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## G = C3⋊D28order 168 = 23·3·7

### The semidirect product of C3 and D28 acting via D28/D14=C2

Aliases: C212D4, C32D28, Dic3⋊D7, D142S3, D423C2, C14.5D6, C6.5D14, C42.5C22, (C6×D7)⋊2C2, C71(C3⋊D4), C2.5(S3×D7), (C7×Dic3)⋊3C2, SmallGroup(168,16)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C42 — C3⋊D28
 Chief series C1 — C7 — C21 — C42 — C6×D7 — C3⋊D28
 Lower central C21 — C42 — C3⋊D28
 Upper central C1 — C2

Generators and relations for C3⋊D28
G = < a,b,c | a3=b28=c2=1, bab-1=cac=a-1, cbc=b-1 >

Character table of C3⋊D28

 class 1 2A 2B 2C 3 4 6A 6B 6C 7A 7B 7C 14A 14B 14C 21A 21B 21C 28A 28B 28C 28D 28E 28F 42A 42B 42C size 1 1 14 42 2 6 2 14 14 2 2 2 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 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 0 2 0 -2 0 0 2 2 2 -2 -2 -2 2 2 2 0 0 0 0 0 0 -2 -2 -2 orthogonal lifted from D4 ρ6 2 2 2 0 -1 0 -1 -1 -1 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from S3 ρ7 2 2 -2 0 -1 0 -1 1 1 2 2 2 2 2 2 -1 -1 -1 0 0 0 0 0 0 -1 -1 -1 orthogonal lifted from D6 ρ8 2 2 0 0 2 2 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D7 ρ9 2 2 0 0 2 -2 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 -ζ76-ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D14 ρ10 2 2 0 0 2 -2 2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 -ζ74-ζ73 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal lifted from D14 ρ11 2 2 0 0 2 2 2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal lifted from D7 ρ12 2 -2 0 0 2 0 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ4ζ75-ζ4ζ72 ζ4ζ74-ζ4ζ73 -ζ4ζ74+ζ4ζ73 -ζ4ζ75+ζ4ζ72 ζ43ζ76-ζ43ζ7 -ζ43ζ76+ζ43ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D28 ρ13 2 -2 0 0 2 0 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ4ζ74+ζ4ζ73 ζ43ζ76-ζ43ζ7 -ζ43ζ76+ζ43ζ7 ζ4ζ74-ζ4ζ73 -ζ4ζ75+ζ4ζ72 ζ4ζ75-ζ4ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D28 ρ14 2 2 0 0 2 -2 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 -ζ75-ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D14 ρ15 2 2 0 0 2 2 2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 ζ76+ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ75+ζ72 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal lifted from D7 ρ16 2 -2 0 0 2 0 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 ζ43ζ76-ζ43ζ7 ζ4ζ75-ζ4ζ72 -ζ4ζ75+ζ4ζ72 -ζ43ζ76+ζ43ζ7 -ζ4ζ74+ζ4ζ73 ζ4ζ74-ζ4ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D28 ρ17 2 -2 0 0 2 0 -2 0 0 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ43ζ76+ζ43ζ7 -ζ4ζ75+ζ4ζ72 ζ4ζ75-ζ4ζ72 ζ43ζ76-ζ43ζ7 ζ4ζ74-ζ4ζ73 -ζ4ζ74+ζ4ζ73 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from D28 ρ18 2 -2 0 0 2 0 -2 0 0 ζ76+ζ7 ζ74+ζ73 ζ75+ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 ζ4ζ74-ζ4ζ73 -ζ43ζ76+ζ43ζ7 ζ43ζ76-ζ43ζ7 -ζ4ζ74+ζ4ζ73 ζ4ζ75-ζ4ζ72 -ζ4ζ75+ζ4ζ72 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from D28 ρ19 2 -2 0 0 2 0 -2 0 0 ζ74+ζ73 ζ75+ζ72 ζ76+ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 ζ75+ζ72 ζ76+ζ7 ζ74+ζ73 -ζ4ζ75+ζ4ζ72 -ζ4ζ74+ζ4ζ73 ζ4ζ74-ζ4ζ73 ζ4ζ75-ζ4ζ72 -ζ43ζ76+ζ43ζ7 ζ43ζ76-ζ43ζ7 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from D28 ρ20 2 -2 0 0 -1 0 1 -√-3 √-3 2 2 2 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 1 1 1 complex lifted from C3⋊D4 ρ21 2 -2 0 0 -1 0 1 √-3 -√-3 2 2 2 -2 -2 -2 -1 -1 -1 0 0 0 0 0 0 1 1 1 complex lifted from C3⋊D4 ρ22 4 -4 0 0 -2 0 2 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 -2ζ76-2ζ7 -2ζ75-2ζ72 -2ζ74-2ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 0 0 0 ζ76+ζ7 ζ75+ζ72 ζ74+ζ73 orthogonal faithful ρ23 4 4 0 0 -2 0 -2 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 2ζ74+2ζ73 2ζ76+2ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 0 0 0 -ζ75-ζ72 -ζ74-ζ73 -ζ76-ζ7 orthogonal lifted from S3×D7 ρ24 4 4 0 0 -2 0 -2 0 0 2ζ76+2ζ7 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 2ζ75+2ζ72 2ζ74+2ζ73 -ζ74-ζ73 -ζ75-ζ72 -ζ76-ζ7 0 0 0 0 0 0 -ζ76-ζ7 -ζ75-ζ72 -ζ74-ζ73 orthogonal lifted from S3×D7 ρ25 4 -4 0 0 -2 0 2 0 0 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 -2ζ75-2ζ72 -2ζ74-2ζ73 -2ζ76-2ζ7 -ζ76-ζ7 -ζ74-ζ73 -ζ75-ζ72 0 0 0 0 0 0 ζ75+ζ72 ζ74+ζ73 ζ76+ζ7 orthogonal faithful ρ26 4 -4 0 0 -2 0 2 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 -2ζ74-2ζ73 -2ζ76-2ζ7 -2ζ75-2ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 0 0 0 ζ74+ζ73 ζ76+ζ7 ζ75+ζ72 orthogonal faithful ρ27 4 4 0 0 -2 0 -2 0 0 2ζ74+2ζ73 2ζ75+2ζ72 2ζ76+2ζ7 2ζ74+2ζ73 2ζ76+2ζ7 2ζ75+2ζ72 -ζ75-ζ72 -ζ76-ζ7 -ζ74-ζ73 0 0 0 0 0 0 -ζ74-ζ73 -ζ76-ζ7 -ζ75-ζ72 orthogonal lifted from S3×D7

