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## G = D52⋊C3order 312 = 23·3·13

### The semidirect product of D52 and C3 acting faithfully

Aliases: D52⋊C3, C521C6, D261C6, C4⋊(C13⋊C6), C13⋊C31D4, C131(C3×D4), C26.3(C2×C6), (C2×C13⋊C6)⋊1C2, (C4×C13⋊C3)⋊1C2, C2.4(C2×C13⋊C6), (C2×C13⋊C3).3C22, SmallGroup(312,10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D52⋊C3
 Chief series C1 — C13 — C26 — C2×C13⋊C3 — C2×C13⋊C6 — D52⋊C3
 Lower central C13 — C26 — D52⋊C3
 Upper central C1 — C2 — C4

Generators and relations for D52⋊C3
G = < a,b,c | a52=b2=c3=1, bab=a-1, cac-1=a9, cbc-1=a8b >

Character table of D52⋊C3

 class 1 2A 2B 2C 3A 3B 4 6A 6B 6C 6D 6E 6F 12A 12B 13A 13B 26A 26B 52A 52B 52C 52D size 1 1 26 26 13 13 2 13 13 26 26 26 26 26 26 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 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 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 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 linear of order 2 ρ5 1 1 1 -1 ζ32 ζ3 -1 ζ32 ζ3 ζ32 ζ65 ζ3 ζ6 ζ6 ζ65 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ6 1 1 1 1 ζ3 ζ32 1 ζ3 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ7 1 1 -1 1 ζ32 ζ3 -1 ζ32 ζ3 ζ6 ζ3 ζ65 ζ32 ζ6 ζ65 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ8 1 1 -1 1 ζ3 ζ32 -1 ζ3 ζ32 ζ65 ζ32 ζ6 ζ3 ζ65 ζ6 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ9 1 1 1 -1 ζ3 ζ32 -1 ζ3 ζ32 ζ3 ζ6 ζ32 ζ65 ζ65 ζ6 1 1 1 1 -1 -1 -1 -1 linear of order 6 ρ10 1 1 1 1 ζ32 ζ3 1 ζ32 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ11 1 1 -1 -1 ζ3 ζ32 1 ζ3 ζ32 ζ65 ζ6 ζ6 ζ65 ζ3 ζ32 1 1 1 1 1 1 1 1 linear of order 6 ρ12 1 1 -1 -1 ζ32 ζ3 1 ζ32 ζ3 ζ6 ζ65 ζ65 ζ6 ζ32 ζ3 1 1 1 1 1 1 1 1 linear of order 6 ρ13 2 -2 0 0 2 2 0 -2 -2 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ14 2 -2 0 0 -1+√-3 -1-√-3 0 1-√-3 1+√-3 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 complex lifted from C3×D4 ρ15 2 -2 0 0 -1-√-3 -1+√-3 0 1+√-3 1-√-3 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 complex lifted from C3×D4 ρ16 6 6 0 0 0 0 -6 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 1-√13/2 1+√13/2 1-√13/2 1+√13/2 orthogonal lifted from C2×C13⋊C6 ρ17 6 6 0 0 0 0 6 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 -1-√13/2 -1+√13/2 -1-√13/2 -1+√13/2 orthogonal lifted from C13⋊C6 ρ18 6 6 0 0 0 0 -6 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 -1-√13/2 -1+√13/2 1+√13/2 1-√13/2 1+√13/2 1-√13/2 orthogonal lifted from C2×C13⋊C6 ρ19 6 6 0 0 0 0 6 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 -1+√13/2 -1-√13/2 -1+√13/2 -1-√13/2 -1+√13/2 -1-√13/2 orthogonal lifted from C13⋊C6 ρ20 6 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 orthogonal faithful ρ21 6 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1-√13/2 -1+√13/2 1-√13/2 1+√13/2 -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 orthogonal faithful ρ22 6 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 orthogonal faithful ρ23 6 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 -1+√13/2 -1-√13/2 1+√13/2 1-√13/2 ζ43ζ1311+ζ43ζ138+ζ43ζ137-ζ43ζ136-ζ43ζ135-ζ43ζ132 -ζ43ζ1312-ζ43ζ1310+ζ43ζ139-ζ43ζ134+ζ43ζ133+ζ43ζ13 ζ4ζ1311+ζ4ζ138+ζ4ζ137-ζ4ζ136-ζ4ζ135-ζ4ζ132 -ζ4ζ1312-ζ4ζ1310+ζ4ζ139-ζ4ζ134+ζ4ζ133+ζ4ζ13 orthogonal faithful

Smallest permutation representation of D52⋊C3
On 52 points
Generators in S52
```(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)
(1 52)(2 51)(3 50)(4 49)(5 48)(6 47)(7 46)(8 45)(9 44)(10 43)(11 42)(12 41)(13 40)(14 39)(15 38)(16 37)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)(25 28)(26 27)
(2 30 10)(3 7 19)(4 36 28)(5 13 37)(6 42 46)(8 48 12)(9 25 21)(11 31 39)(15 43 23)(16 20 32)(17 49 41)(18 26 50)(22 38 34)(24 44 52)(29 33 45)(35 51 47)```

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

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

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

Matrix representation of D52⋊C3 in GL6(𝔽157)

 4 111 4 134 4 4 153 46 103 50 126 46 111 8 111 4 115 138 19 147 46 128 42 151 6 113 2 134 140 136 21 21 155 153 19 155
,
 4 111 4 134 4 4 153 153 23 153 46 153 111 31 107 54 111 4 19 42 153 46 149 46 6 115 29 111 10 138 21 17 23 155 44 151
,
 1 0 0 0 0 0 70 138 69 69 138 70 0 0 0 0 0 1 0 1 0 0 0 0 87 20 155 89 86 88 156 89 155 90 155 89

`G:=sub<GL(6,GF(157))| [4,153,111,19,6,21,111,46,8,147,113,21,4,103,111,46,2,155,134,50,4,128,134,153,4,126,115,42,140,19,4,46,138,151,136,155],[4,153,111,19,6,21,111,153,31,42,115,17,4,23,107,153,29,23,134,153,54,46,111,155,4,46,111,149,10,44,4,153,4,46,138,151],[1,70,0,0,87,156,0,138,0,1,20,89,0,69,0,0,155,155,0,69,0,0,89,90,0,138,0,0,86,155,0,70,1,0,88,89] >;`

D52⋊C3 in GAP, Magma, Sage, TeX

`D_{52}\rtimes C_3`
`% in TeX`

`G:=Group("D52:C3");`
`// GroupNames label`

`G:=SmallGroup(312,10);`
`// by ID`

`G=gap.SmallGroup(312,10);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-2,-13,141,66,7204,464]);`
`// Polycyclic`

`G:=Group<a,b,c|a^52=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^8*b>;`
`// generators/relations`

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