metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic99, C99⋊1C4, C2.D99, C22.D9, C11⋊Dic9, C9⋊Dic11, C18.D11, C66.1S3, C6.1D33, C3.Dic33, C198.1C2, C33.1Dic3, SmallGroup(396,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C99 — Dic99 |
Generators and relations for Dic99
G = < a,b | a198=1, b2=a99, bab-1=a-1 >
Smallest permutation representation of Dic99
►Regular action on 396 points
Generators in S396
(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)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396)
(1 264 100 363)(2 263 101 362)(3 262 102 361)(4 261 103 360)(5 260 104 359)(6 259 105 358)(7 258 106 357)(8 257 107 356)(9 256 108 355)(10 255 109 354)(11 254 110 353)(12 253 111 352)(13 252 112 351)(14 251 113 350)(15 250 114 349)(16 249 115 348)(17 248 116 347)(18 247 117 346)(19 246 118 345)(20 245 119 344)(21 244 120 343)(22 243 121 342)(23 242 122 341)(24 241 123 340)(25 240 124 339)(26 239 125 338)(27 238 126 337)(28 237 127 336)(29 236 128 335)(30 235 129 334)(31 234 130 333)(32 233 131 332)(33 232 132 331)(34 231 133 330)(35 230 134 329)(36 229 135 328)(37 228 136 327)(38 227 137 326)(39 226 138 325)(40 225 139 324)(41 224 140 323)(42 223 141 322)(43 222 142 321)(44 221 143 320)(45 220 144 319)(46 219 145 318)(47 218 146 317)(48 217 147 316)(49 216 148 315)(50 215 149 314)(51 214 150 313)(52 213 151 312)(53 212 152 311)(54 211 153 310)(55 210 154 309)(56 209 155 308)(57 208 156 307)(58 207 157 306)(59 206 158 305)(60 205 159 304)(61 204 160 303)(62 203 161 302)(63 202 162 301)(64 201 163 300)(65 200 164 299)(66 199 165 298)(67 396 166 297)(68 395 167 296)(69 394 168 295)(70 393 169 294)(71 392 170 293)(72 391 171 292)(73 390 172 291)(74 389 173 290)(75 388 174 289)(76 387 175 288)(77 386 176 287)(78 385 177 286)(79 384 178 285)(80 383 179 284)(81 382 180 283)(82 381 181 282)(83 380 182 281)(84 379 183 280)(85 378 184 279)(86 377 185 278)(87 376 186 277)(88 375 187 276)(89 374 188 275)(90 373 189 274)(91 372 190 273)(92 371 191 272)(93 370 192 271)(94 369 193 270)(95 368 194 269)(96 367 195 268)(97 366 196 267)(98 365 197 266)(99 364 198 265)
G:=sub<Sym(396)| (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396), (1,264,100,363)(2,263,101,362)(3,262,102,361)(4,261,103,360)(5,260,104,359)(6,259,105,358)(7,258,106,357)(8,257,107,356)(9,256,108,355)(10,255,109,354)(11,254,110,353)(12,253,111,352)(13,252,112,351)(14,251,113,350)(15,250,114,349)(16,249,115,348)(17,248,116,347)(18,247,117,346)(19,246,118,345)(20,245,119,344)(21,244,120,343)(22,243,121,342)(23,242,122,341)(24,241,123,340)(25,240,124,339)(26,239,125,338)(27,238,126,337)(28,237,127,336)(29,236,128,335)(30,235,129,334)(31,234,130,333)(32,233,131,332)(33,232,132,331)(34,231,133,330)(35,230,134,329)(36,229,135,328)(37,228,136,327)(38,227,137,326)(39,226,138,325)(40,225,139,324)(41,224,140,323)(42,223,141,322)(43,222,142,321)(44,221,143,320)(45,220,144,319)(46,219,145,318)(47,218,146,317)(48,217,147,316)(49,216,148,315)(50,215,149,314)(51,214,150,313)(52,213,151,312)(53,212,152,311)(54,211,153,310)(55,210,154,309)(56,209,155,308)(57,208,156,307)(58,207,157,306)(59,206,158,305)(60,205,159,304)(61,204,160,303)(62,203,161,302)(63,202,162,301)(64,201,163,300)(65,200,164,299)(66,199,165,298)(67,396,166,297)(68,395,167,296)(69,394,168,295)(70,393,169,294)(71,392,170,293)(72,391,171,292)(73,390,172,291)(74,389,173,290)(75,388,174,289)(76,387,175,288)(77,386,176,287)(78,385,177,286)(79,384,178,285)(80,383,179,284)(81,382,180,283)(82,381,181,282)(83,380,182,281)(84,379,183,280)(85,378,184,279)(86,377,185,278)(87,376,186,277)(88,375,187,276)(89,374,188,275)(90,373,189,274)(91,372,190,273)(92,371,191,272)(93,370,192,271)(94,369,193,270)(95,368,194,269)(96,367,195,268)(97,366,196,267)(98,365,197,266)(99,364,198,265)>;
