metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic100, C8.D25, C25⋊1Q16, C50.1D4, C40.1D5, C4.8D50, C5.Dic20, C200.1C2, C10.1D20, C2.3D100, C20.40D10, C100.8C22, Dic50.1C2, SmallGroup(400,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic100
G = < a,b | a200=1, b2=a100, bab-1=a-1 >
Smallest permutation representation of Dic100
►Regular action on 400 points
Generators in S400
(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400)
(1 239 101 339)(2 238 102 338)(3 237 103 337)(4 236 104 336)(5 235 105 335)(6 234 106 334)(7 233 107 333)(8 232 108 332)(9 231 109 331)(10 230 110 330)(11 229 111 329)(12 228 112 328)(13 227 113 327)(14 226 114 326)(15 225 115 325)(16 224 116 324)(17 223 117 323)(18 222 118 322)(19 221 119 321)(20 220 120 320)(21 219 121 319)(22 218 122 318)(23 217 123 317)(24 216 124 316)(25 215 125 315)(26 214 126 314)(27 213 127 313)(28 212 128 312)(29 211 129 311)(30 210 130 310)(31 209 131 309)(32 208 132 308)(33 207 133 307)(34 206 134 306)(35 205 135 305)(36 204 136 304)(37 203 137 303)(38 202 138 302)(39 201 139 301)(40 400 140 300)(41 399 141 299)(42 398 142 298)(43 397 143 297)(44 396 144 296)(45 395 145 295)(46 394 146 294)(47 393 147 293)(48 392 148 292)(49 391 149 291)(50 390 150 290)(51 389 151 289)(52 388 152 288)(53 387 153 287)(54 386 154 286)(55 385 155 285)(56 384 156 284)(57 383 157 283)(58 382 158 282)(59 381 159 281)(60 380 160 280)(61 379 161 279)(62 378 162 278)(63 377 163 277)(64 376 164 276)(65 375 165 275)(66 374 166 274)(67 373 167 273)(68 372 168 272)(69 371 169 271)(70 370 170 270)(71 369 171 269)(72 368 172 268)(73 367 173 267)(74 366 174 266)(75 365 175 265)(76 364 176 264)(77 363 177 263)(78 362 178 262)(79 361 179 261)(80 360 180 260)(81 359 181 259)(82 358 182 258)(83 357 183 257)(84 356 184 256)(85 355 185 255)(86 354 186 254)(87 353 187 253)(88 352 188 252)(89 351 189 251)(90 350 190 250)(91 349 191 249)(92 348 192 248)(93 347 193 247)(94 346 194 246)(95 345 195 245)(96 344 196 244)(97 343 197 243)(98 342 198 242)(99 341 199 241)(100 340 200 240)
G:=sub<Sym(400)| (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400), (1,239,101,339)(2,238,102,338)(3,237,103,337)(4,236,104,336)(5,235,105,335)(6,234,106,334)(7,233,107,333)(8,232,108,332)(9,231,109,331)(10,230,110,330)(11,229,111,329)(12,228,112,328)(13,227,113,327)(14,226,114,326)(15,225,115,325)(16,224,116,324)(17,223,117,323)(18,222,118,322)(19,221,119,321)(20,220,120,320)(21,219,121,319)(22,218,122,318)(23,217,123,317)(24,216,124,316)(25,215,125,315)(26,214,126,314)(27,213,127,313)(28,212,128,312)(29,211,129,311)(30,210,130,310)(31,209,131,309)(32,208,132,308)(33,207,133,307)(34,206,134,306)(35,205,135,305)(36,204,136,304)(37,203,137,303)(38,202,138,302)(39,201,139,301)(40,400,140,300)(41,399,141,299)(42,398,142,298)(43,397,143,297)(44,396,144,296)(45,395,145,295)(46,394,146,294)(47,393,147,293)(48,392,148,292)(49,391,149,291)(50,390,150,290)(51,389,151,289)(52,388,152,288)(53,387,153,287)(54,386,154,286)(55,385,155,285)(56,384,156,284)(57,383,157,283)(58,382,158,282)(59,381,159,281)(60,380,160,280)(61,379,161,279)(62,378,162,278)(63,377,163,277)(64,376,164,276)(65,375,165,275)(66,374,166,274)(67,373,167,273)(68,372,168,272)(69,371,169,271)(70,370,170,270)(71,369,171,269)(72,368,172,268)(73,367,173,267)(74,366,174,266)(75,365,175,265)(76,364,176,264)(77,363,177,263)(78,362,178,262)(79,361,179,261)(80,360,180,260)(81,359,181,259)(82,358,182,258)(83,357,183,257)(84,356,184,256)(85,355,185,255)(86,354,186,254)(87,353,187,253)(88,352,188,252)(89,351,189,251)(90,350,190,250)(91,349,191,249)(92,348,192,248)(93,347,193,247)(94,346,194,246)(95,345,195,245)(96,344,196,244)(97,343,197,243)(98,342,198,242)(99,341,199,241)(100,340,200,240)>;
