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## G = D195order 390 = 2·3·5·13

### Dihedral group

Aliases: D195, C5⋊D39, C3⋊D65, C13⋊D15, C651S3, C391D5, C1951C2, C151D13, sometimes denoted D390 or Dih195 or Dih390, SmallGroup(390,11)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C195 — D195
 Chief series C1 — C13 — C65 — C195 — D195
 Lower central C195 — D195
 Upper central C1

Generators and relations for D195
G = < a,b | a195=b2=1, bab=a-1 >

195C2
65S3
39D5
15D13
13D15
5D39
3D65

Smallest permutation representation of D195
On 195 points
Generators in S195
```(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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195)
(2 195)(3 194)(4 193)(5 192)(6 191)(7 190)(8 189)(9 188)(10 187)(11 186)(12 185)(13 184)(14 183)(15 182)(16 181)(17 180)(18 179)(19 178)(20 177)(21 176)(22 175)(23 174)(24 173)(25 172)(26 171)(27 170)(28 169)(29 168)(30 167)(31 166)(32 165)(33 164)(34 163)(35 162)(36 161)(37 160)(38 159)(39 158)(40 157)(41 156)(42 155)(43 154)(44 153)(45 152)(46 151)(47 150)(48 149)(49 148)(50 147)(51 146)(52 145)(53 144)(54 143)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(61 136)(62 135)(63 134)(64 133)(65 132)(66 131)(67 130)(68 129)(69 128)(70 127)(71 126)(72 125)(73 124)(74 123)(75 122)(76 121)(77 120)(78 119)(79 118)(80 117)(81 116)(82 115)(83 114)(84 113)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)```

`G:=sub<Sym(195)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (2,195)(3,194)(4,193)(5,192)(6,191)(7,190)(8,189)(9,188)(10,187)(11,186)(12,185)(13,184)(14,183)(15,182)(16,181)(17,180)(18,179)(19,178)(20,177)(21,176)(22,175)(23,174)(24,173)(25,172)(26,171)(27,170)(28,169)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,135)(63,134)(64,133)(65,132)(66,131)(67,130)(68,129)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99)>;`

`G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195), (2,195)(3,194)(4,193)(5,192)(6,191)(7,190)(8,189)(9,188)(10,187)(11,186)(12,185)(13,184)(14,183)(15,182)(16,181)(17,180)(18,179)(19,178)(20,177)(21,176)(22,175)(23,174)(24,173)(25,172)(26,171)(27,170)(28,169)(29,168)(30,167)(31,166)(32,165)(33,164)(34,163)(35,162)(36,161)(37,160)(38,159)(39,158)(40,157)(41,156)(42,155)(43,154)(44,153)(45,152)(46,151)(47,150)(48,149)(49,148)(50,147)(51,146)(52,145)(53,144)(54,143)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(61,136)(62,135)(63,134)(64,133)(65,132)(66,131)(67,130)(68,129)(69,128)(70,127)(71,126)(72,125)(73,124)(74,123)(75,122)(76,121)(77,120)(78,119)(79,118)(80,117)(81,116)(82,115)(83,114)(84,113)(85,112)(86,111)(87,110)(88,109)(89,108)(90,107)(91,106)(92,105)(93,104)(94,103)(95,102)(96,101)(97,100)(98,99) );`

`G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195)], [(2,195),(3,194),(4,193),(5,192),(6,191),(7,190),(8,189),(9,188),(10,187),(11,186),(12,185),(13,184),(14,183),(15,182),(16,181),(17,180),(18,179),(19,178),(20,177),(21,176),(22,175),(23,174),(24,173),(25,172),(26,171),(27,170),(28,169),(29,168),(30,167),(31,166),(32,165),(33,164),(34,163),(35,162),(36,161),(37,160),(38,159),(39,158),(40,157),(41,156),(42,155),(43,154),(44,153),(45,152),(46,151),(47,150),(48,149),(49,148),(50,147),(51,146),(52,145),(53,144),(54,143),(55,142),(56,141),(57,140),(58,139),(59,138),(60,137),(61,136),(62,135),(63,134),(64,133),(65,132),(66,131),(67,130),(68,129),(69,128),(70,127),(71,126),(72,125),(73,124),(74,123),(75,122),(76,121),(77,120),(78,119),(79,118),(80,117),(81,116),(82,115),(83,114),(84,113),(85,112),(86,111),(87,110),(88,109),(89,108),(90,107),(91,106),(92,105),(93,104),(94,103),(95,102),(96,101),(97,100),(98,99)])`

99 conjugacy classes

 class 1 2 3 5A 5B 13A ··· 13F 15A 15B 15C 15D 39A ··· 39L 65A ··· 65X 195A ··· 195AV order 1 2 3 5 5 13 ··· 13 15 15 15 15 39 ··· 39 65 ··· 65 195 ··· 195 size 1 195 2 2 2 2 ··· 2 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

99 irreducible representations

 dim 1 1 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 S3 D5 D13 D15 D39 D65 D195 kernel D195 C195 C65 C39 C15 C13 C5 C3 C1 # reps 1 1 1 2 6 4 12 24 48

Matrix representation of D195 in GL2(𝔽1171) generated by

 667 918 253 127
,
 1 0 826 1170
`G:=sub<GL(2,GF(1171))| [667,253,918,127],[1,826,0,1170] >;`

D195 in GAP, Magma, Sage, TeX

`D_{195}`
`% in TeX`

`G:=Group("D195");`
`// GroupNames label`

`G:=SmallGroup(390,11);`
`// by ID`

`G=gap.SmallGroup(390,11);`
`# by ID`

`G:=PCGroup([4,-2,-3,-5,-13,33,290,5763]);`
`// Polycyclic`

`G:=Group<a,b|a^195=b^2=1,b*a*b=a^-1>;`
`// generators/relations`

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