metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic105, C6.D35, C3⋊Dic35, C7⋊Dic15, C105⋊5C4, C14.D15, C10.D21, C2.D105, C70.1S3, C42.1D5, C30.1D7, C21⋊1Dic5, C5⋊2Dic21, C15⋊3Dic7, C35⋊3Dic3, C210.1C2, SmallGroup(420,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C105 — Dic105 |
Generators and relations for Dic105
G = < a,b | a210=1, b2=a105, bab-1=a-1 >
Smallest permutation representation of Dic105
►Regular action on 420 points
Generators in S420
(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)(211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420)
(1 214 106 319)(2 213 107 318)(3 212 108 317)(4 211 109 316)(5 420 110 315)(6 419 111 314)(7 418 112 313)(8 417 113 312)(9 416 114 311)(10 415 115 310)(11 414 116 309)(12 413 117 308)(13 412 118 307)(14 411 119 306)(15 410 120 305)(16 409 121 304)(17 408 122 303)(18 407 123 302)(19 406 124 301)(20 405 125 300)(21 404 126 299)(22 403 127 298)(23 402 128 297)(24 401 129 296)(25 400 130 295)(26 399 131 294)(27 398 132 293)(28 397 133 292)(29 396 134 291)(30 395 135 290)(31 394 136 289)(32 393 137 288)(33 392 138 287)(34 391 139 286)(35 390 140 285)(36 389 141 284)(37 388 142 283)(38 387 143 282)(39 386 144 281)(40 385 145 280)(41 384 146 279)(42 383 147 278)(43 382 148 277)(44 381 149 276)(45 380 150 275)(46 379 151 274)(47 378 152 273)(48 377 153 272)(49 376 154 271)(50 375 155 270)(51 374 156 269)(52 373 157 268)(53 372 158 267)(54 371 159 266)(55 370 160 265)(56 369 161 264)(57 368 162 263)(58 367 163 262)(59 366 164 261)(60 365 165 260)(61 364 166 259)(62 363 167 258)(63 362 168 257)(64 361 169 256)(65 360 170 255)(66 359 171 254)(67 358 172 253)(68 357 173 252)(69 356 174 251)(70 355 175 250)(71 354 176 249)(72 353 177 248)(73 352 178 247)(74 351 179 246)(75 350 180 245)(76 349 181 244)(77 348 182 243)(78 347 183 242)(79 346 184 241)(80 345 185 240)(81 344 186 239)(82 343 187 238)(83 342 188 237)(84 341 189 236)(85 340 190 235)(86 339 191 234)(87 338 192 233)(88 337 193 232)(89 336 194 231)(90 335 195 230)(91 334 196 229)(92 333 197 228)(93 332 198 227)(94 331 199 226)(95 330 200 225)(96 329 201 224)(97 328 202 223)(98 327 203 222)(99 326 204 221)(100 325 205 220)(101 324 206 219)(102 323 207 218)(103 322 208 217)(104 321 209 216)(105 320 210 215)
G:=sub<Sym(420)| (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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420), (1,214,106,319)(2,213,107,318)(3,212,108,317)(4,211,109,316)(5,420,110,315)(6,419,111,314)(7,418,112,313)(8,417,113,312)(9,416,114,311)(10,415,115,310)(11,414,116,309)(12,413,117,308)(13,412,118,307)(14,411,119,306)(15,410,120,305)(16,409,121,304)(17,408,122,303)(18,407,123,302)(19,406,124,301)(20,405,125,300)(21,404,126,299)(22,403,127,298)(23,402,128,297)(24,401,129,296)(25,400,130,295)(26,399,131,294)(27,398,132,293)(28,397,133,292)(29,396,134,291)(30,395,135,290)(31,394,136,289)(32,393,137,288)(33,392,138,287)(34,391,139,286)(35,390,140,285)(36,389,141,284)(37,388,142,283)(38,387,143,282)(39,386,144,281)(40,385,145,280)(41,384,146,279)(42,383,147,278)(43,382,148,277)(44,381,149,276)(45,380,150,275)(46,379,151,274)(47,378,152,273)(48,377