29 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 4 | -2 | 0 | 0 | 2 | 2 | -4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ30 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 4 | 0 | 0 | -4 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
►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 4)(2 3)(5 8)(6 7)(9 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)
(1 21 29)(2 22 30)(3 23 31)(4 24 32)(5 35 11)(6 36 12)(7 33 9)(8 34 10)(13 28 19)(14 25 20)(15 26 17)(16 27 18)
(1 7 16)(2 8 13)(3 5 14)(4 6 15)(9 18 29)(10 19 30)(11 20 31)(12 17 32)(21 33 27)(22 34 28)(23 35 25)(24 36 26)
(1 3)(2 4)(5 16)(6 13)(7 14)(8 15)(9 25)(10 26)(11 27)(12 28)(17 34)(18 35)(19 36)(20 33)(21 31)(22 32)(23 29)(24 30)
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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,34)(18,35)(19,36)(20,33)(21,31)(22,32)(23,29)(24,30)>;
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), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35), (1,21,29)(2,22,30)(3,23,31)(4,24,32)(5,35,11)(6,36,12)(7,33,9)(8,34,10)(13,28,19)(14,25,20)(15,26,17)(16,27,18), (1,7,16)(2,8,13)(3,5,14)(4,6,15)(9,18,29)(10,19,30)(11,20,31)(12,17,32)(21,33,27)(22,34,28)(23,35,25)(24,36,26), (1,3)(2,4)(5,16)(6,13)(7,14)(8,15)(9,25)(10,26)(11,27)(12,28)(17,34)(18,35)(19,36)(20,33)(21,31)(22,32)(23,29)(24,30) );
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)], [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11),(13,14),(15,16),(17,18),(19,20),(21,24),(22,23),(25,28),(26,27),(29,32),(30,31),(33,36),(34,35)], [(1,21,29),(2,22,30),(3,23,31),(4,24,32),(5,35,11),(6,36,12),(7,33,9),(8,34,10),(13,28,19),(14,25,20),(15,26,17),(16,27,18)], [(1,7,16),(2,8,13),(3,5,14),(4,6,15),(9,18,29),(10,19,30),(11,20,31),(12,17,32),(21,33,27),(22,34,28),(23,35,25),(24,36,26)], [(1,3),(2,4),(5,16),(6,13),(7,14),(8,15),(9,25),(10,26),(11,27),(12,28),(17,34),(18,35),(19,36),(20,33),(21,31),(22,32),(23,29),(24,30)]])
D4×C3⋊S3 is a maximal subgroup of
C3⋊S3.5D8 D12⋊D6 Dic6⋊D6 D12⋊5D6 C24⋊8D6 C24⋊7D6 S32×D4 Dic6⋊12D6 D12⋊13D6 C32⋊82+ 1+4 C62.154C23 C12⋊S32 C62⋊23D6
D4×C3⋊S3 is a maximal quotient of
C62⋊6Q8 C62.225C23 C62⋊12D4 C62.227C23 C62.228C23 C62.229C23 C12⋊2Dic6 C62.237C23 C62.238C23 C12⋊3D12 C62.240C23 C24⋊8D6 C24.26D6 C24⋊7D6 C24.32D6 C24.40D6 C24.35D6 C24.28D6 C62⋊13D4 C62.256C23 C62⋊14D4 C62.258C23 C12⋊S32 C62⋊23D6
Matrix representation of D4×C3⋊S3 ►in GL6(ℤ)
| -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 | -1 | -2 |
| 0 | 0 | 0 | 0 | 1 | 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 | -1 | -2 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 1 | 0 | 0 | 0 | 0 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 1 | 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
G:=sub<GL(6,Integers())| [-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,-1,1,0,0,0,0,-2,1],[-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,-1,0,0,0,0,0,-2,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1] >;
D4×C3⋊S3 in GAP, Magma, Sage, TeX
D_4\times C_3\rtimes S_3
% in TeX
G:=Group("D4xC3:S3"); // GroupNames label
G:=SmallGroup(144,172);
// by ID
G=gap.SmallGroup(144,172);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,116,964,3461]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations
Export