metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic106, C53⋊Q8, C4.D53, C212.1C2, C2.3D106, C106.1C22, Dic53.1C2, SmallGroup(424,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic106
G = < a,b | a212=1, b2=a106, bab-1=a-1 >
Smallest permutation representation of Dic106
►Regular action on 424 points
Generators in S424
(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)
(1 378 107 272)(2 377 108 271)(3 376 109 270)(4 375 110 269)(5 374 111 268)(6 373 112 267)(7 372 113 266)(8 371 114 265)(9 370 115 264)(10 369 116 263)(11 368 117 262)(12 367 118 261)(13 366 119 260)(14 365 120 259)(15 364 121 258)(16 363 122 257)(17 362 123 256)(18 361 124 255)(19 360 125 254)(20 359 126 253)(21 358 127 252)(22 357 128 251)(23 356 129 250)(24 355 130 249)(25 354 131 248)(26 353 132 247)(27 352 133 246)(28 351 134 245)(29 350 135 244)(30 349 136 243)(31 348 137 242)(32 347 138 241)(33 346 139 240)(34 345 140 239)(35 344 141 238)(36 343 142 237)(37 342 143 236)(38 341 144 235)(39 340 145 234)(40 339 146 233)(41 338 147 232)(42 337 148 231)(43 336 149 230)(44 335 150 229)(45 334 151 228)(46 333 152 227)(47 332 153 226)(48 331 154 225)(49 330 155 224)(50 329 156 223)(51 328 157 222)(52 327 158 221)(53 326 159 220)(54 325 160 219)(55 324 161 218)(56 323 162 217)(57 322 163 216)(58 321 164 215)(59 320 165 214)(60 319 166 213)(61 318 167 424)(62 317 168 423)(63 316 169 422)(64 315 170 421)(65 314 171 420)(66 313 172 419)(67 312 173 418)(68 311 174 417)(69 310 175 416)(70 309 176 415)(71 308 177 414)(72 307 178 413)(73 306 179 412)(74 305 180 411)(75 304 181 410)(76 303 182 409)(77 302 183 408)(78 301 184 407)(79 300 185 406)(80 299 186 405)(81 298 187 404)(82 297 188 403)(83 296 189 402)(84 295 190 401)(85 294 191 400)(86 293 192 399)(87 292 193 398)(88 291 194 397)(89 290 195 396)(90 289 196 395)(91 288 197 394)(92 287 198 393)(93 286 199 392)(94 285 200 391)(95 284 201 390)(96 283 202 389)(97 282 203 388)(98 281 204 387)(99 280 205 386)(100 279 206 385)(101 278 207 384)(102 277 208 383)(103 276 209 382)(104 275 210 381)(105 274 211 380)(106 273 212 379)
G:=sub<Sym(424)| (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), (1,378,107,272)(2,377,108,271)(3,376,109,270)(4,375,110,269)(5,374,111,268)(6,373,112,267)(7,372,113,266)(8,371,114,265)(9,370,115,264)(10,369,116,263)(11,368,117,262)(12,367,118,261)(13,366,119,260)(14,365,120,259)(15,364,121,258)(16,363,122,257)(17,362,123,256)(18,361,124,255)(19,360,125,254)(20,359,126,253)(21,358,127,252)(22,357,128,251)(23,356,129,250)(24,355,130,249)(25,354,131,248)(26,353,132,247)(27,352,133,246)(28,351,134,245)(29,350,135,244)(30,349,136,243)(31,348,137,242)(32,347,138,241)(33,346,139,240)(34,345,140,239)(35,344,141,238)(36,343,142,237)(37,342,143,236)(38,341,144,235)(39,340,145,234)(40,339,146,233)(41,338,147,232)(42,337,148,231)(43,336,149,230)(44,335,150,229)(45,334,151,228)(46,333,152,227)(47,332,153,226)(48,331,154,225)(49,330,155,224)(50,329,156,223)(51,328,157,222)(52,327,158,221)(53,326,159,220)(54,325,160,219)(55,324,161,218)(56,323,162,217)(57,322,163,216)(58,321,164,215)(59,320,165,214)(60,319,166,213)(61,318,167,424)(62,317,168,423)(63,316,169,422)(64,315,170,421)(65,314,171,420)(66,313,172,419)(67,312,173,418)(68,311,174,417)(69,310,175,416)(70,309,176,415)(71,308,177,414)(72,307,178,413)(73,306,179,412)(74,305,180,411)(75,304,181,410)(76,303,182,409)(77,302,183,408)(78,301,184,407)(79,300,185,406)(80,299,186,405)(81,298,187,404)(82,297,188,403)(83,296,189,402)(84,295,190,401)(85,294,191,400)(86,293,192,399)(87,292,193,398)(88,291,194,397)(89,290,195,396)(90,289,196,395)(91,288,197,394)(92,287,198,393)(93,286,199,392)(94,285,200,391)(95,284,201,390)(96,283,202,389)(97,282,203,388)(98,281,204,387)(99,280,205,386)(100,279,206,385)(101,278,207,384)(102,277,208,383)(103,276,209,382)(104,275,210,381)(105,274,211,380)(106,273,212,379)>;