Smallest permutation representation of C3⋊D28
On 84 points
Generators in S84
```(1 52 78)(2 79 53)(3 54 80)(4 81 55)(5 56 82)(6 83 29)(7 30 84)(8 57 31)(9 32 58)(10 59 33)(11 34 60)(12 61 35)(13 36 62)(14 63 37)(15 38 64)(16 65 39)(17 40 66)(18 67 41)(19 42 68)(20 69 43)(21 44 70)(22 71 45)(23 46 72)(24 73 47)(25 48 74)(26 75 49)(27 50 76)(28 77 51)
(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 7)(2 6)(3 5)(8 28)(9 27)(10 26)(11 25)(12 24)(13 23)(14 22)(15 21)(16 20)(17 19)(29 79)(30 78)(31 77)(32 76)(33 75)(34 74)(35 73)(36 72)(37 71)(38 70)(39 69)(40 68)(41 67)(42 66)(43 65)(44 64)(45 63)(46 62)(47 61)(48 60)(49 59)(50 58)(51 57)(52 84)(53 83)(54 82)(55 81)(56 80)```

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

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

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

C3⋊D28 is a maximal subgroup of   D28⋊S3  D6.D14  D14.D6  S3×D28  Dic7.D6  D7×C3⋊D4  D6⋊D14
C3⋊D28 is a maximal quotient of   C3⋊D56  C6.D28  C21⋊SD16  C3⋊Dic28  D14⋊Dic3  D42⋊C4  C14.Dic6

Matrix representation of C3⋊D28 in GL4(𝔽337) generated by

 1 0 0 0 0 1 0 0 0 0 0 1 0 0 336 336
,
 219 62 0 0 299 257 0 0 0 0 278 139 0 0 198 59
,
 0 193 0 0 227 0 0 0 0 0 0 336 0 0 336 0
`G:=sub<GL(4,GF(337))| [1,0,0,0,0,1,0,0,0,0,0,336,0,0,1,336],[219,299,0,0,62,257,0,0,0,0,278,198,0,0,139,59],[0,227,0,0,193,0,0,0,0,0,0,336,0,0,336,0] >;`

C3⋊D28 in GAP, Magma, Sage, TeX

`C_3\rtimes D_{28}`
`% in TeX`

`G:=Group("C3:D28");`
`// GroupNames label`

`G:=SmallGroup(168,16);`
`// by ID`

`G=gap.SmallGroup(168,16);`
`# by ID`

`G:=PCGroup([5,-2,-2,-2,-3,-7,61,26,168,3604]);`
`// Polycyclic`

`G:=Group<a,b,c|a^3=b^28=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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