,153,272)(49,376,154,271)(50,375,155,270)(51,374,156,269)(52,373,157,268)(53,372,158,267)(54,371,159,266)(55,370,160,265)(56,369,161,264)(57,368,162,263)(58,367,163,262)(59,366,164,261)(60,365,165,260)(61,364,166,259)(62,363,167,258)(63,362,168,257)(64,361,169,256)(65,360,170,255)(66,359,171,254)(67,358,172,253)(68,357,173,252)(69,356,174,251)(70,355,175,250)(71,354,176,249)(72,353,177,248)(73,352,178,247)(74,351,179,246)(75,350,180,245)(76,349,181,244)(77,348,182,243)(78,347,183,242)(79,346,184,241)(80,345,185,240)(81,344,186,239)(82,343,187,238)(83,342,188,237)(84,341,189,236)(85,340,190,235)(86,339,191,234)(87,338,192,233)(88,337,193,232)(89,336,194,231)(90,335,195,230)(91,334,196,229)(92,333,197,228)(93,332,198,227)(94,331,199,226)(95,330,200,225)(96,329,201,224)(97,328,202,223)(98,327,203,222)(99,326,204,221)(100,325,205,220)(101,324,206,219)(102,323,207,218)(103,322,208,217)(104,321,209,216)(105,320,210,215)>;
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)(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420), (1,214,106,319)(2,213,107,318)(3,212,108,317)(4,211,109,316)(5,420,110,315)(6,419,111,314)(7,418,112,313)(8,417,113,312)(9,416,114,311)(10,415,115,310)(11,414,116,309)(12,413,117,308)(13,412,118,307)(14,411,119,306)(15,410,120,305)(16,409,121,304)(17,408,122,303)(18,407,123,302)(19,406,124,301)(20,405,125,300)(21,404,126,299)(22,403,127,298)(23,402,128,297)(24,401,129,296)(25,400,130,295)(26,399,131,294)(27,398,132,293)(28,397,133,292)(29,396,134,291)(30,395,135,290)(31,394,136,289)(32,393,137,288)(33,392,138,287)(34,391,139,286)(35,390,140,285)(36,389,141,284)(37,388,142,283)(38,387,143,282)(39,386,144,281)(40,385,145,280)(41,384,146,279)(42,383,147,278)(43,382,148,277)(44,381,149,276)(45,380,150,275)(46,379,151,274)(47,378,152,273)(48,377,153,272)(49,376,154,271)(50,375,155,270)(51,374,156,269)(52,373,157,268)(53,372,158,267)(54,371,159,266)(55,370,160,265)(56,369,161,264)(57,368,162,263)(58,367,163,262)(59,366,164,261)(60,365,165,260)(61,364,166,259)(62,363,167,258)(63,362,168,257)(64,361,169,256)(65,360,170,255)(66,359,171,254)(67,358,172,253)(68,357,173,252)(69,356,174,251)(70,355,175,250)(71,354,176,249)(72,353,177,248)(73,352,178,247)(74,351,179,246)(75,350,180,245)(76,349,181,244)(77,348,182,243)(78,347,183,242)(79,346,184,241)(80,345,185,240)(81,344,186,239)(82,343,187,238)(83,342,188,237)(84,341,189,236)(85,340,190,235)(86,339,191,234)(87,338,192,233)(88,337,193,232)(89,336,194,231)(90,335,195,230)(91,334,196,229)(92,333,197,228)(93,332,198,227)(94,331,199,226)(95,330,200,225)(96,329,201,224)(97,328,202,223)(98,327,203,222)(99,326,204,221)(100,325,205,220)(101,324,206,219)(102,323,207,218)(103,322,208,217)(104,321,209,216)(105,320,210,215) );
