metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic101, C101⋊2C4, C202.C2, C2.D101, SmallGroup(404,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C101 — Dic101 |
Generators and relations for Dic101
G = < a,b | a202=1, b2=a101, bab-1=a-1 >
Smallest permutation representation of Dic101
►Regular action on 404 points
Generators in S404
(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)(203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404)
(1 388 102 287)(2 387 103 286)(3 386 104 285)(4 385 105 284)(5 384 106 283)(6 383 107 282)(7 382 108 281)(8 381 109 280)(9 380 110 279)(10 379 111 278)(11 378 112 277)(12 377 113 276)(13 376 114 275)(14 375 115 274)(15 374 116 273)(16 373 117 272)(17 372 118 271)(18 371 119 270)(19 370 120 269)(20 369 121 268)(21 368 122 267)(22 367 123 266)(23 366 124 265)(24 365 125 264)(25 364 126 263)(26 363 127 262)(27 362 128 261)(28 361 129 260)(29 360 130 259)(30 359 131 258)(31 358 132 257)(32 357 133 256)(33 356 134 255)(34 355 135 254)(35 354 136 253)(36 353 137 252)(37 352 138 251)(38 351 139 250)(39 350 140 249)(40 349 141 248)(41 348 142 247)(42 347 143 246)(43 346 144 245)(44 345 145 244)(45 344 146 243)(46 343 147 242)(47 342 148 241)(48 341 149 240)(49 340 150 239)(50 339 151 238)(51 338 152 237)(52 337 153 236)(53 336 154 235)(54 335 155 234)(55 334 156 233)(56 333 157 232)(57 332 158 231)(58 331 159 230)(59 330 160 229)(60 329 161 228)(61 328 162 227)(62 327 163 226)(63 326 164 225)(64 325 165 224)(65 324 166 223)(66 323 167 222)(67 322 168 221)(68 321 169 220)(69 320 170 219)(70 319 171 218)(71 318 172 217)(72 317 173 216)(73 316 174 215)(74 315 175 214)(75 314 176 213)(76 313 177 212)(77 312 178 211)(78 311 179 210)(79 310 180 209)(80 309 181 208)(81 308 182 207)(82 307 183 206)(83 306 184 205)(84 305 185 204)(85 304 186 203)(86 303 187 404)(87 302 188 403)(88 301 189 402)(89 300 190 401)(90 299 191 400)(91 298 192 399)(92 297 193 398)(93 296 194 397)(94 295 195 396)(95 294 196 395)(96 293 197 394)(97 292 198 393)(98 291 199 392)(99 290 200 391)(100 289 201 390)(101 288 202 389)
G:=sub<Sym(404)| (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)(203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404), (1,388,102,287)(2,387,103,286)(3,386,104,285)(4,385,105,284)(5,384,106,283)(6,383,107,282)(7,382,108,281)(8,381,109,280)(9,380,110,279)(10,379,111,278)(11,378,112,277)(12,377,113,276)(13,376,114,275)(14,375,115,274)(15,374,116,273)(16,373,117,272)(17,372,118,271)(18,371,119,270)(19,370,120,269)(20,369,121,268)(21,368,122,267)(22,367,123,266)(23,366,124,265)(24,365,125,264)(25,364,126,263)(26,363,127,262)(27,362,128,261)(28,361,129,260)(29,360,130,259)(30,359,131,258)(31,358,132,257)(32,357,133,256)(33,356,134,255)(34,355,135,254)(35,354,136,253)(36,353,137,252)(37,352,138,251)(38,351,139,250)(39,350,140,249)(40,349,141,248)(41,348,142,247)(42,347,143,246)(43,346,144,245)(44,345,145,244)(45,344,146,243)(46,343,147,242)(47,342,148,241)(48,341,149,240)(49,340,150,239)(50,339,151,238)(51,338,152,237)(52,337,153,236)(53,336,154,235)(54,335,155,234)(55,334,156,233)(56,333,157,232)(57,332,158,231)(58,331,159,230)(59,330,160,229)(60,329,161,228)(61,328,162,227)(62,327,163,226)(63,326,164,225)(64,325,165,224)(65,324,166,223)(66,323,167,222)(67,322,168,221)(68,321,169,220)(69,320,170,219)(70,319,171,218)(71,318,172,217)(72,317,173,216)(73,316,174,215)(74,315,175,214)(75,314,176,213)(76,313,177,212)(77,312,178,211)(78,311,179,210)(79,310,180,209)(80,309,181,208)(81,308,182,207)(82,307,183,206)(83,306,184,205)(84,305,185,204)(85,304,186,203)(86,303,187,404)(87,302,188,403)(88,301,189,402)(89,300,190,401)(90,299,191,400)(91,298,192,399)(92,297,193,398)(93,296,194,397)(94,295,195,396)(95,294,196,395)(96,293,197,394)(97,292,198,393)(98,291,199,392)(99,290,200,391)(100,289,201,390)(101,288,202,389)>;
