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## G = D21⋊C4order 168 = 23·3·7

### The semidirect product of D21 and C4 acting via C4/C2=C2

Aliases: D21⋊C4, D42.C2, C14.3D6, C6.3D14, Dic32D7, Dic72S3, C42.3C22, C71(C4×S3), C31(C4×D7), C213(C2×C4), C2.3(S3×D7), (C7×Dic3)⋊2C2, (C3×Dic7)⋊2C2, SmallGroup(168,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C21 — D21⋊C4
 Chief series C1 — C7 — C21 — C42 — C3×Dic7 — D21⋊C4
 Lower central C21 — D21⋊C4
 Upper central C1 — C2

Generators and relations for D21⋊C4
G = < a,b,c | a21=b2=c4=1, bab=a-1, cac-1=a8, cbc-1=a7b >

Character table of D21⋊C4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 6 7A 7B 7C 12A 12B 14A 14B 14C 21A 21B 21C 28A 28B 28C 28D 28E 28F 42A 42B 42C size 1 1 21 21 2 3 3 7 7 2 2 2 2 14 14 2 2 2 4 4 4 6 6 6 6 6 6 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 -i i -i i -1 1 1 1 i -i -1 -1 -1 1 1 1 -i i i -i -i i -1 -1 -1 linear of order 4 ρ6 1 -1 1 -1 1 i -i -i i -1 1 1 1 i -i -1 -1 -1 1 1 1 i -i -i i i -i -1 -1 -1 linear of order 4 ρ7 1 -1 -1 1 1 i -i i -i -1 1 1 1 -i i -1 -1 -1 1 1 1 i -i -i i i -i -1 -1 -1 linear of order 4 ρ8 1 -1 1 -1 1 -i i i -i -1 1 1 1 -i i -1 -1 -1 1 1 1 -i i i -i -i i -1 -1 -1 linear of order 4 ρ9 2 2 0 0 -1 0 0 2 2 -1 2 2 2 -1 -1 2 2 2 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 0 0 -1 0 0 -2 -2 -1 2 2 2 1 1 2 2 2 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from D6 ρ11 2 2 0 0 2 -2 -2 0 0 2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ75-ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D14 ρ12 2 2 0 0 2 2 2 0 0 2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D7 ρ13 2 2 0 0 2 2 2 0 0 2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal lifted from D7 ρ14 2 2 0 0 2 -2 -2 0 0 2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ74-ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal lifted from D14 ρ15 2 2 0 0 2 -2 -2 0 0 2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 -ζ76-ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D14 ρ16 2 2 0 0 2 2 2 0 0 2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal lifted from D7 ρ17 2 -2 0 0 -1 0 0 2i -2i 1 2 2 2 i -i -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 1 1 1 complex lifted from C4×S3 ρ18 2 -2 0 0 -1 0 0 -2i 2i 1 2 2 2 -i i -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 1 1 1 complex lifted from C4×S3 ρ19 2 -2 0 0 2 2i -2i 0 0 -2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ4ζ74+ζ4ζ73 ζ43ζ75+ζ43ζ72 ζ43ζ74+ζ43ζ73 ζ4ζ76+ζ4ζ7 ζ4ζ75+ζ4ζ72 ζ43ζ76+ζ43ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 complex lifted from C4×D7 ρ20 2 -2 0 0 2 -2i 2i 0 0 -2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ43ζ75+ζ43ζ72 ζ4ζ76+ζ4ζ7 ζ4ζ75+ζ4ζ72 ζ43ζ74+ζ43ζ73 ζ43ζ76+ζ43ζ7 ζ4ζ74+ζ4ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 complex lifted from C4×D7 ρ21 2 -2 0 0 2 -2i 2i 0 0 -2 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ43ζ74+ζ43ζ73 ζ4ζ75+ζ4ζ72 ζ4ζ74+ζ4ζ73 ζ43ζ76+ζ43ζ7 ζ43ζ75+ζ43ζ72 ζ4ζ76+ζ4ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 complex lifted from C4×D7 ρ22 2 -2 0 0 2 -2i 2i 0 0 -2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ43ζ76+ζ43ζ7 ζ4ζ74+ζ4ζ73 ζ4ζ76+ζ4ζ7 ζ43ζ75+ζ43ζ72 ζ43ζ74+ζ43ζ73 ζ4ζ75+ζ4ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 complex lifted from C4×D7 ρ23 2 -2 0 0 2 2i -2i 0 0 -2 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ4ζ75+ζ4ζ72 ζ43ζ76+ζ43ζ7 ζ43ζ75+ζ43ζ72 ζ4ζ74+ζ4ζ73 ζ4ζ76+ζ4ζ7 ζ43ζ74+ζ43ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 complex lifted from C4×D7 ρ24 2 -2 0 0 2 2i -2i 0 0 -2 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ4ζ76+ζ4ζ7 ζ43ζ74+ζ43ζ73 ζ43ζ76+ζ43ζ7 ζ4ζ75+ζ4ζ72 ζ4ζ74+ζ4ζ73 ζ43ζ75+ζ43ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 complex lifted from C4×D7 ρ25 4 -4 0 0 -2 0 0 0 0 2 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 0 0 -2ζ75-2ζ72 -2ζ76-2ζ7 -2ζ74-2ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 0 0 0 0 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 orthogonal faithful, Schur index 2 ρ26 4 -4 0 0 -2 0 0 0 0 2 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 0 0 -2ζ76-2ζ7 -2ζ74-2ζ73 -2ζ75-2ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 0 0 0 0 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 orthogonal faithful, Schur index 2 ρ27 4 4 0 0 -2 0 0 0 0 -2 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 0 0 0 0 0 0 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 orthogonal lifted from S3×D7 ρ28 4 -4 0 0 -2 0 0 0 0 2 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 0 0 -2ζ74-2ζ73 -2ζ75-2ζ72 -2ζ76-2ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 0 0 0 0 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 orthogonal faithful, Schur index 2 ρ29 4 4 0 0 -2 0 0 0 0 -2 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 0 0 0 0 0 0 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 orthogonal lifted from S3×D7 ρ30 4 4 0 0 -2 0 0 0 0 -2 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 0 0 0 0 0 0 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 orthogonal lifted from S3×D7

