metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic91, C91⋊3C4, C26.D7, C2.D91, C7⋊Dic13, C14.D13, C13⋊2Dic7, C182.1C2, SmallGroup(364,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C91 — Dic91 |
Generators and relations for Dic91
G = < a,b | a182=1, b2=a91, bab-1=a-1 >
Smallest permutation representation of Dic91
►Regular action on 364 points
Generators in S364
(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)(183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364)
(1 257 92 348)(2 256 93 347)(3 255 94 346)(4 254 95 345)(5 253 96 344)(6 252 97 343)(7 251 98 342)(8 250 99 341)(9 249 100 340)(10 248 101 339)(11 247 102 338)(12 246 103 337)(13 245 104 336)(14 244 105 335)(15 243 106 334)(16 242 107 333)(17 241 108 332)(18 240 109 331)(19 239 110 330)(20 238 111 329)(21 237 112 328)(22 236 113 327)(23 235 114 326)(24 234 115 325)(25 233 116 324)(26 232 117 323)(27 231 118 322)(28 230 119 321)(29 229 120 320)(30 228 121 319)(31 227 122 318)(32 226 123 317)(33 225 124 316)(34 224 125 315)(35 223 126 314)(36 222 127 313)(37 221 128 312)(38 220 129 311)(39 219 130 310)(40 218 131 309)(41 217 132 308)(42 216 133 307)(43 215 134 306)(44 214 135 305)(45 213 136 304)(46 212 137 303)(47 211 138 302)(48 210 139 301)(49 209 140 300)(50 208 141 299)(51 207 142 298)(52 206 143 297)(53 205 144 296)(54 204 145 295)(55 203 146 294)(56 202 147 293)(57 201 148 292)(58 200 149 291)(59 199 150 290)(60 198 151 289)(61 197 152 288)(62 196 153 287)(63 195 154 286)(64 194 155 285)(65 193 156 284)(66 192 157 283)(67 191 158 282)(68 190 159 281)(69 189 160 280)(70 188 161 279)(71 187 162 278)(72 186 163 277)(73 185 164 276)(74 184 165 275)(75 183 166 274)(76 364 167 273)(77 363 168 272)(78 362 169 271)(79 361 170 270)(80 360 171 269)(81 359 172 268)(82 358 173 267)(83 357 174 266)(84 356 175 265)(85 355 176 264)(86 354 177 263)(87 353 178 262)(88 352 179 261)(89 351 180 260)(90 350 181 259)(91 349 182 258)
G:=sub<Sym(364)| (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)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364), (1,257,92,348)(2,256,93,347)(3,255,94,346)(4,254,95,345)(5,253,96,344)(6,252,97,343)(7,251,98,342)(8,250,99,341)(9,249,100,340)(10,248,101,339)(11,247,102,338)(12,246,103,337)(13,245,104,336)(14,244,105,335)(15,243,106,334)(16,242,107,333)(17,241,108,332)(18,240,109,331)(19,239,110,330)(20,238,111,329)(21,237,112,328)(22,236,113,327)(23,235,114,326)(24,234,115,325)(25,233,116,324)(26,232,117,323)(27,231,118,322)(28,230,119,321)(29,229,120,320)(30,228,121,319)(31,227,122,318)(32,226,123,317)(33,225,124,316)(34,224,125,315)(35,223,126,314)(36,222,127,313)(37,221,128,312)(38,220,129,311)(39,219,130,310)(40,218,131,309)(41,217,132,308)(42,216,133,307)(43,215,134,306)(44,214,135,305)(45,213,136,304)(46,212,137,303)(47,211,138,302)(48,210,139,301)(49,209,140,300)(50,208,141,299)(51,207,142,298)(52,206,143,297)(53,205,144,296)(54,204,145,295)(55,203,146,294)(56,202,147,293)(57,201,148,292)(58,200,149,291)(59,199,150,290)(60,198,151,289)(61,197,152,288)(62,196,153,287)(63,195,154,286)(64,194,155,285)(65,193,156,284)(66,192,157,283)(67,191,158,282)(68,190,159,281)(69,189,160,280)(70,188,161,279)(71,187,162,278)(72,186,163,277)(73,185,164,276)(74,184,165,275)(75,183,166,274)(76,364,167,273)(77,363,168,272)(78,362,169,271)(79,361,170,270)(80,360,171,269)(81,359,172,268)(82,358,173,267)(83,357,174,266)(84,356,175,265)(85,355,176,264)(86,354,177,263)(87,353,178,262)(88,352,179,261)(89,351,180,260)(90,350,181,259)(91,349,182,258)>;
