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## G = D57⋊C3order 342 = 2·32·19

### The semidirect product of D57 and C3 acting faithfully

Aliases: D57⋊C3, C571C6, C19⋊C3⋊S3, C19⋊(C3×S3), C3⋊(C19⋊C6), (C3×C19⋊C3)⋊1C2, SmallGroup(342,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C57 — D57⋊C3
 Chief series C1 — C19 — C57 — C3×C19⋊C3 — D57⋊C3
 Lower central C57 — D57⋊C3
 Upper central C1

Generators and relations for D57⋊C3
G = < a,b,c | a57=b2=c3=1, bab=a-1, cac-1=a49, cbc-1=a48b >

57C2
19C3
38C3
19S3
57C6
19C32
3D19
19C3×S3

Character table of D57⋊C3

 class 1 2 3A 3B 3C 3D 3E 6A 6B 19A 19B 19C 57A 57B 57C 57D 57E 57F size 1 57 2 19 19 38 38 57 57 6 6 6 6 6 6 6 6 6 ρ1 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 linear of order 2 ρ3 1 -1 1 ζ3 ζ32 ζ3 ζ32 ζ65 ζ6 1 1 1 1 1 1 1 1 1 linear of order 6 ρ4 1 1 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 1 1 1 1 1 1 1 1 1 linear of order 3 ρ5 1 -1 1 ζ32 ζ3 ζ32 ζ3 ζ6 ζ65 1 1 1 1 1 1 1 1 1 linear of order 6 ρ6 1 1 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 1 1 1 1 1 1 1 1 1 linear of order 3 ρ7 2 0 -1 2 2 -1 -1 0 0 2 2 2 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ8 2 0 -1 -1-√-3 -1+√-3 ζ6 ζ65 0 0 2 2 2 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ9 2 0 -1 -1+√-3 -1-√-3 ζ65 ζ6 0 0 2 2 2 -1 -1 -1 -1 -1 -1 complex lifted from C3×S3 ρ10 6 0 6 0 0 0 0 0 0 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 orthogonal lifted from C19⋊C6 ρ11 6 0 6 0 0 0 0 0 0 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 orthogonal lifted from C19⋊C6 ρ12 6 0 6 0 0 0 0 0 0 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 orthogonal lifted from C19⋊C6 ρ13 6 0 -3 0 0 0 0 0 0 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 -ζ3ζ1917-ζ3ζ1916+ζ3ζ1914-ζ3ζ195+ζ3ζ193+ζ3ζ192-ζ1917-ζ1916-ζ195 ζ3ζ1918+ζ3ζ1912-ζ3ζ1911+ζ3ζ198-ζ3ζ197-ζ3ζ19-ζ1911-ζ197-ζ19 -ζ32ζ1917-ζ32ζ1916+ζ32ζ1914-ζ32ζ195+ζ32ζ193+ζ32ζ192-ζ1917-ζ1916-ζ195 -ζ32ζ1915-ζ32ζ1913-ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194-ζ1915-ζ1913-ζ1910 -ζ3ζ1918-ζ3ζ1912+ζ3ζ1911-ζ3ζ198+ζ3ζ197+ζ3ζ19-ζ1918-ζ1912-ζ198 ζ32ζ1915+ζ32ζ1913+ζ32ζ1910-ζ32ζ199-ζ32ζ196-ζ32ζ194-ζ199-ζ196-ζ194 orthogonal faithful ρ14 6 0 -3 0 0 0 0 0 0 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ3ζ1918+ζ3ζ1912-ζ3ζ1911+ζ3ζ198-ζ3ζ197-ζ3ζ19-ζ1911-ζ197-ζ19 -ζ32ζ1915-ζ32ζ1913-ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194-ζ1915-ζ1913-ζ1910 -ζ3ζ1918-ζ3ζ1912+ζ3ζ1911-ζ3ζ198+ζ3ζ197+ζ3ζ19-ζ1918-ζ1912-ζ198 -ζ3ζ1917-ζ3ζ1916+ζ3ζ1914-ζ3ζ195+ζ3ζ193+ζ3ζ192-ζ1917-ζ1916-ζ195 ζ32ζ1915+ζ32ζ1913+ζ32ζ1910-ζ32ζ199-ζ32ζ196-ζ32ζ194-ζ199-ζ196-ζ194 -ζ32ζ1917-ζ32ζ1916+ζ32ζ1914-ζ32ζ195+ζ32ζ193+ζ32ζ192-ζ1917-ζ1916-ζ195 orthogonal faithful ρ15 6 0 -3 0 0 0 0 0 0 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 -ζ32ζ1917-ζ32ζ1916+ζ32ζ1914-ζ32ζ195+ζ32ζ193+ζ32ζ192-ζ1917-ζ1916-ζ195 -ζ3ζ1918-ζ3ζ1912+ζ3ζ1911-ζ3ζ198+ζ3ζ197+ζ3ζ19-ζ1918-ζ1912-ζ198 -ζ3ζ1917-ζ3ζ1916+ζ3ζ1914-ζ3ζ195+ζ3ζ193+ζ3ζ192-ζ1917-ζ1916-ζ195 ζ32ζ1915+ζ32ζ1913+ζ32ζ1910-ζ32ζ199-ζ32ζ196-ζ32ζ194-ζ199-ζ196-ζ194 ζ3ζ1918+ζ3ζ1912-ζ3ζ1911+ζ3ζ198-ζ3ζ197-ζ3ζ19-ζ1911-ζ197-ζ19 -ζ32ζ1915-ζ32ζ1913-ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194-ζ1915-ζ1913-ζ1910 orthogonal faithful ρ16 6 0 -3 0 0 0 0 0 0 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 -ζ32ζ1915-ζ32ζ1913-ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194-ζ1915-ζ1913-ζ1910 -ζ3ζ1917-ζ3ζ1916+ζ3ζ1914-ζ3ζ195+ζ3ζ193+ζ3ζ192-ζ1917-ζ1916-ζ195 ζ32ζ1915+ζ32ζ1913+ζ32ζ1910-ζ32ζ199-ζ32ζ196-ζ32ζ194-ζ199-ζ196-ζ194 ζ3ζ1918+ζ3ζ1912-ζ3ζ1911+ζ3ζ198-ζ3ζ197-ζ3ζ19-ζ1911-ζ197-ζ19 -ζ32ζ1917-ζ32ζ1916+ζ32ζ1914-ζ32ζ195+ζ32ζ193+ζ32ζ192-ζ1917-ζ1916-ζ195 -ζ3ζ1918-ζ3ζ1912+ζ3ζ1911-ζ3ζ198+ζ3ζ197+ζ3ζ19-ζ1918-ζ1912-ζ198 orthogonal faithful ρ17 6 0 -3 0 0 0 0 0 0 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ32ζ1915+ζ32ζ1913+ζ32ζ1910-ζ32ζ199-ζ32ζ196-ζ32ζ194-ζ199-ζ196-ζ194 -ζ32ζ1917-ζ32ζ1916+ζ32ζ1914-ζ32ζ195+ζ32ζ193+ζ32ζ192-ζ1917-ζ1916-ζ195 -ζ32ζ1915-ζ32ζ1913-ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194-ζ1915-ζ1913-ζ1910 -ζ3ζ1918-ζ3ζ1912+ζ3ζ1911-ζ3ζ198+ζ3ζ197+ζ3ζ19-ζ1918-ζ1912-ζ198 -ζ3ζ1917-ζ3ζ1916+ζ3ζ1914-ζ3ζ195+ζ3ζ193+ζ3ζ192-ζ1917-ζ1916-ζ195 ζ3ζ1918+ζ3ζ1912-ζ3ζ1911+ζ3ζ198-ζ3ζ197-ζ3ζ19-ζ1911-ζ197-ζ19 orthogonal faithful ρ18 6 0 -3 0 0 0 0 0 0 ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 -ζ3ζ1918-ζ3ζ1912+ζ3ζ1911-ζ3ζ198+ζ3ζ197+ζ3ζ19-ζ1918-ζ1912-ζ198 ζ32ζ1915+ζ32ζ1913+ζ32ζ1910-ζ32ζ199-ζ32ζ196-ζ32ζ194-ζ199-ζ196-ζ194 ζ3ζ1918+ζ3ζ1912-ζ3ζ1911+ζ3ζ198-ζ3ζ197-ζ3ζ19-ζ1911-ζ197-ζ19 -ζ32ζ1917-ζ32ζ1916+ζ32ζ1914-ζ32ζ195+ζ32ζ193+ζ32ζ192-ζ1917-ζ1916-ζ195 -ζ32ζ1915-ζ32ζ1913-ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194-ζ1915-ζ1913-ζ1910 -ζ3ζ1917-ζ3ζ1916+ζ3ζ1914-ζ3ζ195+ζ3ζ193+ζ3ζ192-ζ1917-ζ1916-ζ195 orthogonal faithful

