metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic65, C65⋊7C4, C26.D5, C2.D65, C10.D13, C5⋊2Dic13, C13⋊2Dic5, C130.1C2, SmallGroup(260,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — Dic65 |
Generators and relations for Dic65
G = < a,b | a130=1, b2=a65, bab-1=a-1 >
Smallest permutation representation of Dic65
►Regular action on 260 points
Generators in S260
(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)(131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260)
(1 212 66 147)(2 211 67 146)(3 210 68 145)(4 209 69 144)(5 208 70 143)(6 207 71 142)(7 206 72 141)(8 205 73 140)(9 204 74 139)(10 203 75 138)(11 202 76 137)(12 201 77 136)(13 200 78 135)(14 199 79 134)(15 198 80 133)(16 197 81 132)(17 196 82 131)(18 195 83 260)(19 194 84 259)(20 193 85 258)(21 192 86 257)(22 191 87 256)(23 190 88 255)(24 189 89 254)(25 188 90 253)(26 187 91 252)(27 186 92 251)(28 185 93 250)(29 184 94 249)(30 183 95 248)(31 182 96 247)(32 181 97 246)(33 180 98 245)(34 179 99 244)(35 178 100 243)(36 177 101 242)(37 176 102 241)(38 175 103 240)(39 174 104 239)(40 173 105 238)(41 172 106 237)(42 171 107 236)(43 170 108 235)(44 169 109 234)(45 168 110 233)(46 167 111 232)(47 166 112 231)(48 165 113 230)(49 164 114 229)(50 163 115 228)(51 162 116 227)(52 161 117 226)(53 160 118 225)(54 159 119 224)(55 158 120 223)(56 157 121 222)(57 156 122 221)(58 155 123 220)(59 154 124 219)(60 153 125 218)(61 152 126 217)(62 151 127 216)(63 150 128 215)(64 149 129 214)(65 148 130 213)
G:=sub<Sym(260)| (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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260), (1,212,66,147)(2,211,67,146)(3,210,68,145)(4,209,69,144)(5,208,70,143)(6,207,71,142)(7,206,72,141)(8,205,73,140)(9,204,74,139)(10,203,75,138)(11,202,76,137)(12,201,77,136)(13,200,78,135)(14,199,79,134)(15,198,80,133)(16,197,81,132)(17,196,82,131)(18,195,83,260)(19,194,84,259)(20,193,85,258)(21,192,86,257)(22,191,87,256)(23,190,88,255)(24,189,89,254)(25,188,90,253)(26,187,91,252)(27,186,92,251)(28,185,93,250)(29,184,94,249)(30,183,95,248)(31,182,96,247)(32,181,97,246)(33,180,98,245)(34,179,99,244)(35,178,100,243)(36,177,101,242)(37,176,102,241)(38,175,103,240)(39,174,104,239)(40,173,105,238)(41,172,106,237)(42,171,107,236)(43,170,108,235)(44,169,109,234)(45,168,110,233)(46,167,111,232)(47,166,112,231)(48,165,113,230)(49,164,114,229)(50,163,115,228)(51,162,116,227)(52,161,117,226)(53,160,118,225)(54,159,119,224)(55,158,120,223)(56,157,121,222)(57,156,122,221)(58,155,123,220)(59,154,124,219)(60,153,125,218)(61,152,126,217)(62,151,127,216)(63,150,128,215)(64,149,129,214)(65,148,130,213)>;
