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## G = D65⋊C3order 390 = 2·3·5·13

### The semidirect product of D65 and C3 acting faithfully

Aliases: D65⋊C3, C651C6, C13⋊C3⋊D5, C5⋊(C13⋊C6), C13⋊(C3×D5), (C5×C13⋊C3)⋊1C2, SmallGroup(390,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C65 — D65⋊C3
 Chief series C1 — C13 — C65 — C5×C13⋊C3 — D65⋊C3
 Lower central C65 — D65⋊C3
 Upper central C1

Generators and relations for D65⋊C3
G = < a,b,c | a65=b2=c3=1, bab=a-1, cac-1=a61, cbc-1=a60b >

Character table of D65⋊C3

 class 1 2 3A 3B 5A 5B 6A 6B 13A 13B 15A 15B 15C 15D 65A 65B 65C 65D 65E 65F 65G 65H size 1 65 13 13 2 2 65 65 6 6 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 trivial ρ2 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 ζ3 ζ32 1 1 ζ32 ζ3 1 1 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 linear of order 3 ρ4 1 -1 ζ32 ζ3 1 1 ζ65 ζ6 1 1 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 linear of order 6 ρ5 1 1 ζ32 ζ3 1 1 ζ3 ζ32 1 1 ζ32 ζ32 ζ3 ζ3 1 1 1 1 1 1 1 1 linear of order 3 ρ6 1 -1 ζ3 ζ32 1 1 ζ6 ζ65 1 1 ζ3 ζ3 ζ32 ζ32 1 1 1 1 1 1 1 1 linear of order 6 ρ7 2 0 2 2 -1-√5/2 -1+√5/2 0 0 2 2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 orthogonal lifted from D5 ρ8 2 0 2 2 -1+√5/2 -1-√5/2 0 0 2 2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 orthogonal lifted from D5 ρ9 2 0 -1-√-3 -1+√-3 -1+√5/2 -1-√5/2 0 0 2 2 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 complex lifted from C3×D5 ρ10 2 0 -1-√-3 -1+√-3 -1-√5/2 -1+√5/2 0 0 2 2 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 complex lifted from C3×D5 ρ11 2 0 -1+√-3 -1-√-3 -1+√5/2 -1-√5/2 0 0 2 2 ζ3ζ54+ζ3ζ5 ζ3ζ53+ζ3ζ52 ζ32ζ53+ζ32ζ52 ζ32ζ54+ζ32ζ5 -1+√5/2 -1+√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1-√5/2 complex lifted from C3×D5 ρ12 2 0 -1+√-3 -1-√-3 -1-√5/2 -1+√5/2 0 0 2 2 ζ3ζ53+ζ3ζ52 ζ3ζ54+ζ3ζ5 ζ32ζ54+ζ32ζ5 ζ32ζ53+ζ32ζ52 -1-√5/2 -1-√5/2 -1+√5/2 -1-√5/2 -1-√5/2 -1+√5/2 -1+√5/2 -1+√5/2 complex lifted from C3×D5 ρ13 6 0 0 0 6 6 0 0 -1+√13/2 -1-√13/2 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 ρ14 6 0 0 0 6 6 0 0 -1-√13/2 -1+√13/2 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 ρ15 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -1-√13/2 -1+√13/2 0 0 0 0 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 orthogonal faithful ρ16 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -1+√13/2 -1-√13/2 0 0 0 0 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 orthogonal faithful ρ17 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -1+√13/2 -1-√13/2 0 0 0 0 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 orthogonal faithful ρ18 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -1-√13/2 -1+√13/2 0 0 0 0 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 orthogonal faithful ρ19 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -1-√13/2 -1+√13/2 0 0 0 0 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 orthogonal faithful ρ20 6 0 0 0 -3+3√5/2 -3-3√5/2 0 0 -1+√13/2 -1-√13/2 0 0 0 0 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 orthogonal faithful ρ21 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -1-√13/2 -1+√13/2 0 0 0 0 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 orthogonal faithful ρ22 6 0 0 0 -3-3√5/2 -3+3√5/2 0 0 -1+√13/2 -1-√13/2 0 0 0 0 ζ53ζ139+ζ53ζ133+ζ53ζ13+ζ52ζ1312+ζ52ζ1310+ζ52ζ134 ζ53ζ136+ζ53ζ135+ζ53ζ132+ζ52ζ1311+ζ52ζ138+ζ52ζ137 ζ54ζ1311+ζ54ζ138+ζ54ζ137+ζ5ζ136+ζ5ζ135+ζ5ζ132 ζ53ζ1312+ζ53ζ1310+ζ53ζ134+ζ52ζ139+ζ52ζ133+ζ52ζ13 ζ53ζ1311+ζ53ζ138+ζ53ζ137+ζ52ζ136+ζ52ζ135+ζ52ζ132 ζ54ζ139+ζ54ζ133+ζ54ζ13+ζ5ζ1312+ζ5ζ1310+ζ5ζ134 ζ54ζ136+ζ54ζ135+ζ54ζ132+ζ5ζ1311+ζ5ζ138+ζ5ζ137 ζ54ζ1312+ζ54ζ1310+ζ54ζ134+ζ5ζ139+ζ5ζ133+ζ5ζ13 orthogonal faithful