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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400), (1,239,101,339)(2,238,102,338)(3,237,103,337)(4,236,104,336)(5,235,105,335)(6,234,106,334)(7,233,107,333)(8,232,108,332)(9,231,109,331)(10,230,110,330)(11,229,111,329)(12,228,112,328)(13,227,113,327)(14,226,114,326)(15,225,115,325)(16,224,116,324)(17,223,117,323)(18,222,118,322)(19,221,119,321)(20,220,120,320)(21,219,121,319)(22,218,122,318)(23,217,123,317)(24,216,124,316)(25,215,125,315)(26,214,126,314)(27,213,127,313)(28,212,128,312)(29,211,129,311)(30,210,130,310)(31,209,131,309)(32,208,132,308)(33,207,133,307)(34,206,134,306)(35,205,135,305)(36,204,136,304)(37,203,137,303)(38,202,138,302)(39,201,139,301)(40,400,140,300)(41,399,141,299)(42,398,142,298)(43,397,143,297)(44,396,144,296)(45,395,145,295)(46,394,146,294)(47,393,147,293)(48,392,148,292)(49,391,149,291)(50,390,150,290)(51,389,151,289)(52,388,152,288)(53,387,153,287)(54,386,154,286)(55,385,155,285)(56,384,156,284)(57,383,157,283)(58,382,158,282)(59,381,159,281)(60,380,160,280)(61,379,161,279)(62,378,162,278)(63,377,163,277)(64,376,164,276)(65,375,165,275)(66,374,166,274)(67,373,167,273)(68,372,168,272)(69,371,169,271)(70,370,170,270)(71,369,171,269)(72,368,172,268)(73,367,173,267)(74,366,174,266)(75,365,175,265)(76,364,176,264)(77,363,177,263)(78,362,178,262)(79,361,179,261)(80,360,180,260)(81,359,181,259)(82,358,182,258)(83,357,183,257)(84,356,184,256)(85,355,185,255)(86,354,186,254)(87,353,187,253)(88,352,188,252)(89,351,189,251)(90,350,190,250)(91,349,191,249)(92,348,192,248)(93,347,193,247)(94,346,194,246)(95,345,195,245)(96,344,196,244)(97,343,197,243)(98,342,198,242)(99,341,199,241)(100,340,200,240) );
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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)], [(1,239,101,339),(2,238,102,338),(3,237,103,337),(4,236,104,336),(5,235,105,335),(6,234,106,334),(7,233,107,333),(8,232,108,332),(9,231,109,331),(10,230,110,330),(11,229,111,329),(12,228,112,328),(13,227,113,327),(14,226,114,326),(15,225,115,325),(16,224,116,324),(17,223,117,323),(18,222,118,322),(19,221,119,321),(20,220,120,320),(21,219,121,319),(22,218,122,318),(23,217,123,317),(24,216,124,316),(25,215,125,315),(26,214,126,314),(27,213,127,313),(28,212,128,312),(29,211,129,311),(30,210,130,310),(31,209,131,309),(32,208,132,308),(33,207,133,307),(34,206,134,306),(35,205,135,305),(36,204,136,304),(37,203,137,303),(38,202,138,302),(39,201,139,301),(40,400,140,300),(41,399,141,299),(42,398,142,298),(43,397,143,297),(44,396,144,296),(45,395,145,295),(46,394,146,294),(47,393,147,293),(48,392,148,292),(49,391,149,291),(50,390,150,290),(51,389,151,289),(52,388,152,288),(53,387,153,287),(54,386,154,286),(55,385,155,285),(56,384,156,284),(57,383,157,283),(58,382,158,282),(59,381,159,281),(60,380,160,280),(61,379,161,279),(62,378,162,278),(63,377,163,277),(64,376,164,276),(65,375,165,275),(66,374,166,274),(67,373,167,273),(68,372,168,272),(69,371,169,271),(70,370,170,270),(71,369,171,269),(72,368,172,268),(73,367,173,267),(74,366,174,266),(75,365,175,265),(76,364,176,264),(77,363,177,263),(78,362,178,262),(79,361,179,261),(80,360,180,260),(81,359,181,259),(82,358,182,258),(83,357,183,257),(84,356,184,256),(85,355,185,255),(86,354,186,254),(87,353,187,253),(88,352,188,252),(89,351,189,251),(90,350,190,250),(91,349,191,249),(92,348,192,248),(93,347,193,247),(94,346,194,246),(95,345,195,245),(96,344,196,244),(97,343,197,243),(98,342,198,242),(99,341,199,241),(100,340,200,240)]])
103 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 5A | 5B | 8A | 8B | 10A | 10B | 20A | 20B | 20C | 20D | 25A | ··· | 25J | 40A | ··· | 40H | 50A | ··· | 50J | 100A | ··· | 100T | 200A | ··· | 200AN |
| order | 1 | 2 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 10 | 10 | 20 | 20 | 20 | 20 | 25 | ··· | 25 | 40 | ··· | 40 | 50 | ··· | 50 | 100 | ··· | 100 | 200 | ··· | 200 |
| size | 1 | 1 | 2 | 100 | 100 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
103 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | D4 | D5 | Q16 | D10 | D20 | D25 | Dic20 | D50 | D100 | Dic100 |
| kernel | Dic100 | C200 | Dic50 | C50 | C40 | C25 | C20 | C10 | C8 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 10 | 8 | 10 | 20 | 40 |
Matrix representation of Dic100 ►in GL2(𝔽401) generated by
| 284 | 305 |
| 96 | 209 |
| 89 | 278 |
| 335 | 312 |
G:=sub<GL(2,GF(401))| [284,96,305,209],[89,335,278,312] >;
Dic100 in GAP, Magma, Sage, TeX
{\rm Dic}_{100} % in TeX
G:=Group("Dic100"); // GroupNames label
G:=SmallGroup(400,4);
// by ID
G=gap.SmallGroup(400,4);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,79,218,50,4324,628,11525]);
// Polycyclic
G:=Group<a,b|a^200=1,b^2=a^100,b*a*b^-1=a^-1>;
// generators/relations
Export