metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic107, C107⋊C4, C214.C2, C2.D107, SmallGroup(428,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C107 — Dic107 |
Generators and relations for Dic107
G = < a,b | a214=1, b2=a107, bab-1=a-1 >
Smallest permutation representation of Dic107
►Regular action on 428 points
Generators in S428
(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)(215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428)
(1 370 108 263)(2 369 109 262)(3 368 110 261)(4 367 111 260)(5 366 112 259)(6 365 113 258)(7 364 114 257)(8 363 115 256)(9 362 116 255)(10 361 117 254)(11 360 118 253)(12 359 119 252)(13 358 120 251)(14 357 121 250)(15 356 122 249)(16 355 123 248)(17 354 124 247)(18 353 125 246)(19 352 126 245)(20 351 127 244)(21 350 128 243)(22 349 129 242)(23 348 130 241)(24 347 131 240)(25 346 132 239)(26 345 133 238)(27 344 134 237)(28 343 135 236)(29 342 136 235)(30 341 137 234)(31 340 138 233)(32 339 139 232)(33 338 140 231)(34 337 141 230)(35 336 142 229)(36 335 143 228)(37 334 144 227)(38 333 145 226)(39 332 146 225)(40 331 147 224)(41 330 148 223)(42 329 149 222)(43 328 150 221)(44 327 151 220)(45 326 152 219)(46 325 153 218)(47 324 154 217)(48 323 155 216)(49 322 156 215)(50 321 157 428)(51 320 158 427)(52 319 159 426)(53 318 160 425)(54 317 161 424)(55 316 162 423)(56 315 163 422)(57 314 164 421)(58 313 165 420)(59 312 166 419)(60 311 167 418)(61 310 168 417)(62 309 169 416)(63 308 170 415)(64 307 171 414)(65 306 172 413)(66 305 173 412)(67 304 174 411)(68 303 175 410)(69 302 176 409)(70 301 177 408)(71 300 178 407)(72 299 179 406)(73 298 180 405)(74 297 181 404)(75 296 182 403)(76 295 183 402)(77 294 184 401)(78 293 185 400)(79 292 186 399)(80 291 187 398)(81 290 188 397)(82 289 189 396)(83 288 190 395)(84 287 191 394)(85 286 192 393)(86 285 193 392)(87 284 194 391)(88 283 195 390)(89 282 196 389)(90 281 197 388)(91 280 198 387)(92 279 199 386)(93 278 200 385)(94 277 201 384)(95 276 202 383)(96 275 203 382)(97 274 204 381)(98 273 205 380)(99 272 206 379)(100 271 207 378)(101 270 208 377)(102 269 209 376)(103 268 210 375)(104 267 211 374)(105 266 212 373)(106 265 213 372)(107 264 214 371)
G:=sub<Sym(428)| (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)(215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428), (1,370,108,263)(2,369,109,262)(3,368,110,261)(4,367,111,260)(5,366,112,259)(6,365,113,258)(7,364,114,257)(8,363,115,256)(9,362,116,255)(10,361,117,254)(11,360,118,253)(12,359,119,252)(13,358,120,251)(14,357,121,250)(15,356,122,249)(16,355,123,248)(17,354,124,247)(18,353,125,246)(19,352,126,245)(20,351,127,244)(21,350,128,243)(22,349,129,242)(23,348,130,241)(24,347,131,240)(25,346,132,239)(26,345,133,238)(27,344,134,237)(28,343,135,236)(29,342,136,235)(30,341,137,234)(31,340,138,233)(32,339,139,232)(33,338,140,231)(34,337,141,230)(35,336,142,229)(36,335,143,228)(37,334,144,227)(38,333,145,226)(39,332,146,225)(40,331,147,224)(41,330,148,223)(42,329,149,222)(43,328,150,221)(44,327,151,220)(45,326,152,219)(46,325,153,218)(47,324,154,217)(48,323,155,216)(49,322,156,215)(50,321,157,428)(51,320,158,427)(52,319,159,426)(53,318,160,425)(54,317,161,424)(55,316,162,423)(56,315,163,422)(57,314,164,421)(58,313,165,420)(59,312,166,419)(60,311,167,418)(61,310,168,417)(62,309,169,416)(63,308,170,415)(64,307,171,414)(65,306,172,413)(66,305,173,412)(67,304,174,411)(68,303,175,410)(69,302,176,409)(70,301,177,408)(71,300,178,407)(72,299,179,406)(73,298,180,405)(74,297,181,404)(75,296,182,403)(76,295,183,402)(77,294,184,401)(78,293,185,400)(79,292,186,399)(80,291,187,398)(81,290,188,397)(82,289,189,396)(83,288,190,395)(84,287,191,394)(85,286,192,393)(86,285,193,392)(87,284,194,391)(88,283,195,390)(89,282,196,389)(90,281,197,388)(91,280,198,387)(92,279,199,386)(93,278,200,385)(94,277,201,384)(95,276,202,383)(96,275,203,382)(97,274,204,381)(98,273,205,380)(99,272,206,379)(100,271,207,378)(101,270,208,377)(102,269,209,376)(103,268,210,375)(104,267,211,374)(105,266,212,373)(106,265,213,372)(107,264,214,371)>;