Smallest permutation representation of D65⋊C3
On 65 points
Generators in S65
```(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)
(1 65)(2 64)(3 63)(4 62)(5 61)(6 60)(7 59)(8 58)(9 57)(10 56)(11 55)(12 54)(13 53)(14 52)(15 51)(16 50)(17 49)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(25 41)(26 40)(27 39)(28 38)(29 37)(30 36)(31 35)(32 34)
(2 17 62)(3 33 58)(4 49 54)(5 65 50)(6 16 46)(7 32 42)(8 48 38)(9 64 34)(10 15 30)(11 31 26)(12 47 22)(13 63 18)(19 29 59)(20 45 55)(21 61 51)(23 28 43)(24 44 39)(25 60 35)(36 41 56)(37 57 52)```

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

`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), (1,65)(2,64)(3,63)(4,62)(5,61)(6,60)(7,59)(8,58)(9,57)(10,56)(11,55)(12,54)(13,53)(14,52)(15,51)(16,50)(17,49)(18,48)(19,47)(20,46)(21,45)(22,44)(23,43)(24,42)(25,41)(26,40)(27,39)(28,38)(29,37)(30,36)(31,35)(32,34), (2,17,62)(3,33,58)(4,49,54)(5,65,50)(6,16,46)(7,32,42)(8,48,38)(9,64,34)(10,15,30)(11,31,26)(12,47,22)(13,63,18)(19,29,59)(20,45,55)(21,61,51)(23,28,43)(24,44,39)(25,60,35)(36,41,56)(37,57,52) );`

`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)], [(1,65),(2,64),(3,63),(4,62),(5,61),(6,60),(7,59),(8,58),(9,57),(10,56),(11,55),(12,54),(13,53),(14,52),(15,51),(16,50),(17,49),(18,48),(19,47),(20,46),(21,45),(22,44),(23,43),(24,42),(25,41),(26,40),(27,39),(28,38),(29,37),(30,36),(31,35),(32,34)], [(2,17,62),(3,33,58),(4,49,54),(5,65,50),(6,16,46),(7,32,42),(8,48,38),(9,64,34),(10,15,30),(11,31,26),(12,47,22),(13,63,18),(19,29,59),(20,45,55),(21,61,51),(23,28,43),(24,44,39),(25,60,35),(36,41,56),(37,57,52)]])`

Matrix representation of D65⋊C3 in GL6(𝔽1171)

 152 39 239 844 391 126 1045 592 958 805 592 831 340 880 101 453 314 427 744 530 26 718 770 504 667 162 693 1119 881 188 983 896 957 1110 743 1110
,
 152 39 239 844 391 126 1146 1106 932 87 62 1019 870 982 997 112 783 25 61 428 61 214 275 301 957 1110 804 153 830 1110 214 61 214 87 214 214
,
 1 0 0 0 0 0 742 311 741 741 311 742 0 0 0 0 0 1 0 1 0 0 0 0 429 861 1169 431 428 430 1170 431 1169 432 1169 431

`G:=sub<GL(6,GF(1171))| [152,1045,340,744,667,983,39,592,880,530,162,896,239,958,101,26,693,957,844,805,453,718,1119,1110,391,592,314,770,881,743,126,831,427,504,188,1110],[152,1146,870,61,957,214,39,1106,982,428,1110,61,239,932,997,61,804,214,844,87,112,214,153,87,391,62,783,275,830,214,126,1019,25,301,1110,214],[1,742,0,0,429,1170,0,311,0,1,861,431,0,741,0,0,1169,1169,0,741,0,0,431,432,0,311,0,0,428,1169,0,742,1,0,430,431] >;`

D65⋊C3 in GAP, Magma, Sage, TeX

`D_{65}\rtimes C_3`
`% in TeX`

`G:=Group("D65:C3");`
`// GroupNames label`

`G:=SmallGroup(390,3);`
`// by ID`

`G=gap.SmallGroup(390,3);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,-13,290,5763,727]);`
`// Polycyclic`

`G:=Group<a,b,c|a^65=b^2=c^3=1,b*a*b=a^-1,c*a*c^-1=a^61,c*b*c^-1=a^60*b>;`
`// generators/relations`

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