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),(211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420)], [(1,214,106,319),(2,213,107,318),(3,212,108,317),(4,211,109,316),(5,420,110,315),(6,419,111,314),(7,418,112,313),(8,417,113,312),(9,416,114,311),(10,415,115,310),(11,414,116,309),(12,413,117,308),(13,412,118,307),(14,411,119,306),(15,410,120,305),(16,409,121,304),(17,408,122,303),(18,407,123,302),(19,406,124,301),(20,405,125,300),(21,404,126,299),(22,403,127,298),(23,402,128,297),(24,401,129,296),(25,400,130,295),(26,399,131,294),(27,398,132,293),(28,397,133,292),(29,396,134,291),(30,395,135,290),(31,394,136,289),(32,393,137,288),(33,392,138,287),(34,391,139,286),(35,390,140,285),(36,389,141,284),(37,388,142,283),(38,387,143,282),(39,386,144,281),(40,385,145,280),(41,384,146,279),(42,383,147,278),(43,382,148,277),(44,381,149,276),(45,380,150,275),(46,379,151,274),(47,378,152,273),(48,377,153,272),(49,376,154,271),(50,375,155,270),(51,374,156,269),(52,373,157,268),(53,372,158,267),(54,371,159,266),(55,370,160,265),(56,369,161,264),(57,368,162,263),(58,367,163,262),(59,366,164,261),(60,365,165,260),(61,364,166,259),(62,363,167,258),(63,362,168,257),(64,361,169,256),(65,360,170,255),(66,359,171,254),(67,358,172,253),(68,357,173,252),(69,356,174,251),(70,355,175,250),(71,354,176,249),(72,353,177,248),(73,352,178,247),(74,351,179,246),(75,350,180,245),(76,349,181,244),(77,348,182,243),(78,347,183,242),(79,346,184,241),(80,345,185,240),(81,344,186,239),(82,343,187,238),(83,342,188,237),(84,341,189,236),(85,340,190,235),(86,339,191,234),(87,338,192,233),(88,337,193,232),(89,336,194,231),(90,335,195,230),(91,334,196,229),(92,333,197,228),(93,332,198,227),(94,331,199,226),(95,330,200,225),(96,329,201,224),(97,328,202,223),(98,327,203,222),(99,326,204,221),(100,325,205,220),(101,324,206,219),(102,323,207,218),(103,322,208,217),(104,321,209,216),(105,320,210,215)]])
108 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 5A | 5B | 6 | 7A | 7B | 7C | 10A | 10B | 14A | 14B | 14C | 15A | 15B | 15C | 15D | 21A | ··· | 21F | 30A | 30B | 30C | 30D | 35A | ··· | 35L | 42A | ··· | 42F | 70A | ··· | 70L | 105A | ··· | 105X | 210A | ··· | 210X |
| order | 1 | 2 | 3 | 4 | 4 | 5 | 5 | 6 | 7 | 7 | 7 | 10 | 10 | 14 | 14 | 14 | 15 | 15 | 15 | 15 | 21 | ··· | 21 | 30 | 30 | 30 | 30 | 35 | ··· | 35 | 42 | ··· | 42 | 70 | ··· | 70 | 105 | ··· | 105 | 210 | ··· | 210 |
| size | 1 | 1 | 2 | 105 | 105 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
108 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | - | + | + | - | + | - | - | + | - | |
| image | C1 | C2 | C4 | S3 | D5 | Dic3 | D7 | Dic5 | Dic7 | D15 | D21 | Dic15 | D35 | Dic21 | Dic35 | D105 | Dic105 |
| kernel | Dic105 | C210 | C105 | C70 | C42 | C35 | C30 | C21 | C15 | C14 | C10 | C7 | C6 | C5 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 3 | 2 | 3 | 4 | 6 | 4 | 12 | 6 | 12 | 24 | 24 |
Matrix representation of Dic105 ►in GL4(𝔽421) generated by
| 158 | 371 | 0 | 0 |
| 50 | 216 | 0 | 0 |
| 0 | 0 | 225 | 312 |
| 0 | 0 | 218 | 205 |
| 79 | 167 | 0 | 0 |
| 8 | 342 | 0 | 0 |
| 0 | 0 | 215 | 99 |
| 0 | 0 | 222 | 206 |
G:=sub<GL(4,GF(421))| [158,50,0,0,371,216,0,0,0,0,225,218,0,0,312,205],[79,8,0,0,167,342,0,0,0,0,215,222,0,0,99,206] >;
Dic105 in GAP, Magma, Sage, TeX
{\rm Dic}_{105} % in TeX
G:=Group("Dic105"); // GroupNames label
G:=SmallGroup(420,11);
// by ID
G=gap.SmallGroup(420,11);
# by ID
G:=PCGroup([5,-2,-2,-3,-5,-7,10,122,963,9004]);
// Polycyclic
G:=Group<a,b|a^210=1,b^2=a^105,b*a*b^-1=a^-1>;
// generators/relations
Export