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), (1,378,107,272)(2,377,108,271)(3,376,109,270)(4,375,110,269)(5,374,111,268)(6,373,112,267)(7,372,113,266)(8,371,114,265)(9,370,115,264)(10,369,116,263)(11,368,117,262)(12,367,118,261)(13,366,119,260)(14,365,120,259)(15,364,121,258)(16,363,122,257)(17,362,123,256)(18,361,124,255)(19,360,125,254)(20,359,126,253)(21,358,127,252)(22,357,128,251)(23,356,129,250)(24,355,130,249)(25,354,131,248)(26,353,132,247)(27,352,133,246)(28,351,134,245)(29,350,135,244)(30,349,136,243)(31,348,137,242)(32,347,138,241)(33,346,139,240)(34,345,140,239)(35,344,141,238)(36,343,142,237)(37,342,143,236)(38,341,144,235)(39,340,145,234)(40,339,146,233)(41,338,147,232)(42,337,148,231)(43,336,149,230)(44,335,150,229)(45,334,151,228)(46,333,152,227)(47,332,153,226)(48,331,154,225)(49,330,155,224)(50,329,156,223)(51,328,157,222)(52,327,158,221)(53,326,159,220)(54,325,160,219)(55,324,161,218)(56,323,162,217)(57,322,163,216)(58,321,164,215)(59,320,165,214)(60,319,166,213)(61,318,167,424)(62,317,168,423)(63,316,169,422)(64,315,170,421)(65,314,171,420)(66,313,172,419)(67,312,173,418)(68,311,174,417)(69,310,175,416)(70,309,176,415)(71,308,177,414)(72,307,178,413)(73,306,179,412)(74,305,180,411)(75,304,181,410)(76,303,182,409)(77,302,183,408)(78,301,184,407)(79,300,185,406)(80,299,186,405)(81,298,187,404)(82,297,188,403)(83,296,189,402)(84,295,190,401)(85,294,191,400)(86,293,192,399)(87,292,193,398)(88,291,194,397)(89,290,195,396)(90,289,196,395)(91,288,197,394)(92,287,198,393)(93,286,199,392)(94,285,200,391)(95,284,201,390)(96,283,202,389)(97,282,203,388)(98,281,204,387)(99,280,205,386)(100,279,206,385)(101,278,207,384)(102,277,208,383)(103,276,209,382)(104,275,210,381)(105,274,211,380)(106,273,212,379) );
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)], [(1,378,107,272),(2,377,108,271),(3,376,109,270),(4,375,110,269),(5,374,111,268),(6,373,112,267),(7,372,113,266),(8,371,114,265),(9,370,115,264),(10,369,116,263),(11,368,117,262),(12,367,118,261),(13,366,119,260),(14,365,120,259),(15,364,121,258),(16,363,122,257),(17,362,123,256),(18,361,124,255),(19,360,125,254),(20,359,126,253),(21,358,127,252),(22,357,128,251),(23,356,129,250),(24,355,130,249),(25,354,131,248),(26,353,132,247),(27,352,133,246),(28,351,134,245),(29,350,135,244),(30,349,136,243),(31,348,137,242),(32,347,138,241),(33,346,139,240),(34,345,140,239),(35,344,141,238),(36,343,142,237),(37,342,143,236),(38,341,144,235),(39,340,145,234),(40,339,146,233),(41,338,147,232),(42,337,148,231),(43,336,149,230),(44,335,150,229),(45,334,151,228),(46,333,152,227),(47,332,153,226),(48,331,154,225),(49,330,155,224),(50,329,156,223),(51,328,157,222),(52,327,158,221),(53,326,159,220),(54,325,160,219),(55,324,161,218),(56,323,162,217),(57,322,163,216),(58,321,164,215),(59,320,165,214),(60,319,166,213),(61,318,167,424),(62,317,168,423),(63,316,169,422),(64,315,170,421),(65,314,171,420),(66,313,172,419),(67,312,173,418),(68,311,174,417),(69,310,175,416),(70,309,176,415),(71,308,177,414),(72,307,178,413),(73,306,179,412),(74,305,180,411),(75,304,181,410),(76,303,182,409),(77,302,183,408),(78,301,184,407),(79,300,185,406),(80,299,186,405),(81,298,187,404),(82,297,188,403),(83,296,189,402),(84,295,190,401),(85,294,191,400),(86,293,192,399),(87,292,193,398),(88,291,194,397),(89,290,195,396),(90,289,196,395),(91,288,197,394),(92,287,198,393),(93,286,199,392),(94,285,200,391),(95,284,201,390),(96,283,202,389),(97,282,203,388),(98,281,204,387),(99,280,205,386),(100,279,206,385),(101,278,207,384),(102,277,208,383),(103,276,209,382),(104,275,210,381),(105,274,211,380),(106,273,212,379)]])
109 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 53A | ··· | 53Z | 106A | ··· | 106Z | 212A | ··· | 212AZ |
| order | 1 | 2 | 4 | 4 | 4 | 53 | ··· | 53 | 106 | ··· | 106 | 212 | ··· | 212 |
| size | 1 | 1 | 2 | 106 | 106 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
109 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D53 | D106 | Dic106 |
| kernel | Dic106 | Dic53 | C212 | C53 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 26 | 26 | 52 |
Matrix representation of Dic106 ►in GL2(𝔽1061) generated by
| 324 | 185 |
| 876 | 641 |
| 768 | 441 |
| 977 | 293 |
G:=sub<GL(2,GF(1061))| [324,876,185,641],[768,977,441,293] >;
Dic106 in GAP, Magma, Sage, TeX
{\rm Dic}_{106} % in TeX
G:=Group("Dic106"); // GroupNames label
G:=SmallGroup(424,4);
// by ID
G=gap.SmallGroup(424,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-53,16,49,21,6659]);
// Polycyclic
G:=Group<a,b|a^212=1,b^2=a^106,b*a*b^-1=a^-1>;
// generators/relations
Export