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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396), (1,264,100,363)(2,263,101,362)(3,262,102,361)(4,261,103,360)(5,260,104,359)(6,259,105,358)(7,258,106,357)(8,257,107,356)(9,256,108,355)(10,255,109,354)(11,254,110,353)(12,253,111,352)(13,252,112,351)(14,251,113,350)(15,250,114,349)(16,249,115,348)(17,248,116,347)(18,247,117,346)(19,246,118,345)(20,245,119,344)(21,244,120,343)(22,243,121,342)(23,242,122,341)(24,241,123,340)(25,240,124,339)(26,239,125,338)(27,238,126,337)(28,237,127,336)(29,236,128,335)(30,235,129,334)(31,234,130,333)(32,233,131,332)(33,232,132,331)(34,231,133,330)(35,230,134,329)(36,229,135,328)(37,228,136,327)(38,227,137,326)(39,226,138,325)(40,225,139,324)(41,224,140,323)(42,223,141,322)(43,222,142,321)(44,221,143,320)(45,220,144,319)(46,219,145,318)(47,218,146,317)(48,217,147,316)(49,216,148,315)(50,215,149,314)(51,214,150,313)(52,213,151,312)(53,212,152,311)(54,211,153,310)(55,210,154,309)(56,209,155,308)(57,208,156,307)(58,207,157,306)(59,206,158,305)(60,205,159,304)(61,204,160,303)(62,203,161,302)(63,202,162,301)(64,201,163,300)(65,200,164,299)(66,199,165,298)(67,396,166,297)(68,395,167,296)(69,394,168,295)(70,393,169,294)(71,392,170,293)(72,391,171,292)(73,390,172,291)(74,389,173,290)(75,388,174,289)(76,387,175,288)(77,386,176,287)(78,385,177,286)(79,384,178,285)(80,383,179,284)(81,382,180,283)(82,381,181,282)(83,380,182,281)(84,379,183,280)(85,378,184,279)(86,377,185,278)(87,376,186,277)(88,375,187,276)(89,374,188,275)(90,373,189,274)(91,372,190,273)(92,371,191,272)(93,370,192,271)(94,369,193,270)(95,368,194,269)(96,367,195,268)(97,366,196,267)(98,365,197,266)(99,364,198,265) );
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),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396)], [(1,264,100,363),(2,263,101,362),(3,262,102,361),(4,261,103,360),(5,260,104,359),(6,259,105,358),(7,258,106,357),(8,257,107,356),(9,256,108,355),(10,255,109,354),(11,254,110,353),(12,253,111,352),(13,252,112,351),(14,251,113,350),(15,250,114,349),(16,249,115,348),(17,248,116,347),(18,247,117,346),(19,246,118,345),(20,245,119,344),(21,244,120,343),(22,243,121,342),(23,242,122,341),(24,241,123,340),(25,240,124,339),(26,239,125,338),(27,238,126,337),(28,237,127,336),(29,236,128,335),(30,235,129,334),(31,234,130,333),(32,233,131,332),(33,232,132,331),(34,231,133,330),(35,230,134,329),(36,229,135,328),(37,228,136,327),(38,227,137,326),(39,226,138,325),(40,225,139,324),(41,224,140,323),(42,223,141,322),(43,222,142,321),(44,221,143,320),(45,220,144,319),(46,219,145,318),(47,218,146,317),(48,217,147,316),(49,216,148,315),(50,215,149,314),(51,214,150,313),(52,213,151,312),(53,212,152,311),(54,211,153,310),(55,210,154,309),(56,209,155,308),(57,208,156,307),(58,207,157,306),(59,206,158,305),(60,205,159,304),(61,204,160,303),(62,203,161,302),(63,202,162,301),(64,201,163,300),(65,200,164,299),(66,199,165,298),(67,396,166,297),(68,395,167,296),(69,394,168,295),(70,393,169,294),(71,392,170,293),(72,391,171,292),(73,390,172,291),(74,389,173,290),(75,388,174,289),(76,387,175,288),(77,386,176,287),(78,385,177,286),(79,384,178,285),(80,383,179,284),(81,382,180,283),(82,381,181,282),(83,380,182,281),(84,379,183,280),(85,378,184,279),(86,377,185,278),(87,376,186,277),(88,375,187,276),(89,374,188,275),(90,373,189,274),(91,372,190,273),(92,371,191,272),(93,370,192,271),(94,369,193,270),(95,368,194,269),(96,367,195,268),(97,366,196,267),(98,365,197,266),(99,364,198,265)]])
102 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 9A | 9B | 9C | 11A | ··· | 11E | 18A | 18B | 18C | 22A | ··· | 22E | 33A | ··· | 33J | 66A | ··· | 66J | 99A | ··· | 99AD | 198A | ··· | 198AD |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 9 | 9 | 9 | 11 | ··· | 11 | 18 | 18 | 18 | 22 | ··· | 22 | 33 | ··· | 33 | 66 | ··· | 66 | 99 | ··· | 99 | 198 | ··· | 198 |
| size | 1 | 1 | 2 | 99 | 99 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D9 | D11 | Dic9 | Dic11 | D33 | Dic33 | D99 | Dic99 |
| kernel | Dic99 | C198 | C99 | C66 | C33 | C22 | C18 | C11 | C9 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 3 | 5 | 3 | 5 | 10 | 10 | 30 | 30 |
Matrix representation of Dic99 ►in GL3(𝔽397) generated by
| 396 | 0 | 0 |
| 0 | 369 | 230 |
| 0 | 167 | 202 |
| 334 | 0 | 0 |
| 0 | 126 | 336 |
| 0 | 65 | 271 |
G:=sub<GL(3,GF(397))| [396,0,0,0,369,167,0,230,202],[334,0,0,0,126,65,0,336,271] >;
Dic99 in GAP, Magma, Sage, TeX
{\rm Dic}_{99} % in TeX
G:=Group("Dic99"); // GroupNames label
G:=SmallGroup(396,3);
// by ID
G=gap.SmallGroup(396,3);
# by ID
G:=PCGroup([5,-2,-2,-3,-11,-3,10,2102,1002,2403,6604]);
// Polycyclic
G:=Group<a,b|a^198=1,b^2=a^99,b*a*b^-1=a^-1>;
// generators/relations
Export