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)(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364), (1,257,92,348)(2,256,93,347)(3,255,94,346)(4,254,95,345)(5,253,96,344)(6,252,97,343)(7,251,98,342)(8,250,99,341)(9,249,100,340)(10,248,101,339)(11,247,102,338)(12,246,103,337)(13,245,104,336)(14,244,105,335)(15,243,106,334)(16,242,107,333)(17,241,108,332)(18,240,109,331)(19,239,110,330)(20,238,111,329)(21,237,112,328)(22,236,113,327)(23,235,114,326)(24,234,115,325)(25,233,116,324)(26,232,117,323)(27,231,118,322)(28,230,119,321)(29,229,120,320)(30,228,121,319)(31,227,122,318)(32,226,123,317)(33,225,124,316)(34,224,125,315)(35,223,126,314)(36,222,127,313)(37,221,128,312)(38,220,129,311)(39,219,130,310)(40,218,131,309)(41,217,132,308)(42,216,133,307)(43,215,134,306)(44,214,135,305)(45,213,136,304)(46,212,137,303)(47,211,138,302)(48,210,139,301)(49,209,140,300)(50,208,141,299)(51,207,142,298)(52,206,143,297)(53,205,144,296)(54,204,145,295)(55,203,146,294)(56,202,147,293)(57,201,148,292)(58,200,149,291)(59,199,150,290)(60,198,151,289)(61,197,152,288)(62,196,153,287)(63,195,154,286)(64,194,155,285)(65,193,156,284)(66,192,157,283)(67,191,158,282)(68,190,159,281)(69,189,160,280)(70,188,161,279)(71,187,162,278)(72,186,163,277)(73,185,164,276)(74,184,165,275)(75,183,166,274)(76,364,167,273)(77,363,168,272)(78,362,169,271)(79,361,170,270)(80,360,171,269)(81,359,172,268)(82,358,173,267)(83,357,174,266)(84,356,175,265)(85,355,176,264)(86,354,177,263)(87,353,178,262)(88,352,179,261)(89,351,180,260)(90,350,181,259)(91,349,182,258) );
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),(183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364)], [(1,257,92,348),(2,256,93,347),(3,255,94,346),(4,254,95,345),(5,253,96,344),(6,252,97,343),(7,251,98,342),(8,250,99,341),(9,249,100,340),(10,248,101,339),(11,247,102,338),(12,246,103,337),(13,245,104,336),(14,244,105,335),(15,243,106,334),(16,242,107,333),(17,241,108,332),(18,240,109,331),(19,239,110,330),(20,238,111,329),(21,237,112,328),(22,236,113,327),(23,235,114,326),(24,234,115,325),(25,233,116,324),(26,232,117,323),(27,231,118,322),(28,230,119,321),(29,229,120,320),(30,228,121,319),(31,227,122,318),(32,226,123,317),(33,225,124,316),(34,224,125,315),(35,223,126,314),(36,222,127,313),(37,221,128,312),(38,220,129,311),(39,219,130,310),(40,218,131,309),(41,217,132,308),(42,216,133,307),(43,215,134,306),(44,214,135,305),(45,213,136,304),(46,212,137,303),(47,211,138,302),(48,210,139,301),(49,209,140,300),(50,208,141,299),(51,207,142,298),(52,206,143,297),(53,205,144,296),(54,204,145,295),(55,203,146,294),(56,202,147,293),(57,201,148,292),(58,200,149,291),(59,199,150,290),(60,198,151,289),(61,197,152,288),(62,196,153,287),(63,195,154,286),(64,194,155,285),(65,193,156,284),(66,192,157,283),(67,191,158,282),(68,190,159,281),(69,189,160,280),(70,188,161,279),(71,187,162,278),(72,186,163,277),(73,185,164,276),(74,184,165,275),(75,183,166,274),(76,364,167,273),(77,363,168,272),(78,362,169,271),(79,361,170,270),(80,360,171,269),(81,359,172,268),(82,358,173,267),(83,357,174,266),(84,356,175,265),(85,355,176,264),(86,354,177,263),(87,353,178,262),(88,352,179,261),(89,351,180,260),(90,350,181,259),(91,349,182,258)]])
94 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 13A | ··· | 13F | 14A | 14B | 14C | 26A | ··· | 26F | 91A | ··· | 91AJ | 182A | ··· | 182AJ |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 13 | ··· | 13 | 14 | 14 | 14 | 26 | ··· | 26 | 91 | ··· | 91 | 182 | ··· | 182 |
| size | 1 | 1 | 91 | 91 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
94 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | - | + | - | |
| image | C1 | C2 | C4 | D7 | D13 | Dic7 | Dic13 | D91 | Dic91 |
| kernel | Dic91 | C182 | C91 | C26 | C14 | C13 | C7 | C2 | C1 |
| # reps | 1 | 1 | 2 | 3 | 6 | 3 | 6 | 36 | 36 |
Matrix representation of Dic91 ►in GL3(𝔽1093) generated by
| 1092 | 0 | 0 |
| 0 | 512 | 922 |
| 0 | 171 | 404 |
| 530 | 0 | 0 |
| 0 | 219 | 703 |
| 0 | 151 | 874 |
G:=sub<GL(3,GF(1093))| [1092,0,0,0,512,171,0,922,404],[530,0,0,0,219,151,0,703,874] >;
Dic91 in GAP, Magma, Sage, TeX
{\rm Dic}_{91} % in TeX
G:=Group("Dic91"); // GroupNames label
G:=SmallGroup(364,3);
// by ID
G=gap.SmallGroup(364,3);
# by ID
G:=PCGroup([4,-2,-2,-7,-13,8,290,5379]);
// Polycyclic
G:=Group<a,b|a^182=1,b^2=a^91,b*a*b^-1=a^-1>;
// generators/relations
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