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## G = C21⋊D4order 168 = 23·3·7

### 1st semidirect product of C21 and D4 acting via D4/C2=C22

Aliases: C211D4, D61D7, D141S3, C6.4D14, C14.4D6, Dic214C2, C42.4C22, (C6×D7)⋊1C2, C72(C3⋊D4), C32(C7⋊D4), C2.4(S3×D7), (S3×C14)⋊1C2, SmallGroup(168,15)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C21⋊D4
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C21⋊D4
 Lower central C21 — C42 — C21⋊D4
 Upper central C1 — C2

Generators and relations for C21⋊D4
G = < a,b,c | a21=b4=c2=1, bab-1=a-1, cac=a13, cbc=b-1 >

Character table of C21⋊D4

 class 1 2A 2B 2C 3 4 6A 6B 6C 7A 7B 7C 14A 14B 14C 14D 14E 14F 14G 14H 14I 21A 21B 21C 42A 42B 42C size 1 1 6 14 2 42 2 14 14 2 2 2 2 2 2 6 6 6 6 6 6 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 2 0 -2 -1 0 -1 1 1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from D6 ρ6 2 2 0 2 -1 0 -1 -1 -1 2 2 2 2 2 2 0 0 0 0 0 0 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ7 2 -2 0 0 2 0 -2 0 0 2 2 2 -2 -2 -2 0 0 0 0 0 0 2 2 2 -2 -2 -2 orthogonal lifted from D4 ρ8 2 2 -2 0 2 0 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ9 2 2 2 0 2 0 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ10 2 2 2 0 2 0 2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ11 2 2 -2 0 2 0 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ12 2 2 -2 0 2 0 2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ13 2 2 2 0 2 0 2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ14 2 -2 0 0 -1 0 1 √-3 -√-3 2 2 2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 complex lifted from C3⋊D4 ρ15 2 -2 0 0 2 0 -2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75+ζ72 ζ75-ζ72 -ζ76+ζ7 ζ74-ζ73 -ζ74+ζ73 ζ76-ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex lifted from C7⋊D4 ρ16 2 -2 0 0 2 0 -2 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ75-ζ72 -ζ75+ζ72 ζ76-ζ7 -ζ74+ζ73 ζ74-ζ73 -ζ76+ζ7 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 complex lifted from C7⋊D4 ρ17 2 -2 0 0 2 0 -2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74+ζ73 ζ74-ζ73 ζ75-ζ72 ζ76-ζ7 -ζ76+ζ7 -ζ75+ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex lifted from C7⋊D4 ρ18 2 -2 0 0 2 0 -2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76+ζ7 ζ76-ζ7 ζ74-ζ73 -ζ75+ζ72 ζ75-ζ72 -ζ74+ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex lifted from C7⋊D4 ρ19 2 -2 0 0 2 0 -2 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ74-ζ73 -ζ74+ζ73 -ζ75+ζ72 -ζ76+ζ7 ζ76-ζ7 ζ75-ζ72 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 complex lifted from C7⋊D4 ρ20 2 -2 0 0 -1 0 1 -√-3 √-3 2 2 2 -2 -2 -2 0 0 0 0 0 0 -1 -1 -1 1 1 1 complex lifted from C3⋊D4 ρ21 2 -2 0 0 2 0 -2 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ76-ζ7 -ζ76+ζ7 -ζ74+ζ73 ζ75-ζ72 -ζ75+ζ72 ζ74-ζ73 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 complex lifted from C7⋊D4 ρ22 4 4 0 0 -2 0 -2 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 0 0 0 0 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from S3×D7 ρ23 4 4 0 0 -2 0 -2 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 0 0 0 0 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from S3×D7 ρ24 4 4 0 0 -2 0 -2 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 0 0 0 0 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from S3×D7 ρ25 4 -4 0 0 -2 0 2 0 0 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 0 0 0 0 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 symplectic faithful, Schur index 2 ρ26 4 -4 0 0 -2 0 2 0 0 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 0 0 0 0 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 symplectic faithful, Schur index 2 ρ27 4 -4 0 0 -2 0 2 0 0 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 0 0 0 0 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 symplectic faithful, Schur index 2