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)(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260), (1,212,66,147)(2,211,67,146)(3,210,68,145)(4,209,69,144)(5,208,70,143)(6,207,71,142)(7,206,72,141)(8,205,73,140)(9,204,74,139)(10,203,75,138)(11,202,76,137)(12,201,77,136)(13,200,78,135)(14,199,79,134)(15,198,80,133)(16,197,81,132)(17,196,82,131)(18,195,83,260)(19,194,84,259)(20,193,85,258)(21,192,86,257)(22,191,87,256)(23,190,88,255)(24,189,89,254)(25,188,90,253)(26,187,91,252)(27,186,92,251)(28,185,93,250)(29,184,94,249)(30,183,95,248)(31,182,96,247)(32,181,97,246)(33,180,98,245)(34,179,99,244)(35,178,100,243)(36,177,101,242)(37,176,102,241)(38,175,103,240)(39,174,104,239)(40,173,105,238)(41,172,106,237)(42,171,107,236)(43,170,108,235)(44,169,109,234)(45,168,110,233)(46,167,111,232)(47,166,112,231)(48,165,113,230)(49,164,114,229)(50,163,115,228)(51,162,116,227)(52,161,117,226)(53,160,118,225)(54,159,119,224)(55,158,120,223)(56,157,121,222)(57,156,122,221)(58,155,123,220)(59,154,124,219)(60,153,125,218)(61,152,126,217)(62,151,127,216)(63,150,128,215)(64,149,129,214)(65,148,130,213) );
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),(131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260)], [(1,212,66,147),(2,211,67,146),(3,210,68,145),(4,209,69,144),(5,208,70,143),(6,207,71,142),(7,206,72,141),(8,205,73,140),(9,204,74,139),(10,203,75,138),(11,202,76,137),(12,201,77,136),(13,200,78,135),(14,199,79,134),(15,198,80,133),(16,197,81,132),(17,196,82,131),(18,195,83,260),(19,194,84,259),(20,193,85,258),(21,192,86,257),(22,191,87,256),(23,190,88,255),(24,189,89,254),(25,188,90,253),(26,187,91,252),(27,186,92,251),(28,185,93,250),(29,184,94,249),(30,183,95,248),(31,182,96,247),(32,181,97,246),(33,180,98,245),(34,179,99,244),(35,178,100,243),(36,177,101,242),(37,176,102,241),(38,175,103,240),(39,174,104,239),(40,173,105,238),(41,172,106,237),(42,171,107,236),(43,170,108,235),(44,169,109,234),(45,168,110,233),(46,167,111,232),(47,166,112,231),(48,165,113,230),(49,164,114,229),(50,163,115,228),(51,162,116,227),(52,161,117,226),(53,160,118,225),(54,159,119,224),(55,158,120,223),(56,157,121,222),(57,156,122,221),(58,155,123,220),(59,154,124,219),(60,153,125,218),(61,152,126,217),(62,151,127,216),(63,150,128,215),(64,149,129,214),(65,148,130,213)]])
68 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 10A | 10B | 13A | ··· | 13F | 26A | ··· | 26F | 65A | ··· | 65X | 130A | ··· | 130X |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 10 | 10 | 13 | ··· | 13 | 26 | ··· | 26 | 65 | ··· | 65 | 130 | ··· | 130 |
| size | 1 | 1 | 65 | 65 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
68 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | D5 | Dic5 | D13 | Dic13 | D65 | Dic65 |
| kernel | Dic65 | C130 | C65 | C26 | C13 | C10 | C5 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 24 | 24 |
Matrix representation of Dic65 ►in GL3(𝔽521) generated by
| 520 | 0 | 0 |
| 0 | 406 | 46 |
| 0 | 475 | 109 |
| 286 | 0 | 0 |
| 0 | 12 | 192 |
| 0 | 409 | 509 |
G:=sub<GL(3,GF(521))| [520,0,0,0,406,475,0,46,109],[286,0,0,0,12,409,0,192,509] >;
Dic65 in GAP, Magma, Sage, TeX
{\rm Dic}_{65} % in TeX
G:=Group("Dic65"); // GroupNames label
G:=SmallGroup(260,3);
// by ID
G=gap.SmallGroup(260,3);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,8,194,3843]);
// Polycyclic
G:=Group<a,b|a^130=1,b^2=a^65,b*a*b^-1=a^-1>;
// generators/relations
Export