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)(215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428), (1,370,108,263)(2,369,109,262)(3,368,110,261)(4,367,111,260)(5,366,112,259)(6,365,113,258)(7,364,114,257)(8,363,115,256)(9,362,116,255)(10,361,117,254)(11,360,118,253)(12,359,119,252)(13,358,120,251)(14,357,121,250)(15,356,122,249)(16,355,123,248)(17,354,124,247)(18,353,125,246)(19,352,126,245)(20,351,127,244)(21,350,128,243)(22,349,129,242)(23,348,130,241)(24,347,131,240)(25,346,132,239)(26,345,133,238)(27,344,134,237)(28,343,135,236)(29,342,136,235)(30,341,137,234)(31,340,138,233)(32,339,139,232)(33,338,140,231)(34,337,141,230)(35,336,142,229)(36,335,143,228)(37,334,144,227)(38,333,145,226)(39,332,146,225)(40,331,147,224)(41,330,148,223)(42,329,149,222)(43,328,150,221)(44,327,151,220)(45,326,152,219)(46,325,153,218)(47,324,154,217)(48,323,155,216)(49,322,156,215)(50,321,157,428)(51,320,158,427)(52,319,159,426)(53,318,160,425)(54,317,161,424)(55,316,162,423)(56,315,163,422)(57,314,164,421)(58,313,165,420)(59,312,166,419)(60,311,167,418)(61,310,168,417)(62,309,169,416)(63,308,170,415)(64,307,171,414)(65,306,172,413)(66,305,173,412)(67,304,174,411)(68,303,175,410)(69,302,176,409)(70,301,177,408)(71,300,178,407)(72,299,179,406)(73,298,180,405)(74,297,181,404)(75,296,182,403)(76,295,183,402)(77,294,184,401)(78,293,185,400)(79,292,186,399)(80,291,187,398)(81,290,188,397)(82,289,189,396)(83,288,190,395)(84,287,191,394)(85,286,192,393)(86,285,193,392)(87,284,194,391)(88,283,195,390)(89,282,196,389)(90,281,197,388)(91,280,198,387)(92,279,199,386)(93,278,200,385)(94,277,201,384)(95,276,202,383)(96,275,203,382)(97,274,204,381)(98,273,205,380)(99,272,206,379)(100,271,207,378)(101,270,208,377)(102,269,209,376)(103,268,210,375)(104,267,211,374)(105,266,212,373)(106,265,213,372)(107,264,214,371) );
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),(215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428)], [(1,370,108,263),(2,369,109,262),(3,368,110,261),(4,367,111,260),(5,366,112,259),(6,365,113,258),(7,364,114,257),(8,363,115,256),(9,362,116,255),(10,361,117,254),(11,360,118,253),(12,359,119,252),(13,358,120,251),(14,357,121,250),(15,356,122,249),(16,355,123,248),(17,354,124,247),(18,353,125,246),(19,352,126,245),(20,351,127,244),(21,350,128,243),(22,349,129,242),(23,348,130,241),(24,347,131,240),(25,346,132,239),(26,345,133,238),(27,344,134,237),(28,343,135,236),(29,342,136,235),(30,341,137,234),(31,340,138,233),(32,339,139,232),(33,338,140,231),(34,337,141,230),(35,336,142,229),(36,335,143,228),(37,334,144,227),(38,333,145,226),(39,332,146,225),(40,331,147,224),(41,330,148,223),(42,329,149,222),(43,328,150,221),(44,327,151,220),(45,326,152,219),(46,325,153,218),(47,324,154,217),(48,323,155,216),(49,322,156,215),(50,321,157,428),(51,320,158,427),(52,319,159,426),(53,318,160,425),(54,317,161,424),(55,316,162,423),(56,315,163,422),(57,314,164,421),(58,313,165,420),(59,312,166,419),(60,311,167,418),(61,310,168,417),(62,309,169,416),(63,308,170,415),(64,307,171,414),(65,306,172,413),(66,305,173,412),(67,304,174,411),(68,303,175,410),(69,302,176,409),(70,301,177,408),(71,300,178,407),(72,299,179,406),(73,298,180,405),(74,297,181,404),(75,296,182,403),(76,295,183,402),(77,294,184,401),(78,293,185,400),(79,292,186,399),(80,291,187,398),(81,290,188,397),(82,289,189,396),(83,288,190,395),(84,287,191,394),(85,286,192,393),(86,285,193,392),(87,284,194,391),(88,283,195,390),(89,282,196,389),(90,281,197,388),(91,280,198,387),(92,279,199,386),(93,278,200,385),(94,277,201,384),(95,276,202,383),(96,275,203,382),(97,274,204,381),(98,273,205,380),(99,272,206,379),(100,271,207,378),(101,270,208,377),(102,269,209,376),(103,268,210,375),(104,267,211,374),(105,266,212,373),(106,265,213,372),(107,264,214,371)]])
110 conjugacy classes
| class | 1 | 2 | 4A | 4B | 107A | ··· | 107BA | 214A | ··· | 214BA |
| order | 1 | 2 | 4 | 4 | 107 | ··· | 107 | 214 | ··· | 214 |
| size | 1 | 1 | 107 | 107 | 2 | ··· | 2 | 2 | ··· | 2 |
110 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D107 | Dic107 |
| kernel | Dic107 | C214 | C107 | C2 | C1 |
| # reps | 1 | 1 | 2 | 53 | 53 |
Matrix representation of Dic107 ►in GL3(𝔽857) generated by
| 856 | 0 | 0 |
| 0 | 615 | 856 |
| 0 | 1 | 0 |
| 650 | 0 | 0 |
| 0 | 94 | 150 |
| 0 | 541 | 763 |
G:=sub<GL(3,GF(857))| [856,0,0,0,615,1,0,856,0],[650,0,0,0,94,541,0,150,763] >;
Dic107 in GAP, Magma, Sage, TeX
{\rm Dic}_{107} % in TeX
G:=Group("Dic107"); // GroupNames label
G:=SmallGroup(428,1);
// by ID
G=gap.SmallGroup(428,1);
# by ID
G:=PCGroup([3,-2,-2,-107,6,3818]);
// Polycyclic
G:=Group<a,b|a^214=1,b^2=a^107,b*a*b^-1=a^-1>;
// generators/relations
Export