metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D195, C5⋊D39, C3⋊D65, C13⋊D15, C65⋊1S3, C39⋊1D5, C195⋊1C2, C15⋊1D13, sometimes denoted D390 or Dih195 or Dih390, SmallGroup(390,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C195 — D195 |
Generators and relations for D195
G = < a,b | a195=b2=1, bab=a-1 >
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
Export