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## G = D75order 150 = 2·3·52

### Dihedral group

Aliases: D75, C25⋊S3, C3⋊D25, C751C2, C5.D15, C15.1D5, sometimes denoted D150 or Dih75 or Dih150, SmallGroup(150,3)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C75 — D75
 Chief series C1 — C5 — C25 — C75 — D75
 Lower central C75 — D75
 Upper central C1

Generators and relations for D75
G = < a,b | a75=b2=1, bab=a-1 >

75C2
25S3
15D5
5D15
3D25

Smallest permutation representation of D75
On 75 points
Generators in S75
```(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)
(2 75)(3 74)(4 73)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)```

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

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

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

D75 is a maximal subgroup of   S3×D25  D225  C3⋊D75
D75 is a maximal quotient of   Dic75  D225  C3⋊D75

39 conjugacy classes

 class 1 2 3 5A 5B 15A 15B 15C 15D 25A ··· 25J 75A ··· 75T order 1 2 3 5 5 15 15 15 15 25 ··· 25 75 ··· 75 size 1 75 2 2 2 2 2 2 2 2 ··· 2 2 ··· 2

39 irreducible representations

 dim 1 1 2 2 2 2 2 type + + + + + + + image C1 C2 S3 D5 D15 D25 D75 kernel D75 C75 C25 C15 C5 C3 C1 # reps 1 1 1 2 4 10 20

Matrix representation of D75 in GL2(𝔽151) generated by

 112 91 60 131
,
 1 0 123 150
`G:=sub<GL(2,GF(151))| [112,60,91,131],[1,123,0,150] >;`

D75 in GAP, Magma, Sage, TeX

`D_{75}`
`% in TeX`

`G:=Group("D75");`
`// GroupNames label`

`G:=SmallGroup(150,3);`
`// by ID`

`G=gap.SmallGroup(150,3);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,-5,33,650,250,1923]);`
`// Polycyclic`

`G:=Group<a,b|a^75=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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