metabelian, supersoluble, monomial, A-group
Aliases: D21⋊S3, C21⋊3D6, C32⋊2D14, C7⋊2S32, C3⋊S3⋊D7, C3⋊3(S3×D7), (C3×D21)⋊3C2, (C3×C21)⋊4C22, (C7×C3⋊S3)⋊2C2, SmallGroup(252,37)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C21 — D21⋊S3 |
Generators and relations for D21⋊S3
G = < a,b,c,d | a21=b2=c3=d2=1, bab=a-1, ac=ca, dad=a8, bc=cb, dbd=a7b, dcd=c-1 >
Character table of D21⋊S3
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 6A | 6B | 7A | 7B | 7C | 14A | 14B | 14C | 21A | 21B | 21C | 21D | 21E | 21F | 21G | 21H | 21I | 21J | 21K | 21L | |
| size | 1 | 9 | 21 | 21 | 2 | 2 | 4 | 42 | 42 | 2 | 2 | 2 | 18 | 18 | 18 | 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 | 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 | 2 | 0 | 2 | 0 | 2 | -1 | -1 | -1 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 0 | -2 | -1 | 2 | -1 | 0 | 1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ7 | 2 | 0 | 0 | 2 | -1 | 2 | -1 | 0 | -1 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | -2 | 0 | 2 | -1 | -1 | 1 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | orthogonal lifted from D6 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | orthogonal lifted from D7 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | orthogonal lifted from D7 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | ζ74+ζ73 | orthogonal lifted from D14 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ76+ζ7 | ζ75+ζ72 | ζ74+ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ75+ζ72 | orthogonal lifted from D14 |
| ρ13 | 2 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | orthogonal lifted from D14 |
| ρ14 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | ζ74+ζ73 | ζ76+ζ7 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ76+ζ7 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ75+ζ72 | ζ74+ζ73 | ζ76+ζ7 | ζ76+ζ7 | orthogonal lifted from D7 |
| ρ15 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 4 | 4 | 4 | 0 | 0 | 0 | 1 | 1 | 1 | -2 | -2 | -2 | 1 | 1 | -2 | -2 | -2 | 1 | orthogonal lifted from S32 |
| ρ16 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ17 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ18 | 4 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ19 | 4 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | 2ζ76+2ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75-ζ72 | orthogonal lifted from S3×D7 |
| ρ20 | 4 | 0 | 0 | 0 | 4 | -2 | -2 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | -ζ76-ζ7 | orthogonal lifted from S3×D7 |
| ρ21 | 4 | 0 | 0 | 0 | -2 | 4 | -2 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 2ζ76+2ζ7 | 2ζ74+2ζ73 | 2ζ75+2ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74-ζ73 | orthogonal lifted from S3×D7 |
| ρ22 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | -ζ76+2ζ7 | -ζ74+2ζ73 | -ζ75+2ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | 2ζ76-ζ7 | 2ζ75-ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | 2ζ74-ζ73 | complex faithful |
| ρ23 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | -ζ74+2ζ73 | -ζ75+2ζ72 | 2ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | 2ζ74-ζ73 | -ζ76+2ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | 2ζ75-ζ72 | complex faithful |
| ρ24 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 0 | 0 | 0 | 2ζ76-ζ7 | 2ζ74-ζ73 | 2ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76+2ζ7 | -ζ75+2ζ72 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ74+2ζ73 | complex faithful |
| ρ25 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | 2ζ75-ζ72 | -ζ76+2ζ7 | -ζ74+2ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | -ζ75+2ζ72 | 2ζ74-ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | 2ζ76-ζ7 | complex faithful |
| ρ26 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 2ζ74+2ζ73 | 0 | 0 | 0 | 2ζ74-ζ73 | 2ζ75-ζ72 | -ζ76+2ζ7 | -ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74+2ζ73 | 2ζ76-ζ7 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ75-ζ72 | -ζ75+2ζ72 | complex faithful |
| ρ27 | 4 | 0 | 0 | 0 | -2 | -2 | 1 | 0 | 0 | 2ζ74+2ζ73 | 2ζ76+2ζ7 | 2ζ75+2ζ72 | 0 | 0 | 0 | -ζ75+2ζ72 | 2ζ76-ζ7 | 2ζ74-ζ73 | -ζ75-ζ72 | -ζ76-ζ7 | -ζ74-ζ73 | 2ζ75-ζ72 | -ζ74+2ζ73 | -ζ75-ζ72 | -ζ74-ζ73 | -ζ76-ζ7 | -ζ76+2ζ7 | complex faithful |
►On 42 points
Generators in S42
(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)
(1 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 42)(10 41)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)(21 30)
(1 8 15)(2 9 16)(3 10 17)(4 11 18)(5 12 19)(6 13 20)(7 14 21)(22 36 29)(23 37 30)(24 38 31)(25 39 32)(26 40 33)(27 41 34)(28 42 35)
(2 9)(3 17)(5 12)(6 20)(8 15)(11 18)(14 21)(22 29)(23 37)(25 32)(26 40)(28 35)(31 38)(34 41)
G:=sub<Sym(42)| (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30), (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (2,9)(3,17)(5,12)(6,20)(8,15)(11,18)(14,21)(22,29)(23,37)(25,32)(26,40)(28,35)(31,38)(34,41)>;
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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,42)(10,41)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30), (1,8,15)(2,9,16)(3,10,17)(4,11,18)(5,12,19)(6,13,20)(7,14,21)(22,36,29)(23,37,30)(24,38,31)(25,39,32)(26,40,33)(27,41,34)(28,42,35), (2,9)(3,17)(5,12)(6,20)(8,15)(11,18)(14,21)(22,29)(23,37)(25,32)(26,40)(28,35)(31,38)(34,41) );
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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,42),(10,41),(11,40),(12,39),(13,38),(14,37),(15,36),(16,35),(17,34),(18,33),(19,32),(20,31),(21,30)], [(1,8,15),(2,9,16),(3,10,17),(4,11,18),(5,12,19),(6,13,20),(7,14,21),(22,36,29),(23,37,30),(24,38,31),(25,39,32),(26,40,33),(27,41,34),(28,42,35)], [(2,9),(3,17),(5,12),(6,20),(8,15),(11,18),(14,21),(22,29),(23,37),(25,32),(26,40),(28,35),(31,38),(34,41)]])
Matrix representation of D21⋊S3 ►in GL6(𝔽43)
| 16 | 0 | 0 | 0 | 0 | 0 |
| 8 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 0 | 0 | 0 | 0 | 1 | 42 |
| 30 | 39 | 0 | 0 | 0 | 0 |
| 42 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 42 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 42 |
| 0 | 0 | 0 | 0 | 0 | 42 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 1 | 0 | 0 |
| 0 | 0 | 42 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 42 | 0 | 0 | 0 | 0 | 0 |
| 0 | 42 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(43))| [16,8,0,0,0,0,0,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,42,42],[30,42,0,0,0,0,39,13,0,0,0,0,0,0,42,0,0,0,0,0,0,42,0,0,0,0,0,0,1,0,0,0,0,0,42,42],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,42,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[42,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
D21⋊S3 in GAP, Magma, Sage, TeX
D_{21}\rtimes S_3 % in TeX
G:=Group("D21:S3"); // GroupNames label
G:=SmallGroup(252,37);
// by ID
G=gap.SmallGroup(252,37);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-7,122,67,248,5404]);
// Polycyclic
G:=Group<a,b,c,d|a^21=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^8,b*c=c*b,d*b*d=a^7*b,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of D21⋊S3 in TeX
Character table of D21⋊S3 in TeX