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## G = D21⋊S3order 252 = 22·32·7

### The semidirect product of D21 and S3 acting via S3/C3=C2

Aliases: D21⋊S3, C213D6, C322D14, C72S32, C3⋊S3⋊D7, C33(S3×D7), (C3×D21)⋊3C2, (C3×C21)⋊4C22, (C7×C3⋊S3)⋊2C2, SmallGroup(252,37)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C21 — D21⋊S3
 Chief series C1 — C7 — C21 — C3×C21 — C3×D21 — D21⋊S3
 Lower central C3×C21 — D21⋊S3
 Upper central C1

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

Smallest permutation representation of D21⋊S3
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`

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