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## G = D78order 156 = 22·3·13

### Dihedral group

Aliases: D78, C2×D39, C26⋊S3, C6⋊D13, C132D6, C32D26, C781C2, C392C22, sometimes denoted D156 or Dih78 or Dih156, SmallGroup(156,17)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C39 — D78
 Chief series C1 — C13 — C39 — D39 — D78
 Lower central C39 — D78
 Upper central C1 — C2

Generators and relations for D78
G = < a,b | a78=b2=1, bab=a-1 >

39C2
39C2
39C22
13S3
13S3
3D13
3D13
13D6
3D26

Smallest permutation representation of D78
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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 54)(26 53)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)```

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

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

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

D78 is a maximal subgroup of   D78.C2  C3⋊D52  C13⋊D12  D156  C397D4  C2×S3×D13
D78 is a maximal quotient of   Dic78  D156  C397D4

42 conjugacy classes

 class 1 2A 2B 2C 3 6 13A ··· 13F 26A ··· 26F 39A ··· 39L 78A ··· 78L order 1 2 2 2 3 6 13 ··· 13 26 ··· 26 39 ··· 39 78 ··· 78 size 1 1 39 39 2 2 2 ··· 2 2 ··· 2 2 ··· 2 2 ··· 2

42 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 S3 D6 D13 D26 D39 D78 kernel D78 D39 C78 C26 C13 C6 C3 C2 C1 # reps 1 2 1 1 1 6 6 12 12

Matrix representation of D78 in GL2(𝔽79) generated by

 76 12 44 8
,
 63 15 62 16
`G:=sub<GL(2,GF(79))| [76,44,12,8],[63,62,15,16] >;`

D78 in GAP, Magma, Sage, TeX

`D_{78}`
`% in TeX`

`G:=Group("D78");`
`// GroupNames label`

`G:=SmallGroup(156,17);`
`// by ID`

`G=gap.SmallGroup(156,17);`
`# by ID`

`G:=PCGroup([4,-2,-2,-3,-13,98,2307]);`
`// Polycyclic`

`G:=Group<a,b|a^78=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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