metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic83, C83⋊C4, C2.D83, C166.C2, SmallGroup(332,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C83 — Dic83 |
Generators and relations for Dic83
G = < a,b | a166=1, b2=a83, bab-1=a-1 >
Smallest permutation representation of Dic83
►Regular action on 332 points
Generators in S332
(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)(167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332)
(1 241 84 324)(2 240 85 323)(3 239 86 322)(4 238 87 321)(5 237 88 320)(6 236 89 319)(7 235 90 318)(8 234 91 317)(9 233 92 316)(10 232 93 315)(11 231 94 314)(12 230 95 313)(13 229 96 312)(14 228 97 311)(15 227 98 310)(16 226 99 309)(17 225 100 308)(18 224 101 307)(19 223 102 306)(20 222 103 305)(21 221 104 304)(22 220 105 303)(23 219 106 302)(24 218 107 301)(25 217 108 300)(26 216 109 299)(27 215 110 298)(28 214 111 297)(29 213 112 296)(30 212 113 295)(31 211 114 294)(32 210 115 293)(33 209 116 292)(34 208 117 291)(35 207 118 290)(36 206 119 289)(37 205 120 288)(38 204 121 287)(39 203 122 286)(40 202 123 285)(41 201 124 284)(42 200 125 283)(43 199 126 282)(44 198 127 281)(45 197 128 280)(46 196 129 279)(47 195 130 278)(48 194 131 277)(49 193 132 276)(50 192 133 275)(51 191 134 274)(52 190 135 273)(53 189 136 272)(54 188 137 271)(55 187 138 270)(56 186 139 269)(57 185 140 268)(58 184 141 267)(59 183 142 266)(60 182 143 265)(61 181 144 264)(62 180 145 263)(63 179 146 262)(64 178 147 261)(65 177 148 260)(66 176 149 259)(67 175 150 258)(68 174 151 257)(69 173 152 256)(70 172 153 255)(71 171 154 254)(72 170 155 253)(73 169 156 252)(74 168 157 251)(75 167 158 250)(76 332 159 249)(77 331 160 248)(78 330 161 247)(79 329 162 246)(80 328 163 245)(81 327 164 244)(82 326 165 243)(83 325 166 242)
G:=sub<Sym(332)| (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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332), (1,241,84,324)(2,240,85,323)(3,239,86,322)(4,238,87,321)(5,237,88,320)(6,236,89,319)(7,235,90,318)(8,234,91,317)(9,233,92,316)(10,232,93,315)(11,231,94,314)(12,230,95,313)(13,229,96,312)(14,228,97,311)(15,227,98,310)(16,226,99,309)(17,225,100,308)(18,224,101,307)(19,223,102,306)(20,222,103,305)(21,221,104,304)(22,220,105,303)(23,219,106,302)(24,218,107,301)(25,217,108,300)(26,216,109,299)(27,215,110,298)(28,214,111,297)(29,213,112,296)(30,212,113,295)(31,211,114,294)(32,210,115,293)(33,209,116,292)(34,208,117,291)(35,207,118,290)(36,206,119,289)(37,205,120,288)(38,204,121,287)(39,203,122,286)(40,202,123,285)(41,201,124,284)(42,200,125,283)(43,199,126,282)(44,198,127,281)(45,197,128,280)(46,196,129,279)(47,195,130,278)(48,194,131,277)(49,193,132,276)(50,192,133,275)(51,191,134,274)(52,190,135,273)(53,189,136,272)(54,188,137,271)(55,187,138,270)(56,186,139,269)(57,185,140,268)(58,184,141,267)(59,183,142,266)(60,182,143,265)(61,181,144,264)(62,180,145,263)(63,179,146,262)(64,178,147,261)(65,177,148,260)(66,176,149,259)(67,175,150,258)(68,174,151,257)(69,173,152,256)(70,172,153,255)(71,171,154,254)(72,170,155,253)(73,169,156,252)(74,168,157,251)(75,167,158,250)(76,332,159,249)(77,331,160,248)(78,330,161,247)(79,329,162,246)(80,328,163,245)(81,327,164,244)(82,326,165,243)(83,325,166,242)>;