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)(203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404), (1,388,102,287)(2,387,103,286)(3,386,104,285)(4,385,105,284)(5,384,106,283)(6,383,107,282)(7,382,108,281)(8,381,109,280)(9,380,110,279)(10,379,111,278)(11,378,112,277)(12,377,113,276)(13,376,114,275)(14,375,115,274)(15,374,116,273)(16,373,117,272)(17,372,118,271)(18,371,119,270)(19,370,120,269)(20,369,121,268)(21,368,122,267)(22,367,123,266)(23,366,124,265)(24,365,125,264)(25,364,126,263)(26,363,127,262)(27,362,128,261)(28,361,129,260)(29,360,130,259)(30,359,131,258)(31,358,132,257)(32,357,133,256)(33,356,134,255)(34,355,135,254)(35,354,136,253)(36,353,137,252)(37,352,138,251)(38,351,139,250)(39,350,140,249)(40,349,141,248)(41,348,142,247)(42,347,143,246)(43,346,144,245)(44,345,145,244)(45,344,146,243)(46,343,147,242)(47,342,148,241)(48,341,149,240)(49,340,150,239)(50,339,151,238)(51,338,152,237)(52,337,153,236)(53,336,154,235)(54,335,155,234)(55,334,156,233)(56,333,157,232)(57,332,158,231)(58,331,159,230)(59,330,160,229)(60,329,161,228)(61,328,162,227)(62,327,163,226)(63,326,164,225)(64,325,165,224)(65,324,166,223)(66,323,167,222)(67,322,168,221)(68,321,169,220)(69,320,170,219)(70,319,171,218)(71,318,172,217)(72,317,173,216)(73,316,174,215)(74,315,175,214)(75,314,176,213)(76,313,177,212)(77,312,178,211)(78,311,179,210)(79,310,180,209)(80,309,181,208)(81,308,182,207)(82,307,183,206)(83,306,184,205)(84,305,185,204)(85,304,186,203)(86,303,187,404)(87,302,188,403)(88,301,189,402)(89,300,190,401)(90,299,191,400)(91,298,192,399)(92,297,193,398)(93,296,194,397)(94,295,195,396)(95,294,196,395)(96,293,197,394)(97,292,198,393)(98,291,199,392)(99,290,200,391)(100,289,201,390)(101,288,202,389) );
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),(203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404)], [(1,388,102,287),(2,387,103,286),(3,386,104,285),(4,385,105,284),(5,384,106,283),(6,383,107,282),(7,382,108,281),(8,381,109,280),(9,380,110,279),(10,379,111,278),(11,378,112,277),(12,377,113,276),(13,376,114,275),(14,375,115,274),(15,374,116,273),(16,373,117,272),(17,372,118,271),(18,371,119,270),(19,370,120,269),(20,369,121,268),(21,368,122,267),(22,367,123,266),(23,366,124,265),(24,365,125,264),(25,364,126,263),(26,363,127,262),(27,362,128,261),(28,361,129,260),(29,360,130,259),(30,359,131,258),(31,358,132,257),(32,357,133,256),(33,356,134,255),(34,355,135,254),(35,354,136,253),(36,353,137,252),(37,352,138,251),(38,351,139,250),(39,350,140,249),(40,349,141,248),(41,348,142,247),(42,347,143,246),(43,346,144,245),(44,345,145,244),(45,344,146,243),(46,343,147,242),(47,342,148,241),(48,341,149,240),(49,340,150,239),(50,339,151,238),(51,338,152,237),(52,337,153,236),(53,336,154,235),(54,335,155,234),(55,334,156,233),(56,333,157,232),(57,332,158,231),(58,331,159,230),(59,330,160,229),(60,329,161,228),(61,328,162,227),(62,327,163,226),(63,326,164,225),(64,325,165,224),(65,324,166,223),(66,323,167,222),(67,322,168,221),(68,321,169,220),(69,320,170,219),(70,319,171,218),(71,318,172,217),(72,317,173,216),(73,316,174,215),(74,315,175,214),(75,314,176,213),(76,313,177,212),(77,312,178,211),(78,311,179,210),(79,310,180,209),(80,309,181,208),(81,308,182,207),(82,307,183,206),(83,306,184,205),(84,305,185,204),(85,304,186,203),(86,303,187,404),(87,302,188,403),(88,301,189,402),(89,300,190,401),(90,299,191,400),(91,298,192,399),(92,297,193,398),(93,296,194,397),(94,295,195,396),(95,294,196,395),(96,293,197,394),(97,292,198,393),(98,291,199,392),(99,290,200,391),(100,289,201,390),(101,288,202,389)]])
104 conjugacy classes
| class | 1 | 2 | 4A | 4B | 101A | ··· | 101AX | 202A | ··· | 202AX |
| order | 1 | 2 | 4 | 4 | 101 | ··· | 101 | 202 | ··· | 202 |
| size | 1 | 1 | 101 | 101 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D101 | Dic101 |
| kernel | Dic101 | C202 | C101 | C2 | C1 |
| # reps | 1 | 1 | 2 | 50 | 50 |
Matrix representation of Dic101 ►in GL2(𝔽809) generated by
| 220 | 1 |
| 808 | 0 |
| 323 | 385 |
| 517 | 486 |
G:=sub<GL(2,GF(809))| [220,808,1,0],[323,517,385,486] >;
Dic101 in GAP, Magma, Sage, TeX
{\rm Dic}_{101} % in TeX
G:=Group("Dic101"); // GroupNames label
G:=SmallGroup(404,1);
// by ID
G=gap.SmallGroup(404,1);
# by ID
G:=PCGroup([3,-2,-2,-101,6,3602]);
// Polycyclic
G:=Group<a,b|a^202=1,b^2=a^101,b*a*b^-1=a^-1>;
// generators/relations
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