Smallest permutation representation of D57⋊C3
On 57 points
Generators in S57
```(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)
(1 57)(2 56)(3 55)(4 54)(5 53)(6 52)(7 51)(8 50)(9 49)(10 48)(11 47)(12 46)(13 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 36)(23 35)(24 34)(25 33)(26 32)(27 31)(28 30)
(2 8 50)(3 15 42)(4 22 34)(5 29 26)(6 36 18)(7 43 10)(9 57 51)(11 14 35)(12 21 27)(13 28 19)(16 49 52)(17 56 44)(23 41 53)(24 48 45)(25 55 37)(30 33 54)(31 40 46)(32 47 38)```

`G:=sub<Sym(57)| (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), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30), (2,8,50)(3,15,42)(4,22,34)(5,29,26)(6,36,18)(7,43,10)(9,57,51)(11,14,35)(12,21,27)(13,28,19)(16,49,52)(17,56,44)(23,41,53)(24,48,45)(25,55,37)(30,33,54)(31,40,46)(32,47,38)>;`

`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), (1,57)(2,56)(3,55)(4,54)(5,53)(6,52)(7,51)(8,50)(9,49)(10,48)(11,47)(12,46)(13,45)(14,44)(15,43)(16,42)(17,41)(18,40)(19,39)(20,38)(21,37)(22,36)(23,35)(24,34)(25,33)(26,32)(27,31)(28,30), (2,8,50)(3,15,42)(4,22,34)(5,29,26)(6,36,18)(7,43,10)(9,57,51)(11,14,35)(12,21,27)(13,28,19)(16,49,52)(17,56,44)(23,41,53)(24,48,45)(25,55,37)(30,33,54)(31,40,46)(32,47,38) );`

`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)], [(1,57),(2,56),(3,55),(4,54),(5,53),(6,52),(7,51),(8,50),(9,49),(10,48),(11,47),(12,46),(13,45),(14,44),(15,43),(16,42),(17,41),(18,40),(19,39),(20,38),(21,37),(22,36),(23,35),(24,34),(25,33),(26,32),(27,31),(28,30)], [(2,8,50),(3,15,42),(4,22,34),(5,29,26),(6,36,18),(7,43,10),(9,57,51),(11,14,35),(12,21,27),(13,28,19),(16,49,52),(17,56,44),(23,41,53),(24,48,45),(25,55,37),(30,33,54),(31,40,46),(32,47,38)]])`

Matrix representation of D57⋊C3 in GL6(𝔽229)

 152 46 184 198 66 181 99 209 42 112 24 160 132 32 180 224 62 37 62 222 151 214 17 124 225 175 52 16 213 22 182 161 178 202 192 119
,
 196 31 45 150 195 140 106 83 188 8 147 70 17 96 176 101 68 225 83 115 9 198 12 144 25 119 164 219 82 37 62 168 62 27 172 181
,
 1 127 223 125 103 108 0 0 1 0 0 0 103 125 223 127 1 1 0 0 0 0 0 1 29 196 51 75 131 102 28 69 177 178 29 103

`G:=sub<GL(6,GF(229))| [152,99,132,62,225,182,46,209,32,222,175,161,184,42,180,151,52,178,198,112,224,214,16,202,66,24,62,17,213,192,181,160,37,124,22,119],[196,106,17,83,25,62,31,83,96,115,119,168,45,188,176,9,164,62,150,8,101,198,219,27,195,147,68,12,82,172,140,70,225,144,37,181],[1,0,103,0,29,28,127,0,125,0,196,69,223,1,223,0,51,177,125,0,127,0,75,178,103,0,1,0,131,29,108,0,1,1,102,103] >;`

D57⋊C3 in GAP, Magma, Sage, TeX

`D_{57}\rtimes C_3`
`% in TeX`

`G:=Group("D57:C3");`
`// GroupNames label`

`G:=SmallGroup(342,11);`
`// by ID`

`G=gap.SmallGroup(342,11);`
`# by ID`

`G:=PCGroup([4,-2,-3,-3,-19,146,5187,1015]);`
`// Polycyclic`

`G:=Group<a,b,c|a^57=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^49,c*b*c^-1=a^48*b>;`
`// generators/relations`

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