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)(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332), (1,241,84,324)(2,240,85,323)(3,239,86,322)(4,238,87,321)(5,237,88,320)(6,236,89,319)(7,235,90,318)(8,234,91,317)(9,233,92,316)(10,232,93,315)(11,231,94,314)(12,230,95,313)(13,229,96,312)(14,228,97,311)(15,227,98,310)(16,226,99,309)(17,225,100,308)(18,224,101,307)(19,223,102,306)(20,222,103,305)(21,221,104,304)(22,220,105,303)(23,219,106,302)(24,218,107,301)(25,217,108,300)(26,216,109,299)(27,215,110,298)(28,214,111,297)(29,213,112,296)(30,212,113,295)(31,211,114,294)(32,210,115,293)(33,209,116,292)(34,208,117,291)(35,207,118,290)(36,206,119,289)(37,205,120,288)(38,204,121,287)(39,203,122,286)(40,202,123,285)(41,201,124,284)(42,200,125,283)(43,199,126,282)(44,198,127,281)(45,197,128,280)(46,196,129,279)(47,195,130,278)(48,194,131,277)(49,193,132,276)(50,192,133,275)(51,191,134,274)(52,190,135,273)(53,189,136,272)(54,188,137,271)(55,187,138,270)(56,186,139,269)(57,185,140,268)(58,184,141,267)(59,183,142,266)(60,182,143,265)(61,181,144,264)(62,180,145,263)(63,179,146,262)(64,178,147,261)(65,177,148,260)(66,176,149,259)(67,175,150,258)(68,174,151,257)(69,173,152,256)(70,172,153,255)(71,171,154,254)(72,170,155,253)(73,169,156,252)(74,168,157,251)(75,167,158,250)(76,332,159,249)(77,331,160,248)(78,330,161,247)(79,329,162,246)(80,328,163,245)(81,327,164,244)(82,326,165,243)(83,325,166,242) );
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),(167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332)], [(1,241,84,324),(2,240,85,323),(3,239,86,322),(4,238,87,321),(5,237,88,320),(6,236,89,319),(7,235,90,318),(8,234,91,317),(9,233,92,316),(10,232,93,315),(11,231,94,314),(12,230,95,313),(13,229,96,312),(14,228,97,311),(15,227,98,310),(16,226,99,309),(17,225,100,308),(18,224,101,307),(19,223,102,306),(20,222,103,305),(21,221,104,304),(22,220,105,303),(23,219,106,302),(24,218,107,301),(25,217,108,300),(26,216,109,299),(27,215,110,298),(28,214,111,297),(29,213,112,296),(30,212,113,295),(31,211,114,294),(32,210,115,293),(33,209,116,292),(34,208,117,291),(35,207,118,290),(36,206,119,289),(37,205,120,288),(38,204,121,287),(39,203,122,286),(40,202,123,285),(41,201,124,284),(42,200,125,283),(43,199,126,282),(44,198,127,281),(45,197,128,280),(46,196,129,279),(47,195,130,278),(48,194,131,277),(49,193,132,276),(50,192,133,275),(51,191,134,274),(52,190,135,273),(53,189,136,272),(54,188,137,271),(55,187,138,270),(56,186,139,269),(57,185,140,268),(58,184,141,267),(59,183,142,266),(60,182,143,265),(61,181,144,264),(62,180,145,263),(63,179,146,262),(64,178,147,261),(65,177,148,260),(66,176,149,259),(67,175,150,258),(68,174,151,257),(69,173,152,256),(70,172,153,255),(71,171,154,254),(72,170,155,253),(73,169,156,252),(74,168,157,251),(75,167,158,250),(76,332,159,249),(77,331,160,248),(78,330,161,247),(79,329,162,246),(80,328,163,245),(81,327,164,244),(82,326,165,243),(83,325,166,242)]])
86 conjugacy classes
| class | 1 | 2 | 4A | 4B | 83A | ··· | 83AO | 166A | ··· | 166AO |
| order | 1 | 2 | 4 | 4 | 83 | ··· | 83 | 166 | ··· | 166 |
| size | 1 | 1 | 83 | 83 | 2 | ··· | 2 | 2 | ··· | 2 |
86 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D83 | Dic83 |
| kernel | Dic83 | C166 | C83 | C2 | C1 |
| # reps | 1 | 1 | 2 | 41 | 41 |
Matrix representation of Dic83 ►in GL2(𝔽997) generated by
| 420 | 1 |
| 996 | 0 |
| 362 | 666 |
| 170 | 635 |
G:=sub<GL(2,GF(997))| [420,996,1,0],[362,170,666,635] >;
Dic83 in GAP, Magma, Sage, TeX
{\rm Dic}_{83} % in TeX
G:=Group("Dic83"); // GroupNames label
G:=SmallGroup(332,1);
// by ID
G=gap.SmallGroup(332,1);
# by ID
G:=PCGroup([3,-2,-2,-83,6,2954]);
// Polycyclic
G:=Group<a,b|a^166=1,b^2=a^83,b*a*b^-1=a^-1>;
// generators/relations
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