Smallest permutation representation of C21⋊D4
On 84 points
Generators in S84
```(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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63)(64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)
(1 46 38 84)(2 45 39 83)(3 44 40 82)(4 43 41 81)(5 63 42 80)(6 62 22 79)(7 61 23 78)(8 60 24 77)(9 59 25 76)(10 58 26 75)(11 57 27 74)(12 56 28 73)(13 55 29 72)(14 54 30 71)(15 53 31 70)(16 52 32 69)(17 51 33 68)(18 50 34 67)(19 49 35 66)(20 48 36 65)(21 47 37 64)
(2 14)(3 6)(4 19)(5 11)(7 16)(9 21)(10 13)(12 18)(17 20)(22 40)(23 32)(25 37)(26 29)(27 42)(28 34)(30 39)(33 36)(35 41)(43 66)(44 79)(45 71)(46 84)(47 76)(48 68)(49 81)(50 73)(51 65)(52 78)(53 70)(54 83)(55 75)(56 67)(57 80)(58 72)(59 64)(60 77)(61 69)(62 82)(63 74)```

`G:=sub<Sym(84)| (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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,46,38,84)(2,45,39,83)(3,44,40,82)(4,43,41,81)(5,63,42,80)(6,62,22,79)(7,61,23,78)(8,60,24,77)(9,59,25,76)(10,58,26,75)(11,57,27,74)(12,56,28,73)(13,55,29,72)(14,54,30,71)(15,53,31,70)(16,52,32,69)(17,51,33,68)(18,50,34,67)(19,49,35,66)(20,48,36,65)(21,47,37,64), (2,14)(3,6)(4,19)(5,11)(7,16)(9,21)(10,13)(12,18)(17,20)(22,40)(23,32)(25,37)(26,29)(27,42)(28,34)(30,39)(33,36)(35,41)(43,66)(44,79)(45,71)(46,84)(47,76)(48,68)(49,81)(50,73)(51,65)(52,78)(53,70)(54,83)(55,75)(56,67)(57,80)(58,72)(59,64)(60,77)(61,69)(62,82)(63,74)>;`

`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)(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63)(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84), (1,46,38,84)(2,45,39,83)(3,44,40,82)(4,43,41,81)(5,63,42,80)(6,62,22,79)(7,61,23,78)(8,60,24,77)(9,59,25,76)(10,58,26,75)(11,57,27,74)(12,56,28,73)(13,55,29,72)(14,54,30,71)(15,53,31,70)(16,52,32,69)(17,51,33,68)(18,50,34,67)(19,49,35,66)(20,48,36,65)(21,47,37,64), (2,14)(3,6)(4,19)(5,11)(7,16)(9,21)(10,13)(12,18)(17,20)(22,40)(23,32)(25,37)(26,29)(27,42)(28,34)(30,39)(33,36)(35,41)(43,66)(44,79)(45,71)(46,84)(47,76)(48,68)(49,81)(50,73)(51,65)(52,78)(53,70)(54,83)(55,75)(56,67)(57,80)(58,72)(59,64)(60,77)(61,69)(62,82)(63,74) );`

`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),(43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63),(64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)], [(1,46,38,84),(2,45,39,83),(3,44,40,82),(4,43,41,81),(5,63,42,80),(6,62,22,79),(7,61,23,78),(8,60,24,77),(9,59,25,76),(10,58,26,75),(11,57,27,74),(12,56,28,73),(13,55,29,72),(14,54,30,71),(15,53,31,70),(16,52,32,69),(17,51,33,68),(18,50,34,67),(19,49,35,66),(20,48,36,65),(21,47,37,64)], [(2,14),(3,6),(4,19),(5,11),(7,16),(9,21),(10,13),(12,18),(17,20),(22,40),(23,32),(25,37),(26,29),(27,42),(28,34),(30,39),(33,36),(35,41),(43,66),(44,79),(45,71),(46,84),(47,76),(48,68),(49,81),(50,73),(51,65),(52,78),(53,70),(54,83),(55,75),(56,67),(57,80),(58,72),(59,64),(60,77),(61,69),(62,82),(63,74)]])`

C21⋊D4 is a maximal subgroup of   D285S3  D6.D14  D125D7  C28⋊D6  C42.C23  D7×C3⋊D4  S3×C7⋊D4
C21⋊D4 is a maximal quotient of   C21⋊D8  C28.D6  C42.D4  C21⋊Q16  D14⋊Dic3  D6⋊Dic7  Dic21⋊C4

Matrix representation of C21⋊D4 in GL4(𝔽337) generated by

 0 194 0 0 33 227 0 0 0 0 208 0 0 0 84 128
,
 0 194 0 0 304 0 0 0 0 0 206 205 0 0 33 131
,
 0 194 0 0 304 0 0 0 0 0 1 0 0 0 238 336
`G:=sub<GL(4,GF(337))| [0,33,0,0,194,227,0,0,0,0,208,84,0,0,0,128],[0,304,0,0,194,0,0,0,0,0,206,33,0,0,205,131],[0,304,0,0,194,0,0,0,0,0,1,238,0,0,0,336] >;`

C21⋊D4 in GAP, Magma, Sage, TeX

`C_{21}\rtimes D_4`
`% in TeX`

`G:=Group("C21:D4");`
`// GroupNames label`

`G:=SmallGroup(168,15);`
`// by ID`

`G=gap.SmallGroup(168,15);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-7,61,168,3604]);`
`// Polycyclic`

`G:=Group<a,b,c|a^21=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^13,c*b*c=b^-1>;`
`// generators/relations`

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