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## G = D39⋊S3order 468 = 22·32·13

### The semidirect product of D39 and S3 acting via S3/C3=C2

Aliases: D39⋊S3, C393D6, C322D26, C132S32, C3⋊S3⋊D13, C33(S3×D13), (C3×D39)⋊3C2, (C3×C39)⋊4C22, (C13×C3⋊S3)⋊2C2, SmallGroup(468,46)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C39 — D39⋊S3
 Chief series C1 — C13 — C39 — C3×C39 — C3×D39 — D39⋊S3
 Lower central C3×C39 — D39⋊S3
 Upper central C1

Generators and relations for D39⋊S3
G = < a,b,c,d | a39=b2=c3=d2=1, bab=a-1, ac=ca, dad=a14, bc=cb, dbd=a13b, dcd=c-1 >

9C2
39C2
39C2
2C3
117C22
3S3
3S3
6S3
13S3
13S3
39C6
39C6
3D13
3D13
9C26
2C39
39D6
39D6
13C3×S3
13C3×S3
9D26
13S32

Smallest permutation representation of D39⋊S3
On 78 points
Generators in S78
```(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)
(1 60)(2 59)(3 58)(4 57)(5 56)(6 55)(7 54)(8 53)(9 52)(10 51)(11 50)(12 49)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 42)(20 41)(21 40)(22 78)(23 77)(24 76)(25 75)(26 74)(27 73)(28 72)(29 71)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)
(1 14 27)(2 15 28)(3 16 29)(4 17 30)(5 18 31)(6 19 32)(7 20 33)(8 21 34)(9 22 35)(10 23 36)(11 24 37)(12 25 38)(13 26 39)(40 66 53)(41 67 54)(42 68 55)(43 69 56)(44 70 57)(45 71 58)(46 72 59)(47 73 60)(48 74 61)(49 75 62)(50 76 63)(51 77 64)(52 78 65)
(2 15)(3 29)(5 18)(6 32)(8 21)(9 35)(11 24)(12 38)(14 27)(17 30)(20 33)(23 36)(26 39)(41 54)(42 68)(44 57)(45 71)(47 60)(48 74)(50 63)(51 77)(53 66)(56 69)(59 72)(62 75)(65 78)```

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

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

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

45 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 6A 6B 13A ··· 13F 26A ··· 26F 39A ··· 39X order 1 2 2 2 3 3 3 6 6 13 ··· 13 26 ··· 26 39 ··· 39 size 1 9 39 39 2 2 4 78 78 2 ··· 2 18 ··· 18 4 ··· 4

45 irreducible representations

 dim 1 1 1 2 2 2 2 4 4 4 type + + + + + + + + + image C1 C2 C2 S3 D6 D13 D26 S32 S3×D13 D39⋊S3 kernel D39⋊S3 C3×D39 C13×C3⋊S3 D39 C39 C3⋊S3 C32 C13 C3 C1 # reps 1 2 1 2 2 6 6 1 12 12

Matrix representation of D39⋊S3 in GL6(𝔽79)

 57 33 0 0 0 0 73 70 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 78 0 0 0 0 1 78
,
 21 62 0 0 0 0 77 58 0 0 0 0 0 0 78 0 0 0 0 0 0 78 0 0 0 0 0 0 1 78 0 0 0 0 0 78
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 78 1 0 0 0 0 78 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 78 0 0 0 0 0 0 78 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(79))| [57,73,0,0,0,0,33,70,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,78,78],[21,77,0,0,0,0,62,58,0,0,0,0,0,0,78,0,0,0,0,0,0,78,0,0,0,0,0,0,1,0,0,0,0,0,78,78],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,78,78,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[78,0,0,0,0,0,0,78,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

D39⋊S3 in GAP, Magma, Sage, TeX

`D_{39}\rtimes S_3`
`% in TeX`

`G:=Group("D39:S3");`
`// GroupNames label`

`G:=SmallGroup(468,46);`
`// by ID`

`G=gap.SmallGroup(468,46);`
`# by ID`

`G:=PCGroup([5,-2,-2,-3,-3,-13,122,67,248,10804]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^39=b^2=c^3=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^14,b*c=c*b,d*b*d=a^13*b,d*c*d=c^-1>;`
`// generators/relations`

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