Smallest permutation representation of D21⋊C4
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 21)(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(22 38)(23 37)(24 36)(25 35)(26 34)(27 33)(28 32)(29 31)(39 42)(40 41)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(59 63)(60 62)(64 74)(65 73)(66 72)(67 71)(68 70)(75 84)(76 83)(77 82)(78 81)(79 80)
(1 80 41 51)(2 67 42 59)(3 75 22 46)(4 83 23 54)(5 70 24 62)(6 78 25 49)(7 65 26 57)(8 73 27 44)(9 81 28 52)(10 68 29 60)(11 76 30 47)(12 84 31 55)(13 71 32 63)(14 79 33 50)(15 66 34 58)(16 74 35 45)(17 82 36 53)(18 69 37 61)(19 77 38 48)(20 64 39 56)(21 72 40 43)```

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

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

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

D21⋊C4 is a maximal subgroup of   D84⋊C2  D21⋊Q8  D14.D6  C4×S3×D7  Dic7.D6  Dic3.D14  D6⋊D14
D21⋊C4 is a maximal quotient of   D21⋊C8  D42.C4  Dic3×Dic7  D42⋊C4  Dic21⋊C4

Matrix representation of D21⋊C4 in GL4(𝔽337) generated by

 0 1 0 0 336 336 0 0 0 0 34 194 0 0 178 193
,
 0 1 0 0 1 0 0 0 0 0 0 228 0 0 34 0
,
 336 0 0 0 1 1 0 0 0 0 148 0 0 0 0 148
`G:=sub<GL(4,GF(337))| [0,336,0,0,1,336,0,0,0,0,34,178,0,0,194,193],[0,1,0,0,1,0,0,0,0,0,0,34,0,0,228,0],[336,1,0,0,0,1,0,0,0,0,148,0,0,0,0,148] >;`

D21⋊C4 in GAP, Magma, Sage, TeX

`D_{21}\rtimes C_4`
`% in TeX`

`G:=Group("D21:C4");`
`// GroupNames label`

`G:=SmallGroup(168,14);`
`// by ID`

`G=gap.SmallGroup(168,14);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-7,20,26,168,3604]);`
`// Polycyclic`

`G:=Group<a,b,c|a^21=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^8,c*b*c^-1=a^7*b>;`
`// generators/relations`

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