metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic108, C8.D27, C27⋊1Q16, C72.1S3, C24.1D9, C54.1D4, C6.1D36, C4.8D54, C3.Dic36, C9.Dic12, C216.1C2, C2.3D108, C18.1D12, C36.51D6, C12.51D18, C108.8C22, Dic54.1C2, SmallGroup(432,4)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic108
G = < a,b | a216=1, b2=a108, bab-1=a-1 >
Smallest permutation representation of Dic108
►Regular action on 432 points
Generators in S432
(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)(217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432)
(1 231 109 339)(2 230 110 338)(3 229 111 337)(4 228 112 336)(5 227 113 335)(6 226 114 334)(7 225 115 333)(8 224 116 332)(9 223 117 331)(10 222 118 330)(11 221 119 329)(12 220 120 328)(13 219 121 327)(14 218 122 326)(15 217 123 325)(16 432 124 324)(17 431 125 323)(18 430 126 322)(19 429 127 321)(20 428 128 320)(21 427 129 319)(22 426 130 318)(23 425 131 317)(24 424 132 316)(25 423 133 315)(26 422 134 314)(27 421 135 313)(28 420 136 312)(29 419 137 311)(30 418 138 310)(31 417 139 309)(32 416 140 308)(33 415 141 307)(34 414 142 306)(35 413 143 305)(36 412 144 304)(37 411 145 303)(38 410 146 302)(39 409 147 301)(40 408 148 300)(41 407 149 299)(42 406 150 298)(43 405 151 297)(44 404 152 296)(45 403 153 295)(46 402 154 294)(47 401 155 293)(48 400 156 292)(49 399 157 291)(50 398 158 290)(51 397 159 289)(52 396 160 288)(53 395 161 287)(54 394 162 286)(55 393 163 285)(56 392 164 284)(57 391 165 283)(58 390 166 282)(59 389 167 281)(60 388 168 280)(61 387 169 279)(62 386 170 278)(63 385 171 277)(64 384 172 276)(65 383 173 275)(66 382 174 274)(67 381 175 273)(68 380 176 272)(69 379 177 271)(70 378 178 270)(71 377 179 269)(72 376 180 268)(73 375 181 267)(74 374 182 266)(75 373 183 265)(76 372 184 264)(77 371 185 263)(78 370 186 262)(79 369 187 261)(80 368 188 260)(81 367 189 259)(82 366 190 258)(83 365 191 257)(84 364 192 256)(85 363 193 255)(86 362 194 254)(87 361 195 253)(88 360 196 252)(89 359 197 251)(90 358 198 250)(91 357 199 249)(92 356 200 248)(93 355 201 247)(94 354 202 246)(95 353 203 245)(96 352 204 244)(97 351 205 243)(98 350 206 242)(99 349 207 241)(100 348 208 240)(101 347 209 239)(102 346 210 238)(103 345 211 237)(104 344 212 236)(105 343 213 235)(106 342 214 234)(107 341 215 233)(108 340 216 232)
G:=sub<Sym(432)| (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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,231,109,339)(2,230,110,338)(3,229,111,337)(4,228,112,336)(5,227,113,335)(6,226,114,334)(7,225,115,333)(8,224,116,332)(9,223,117,331)(10,222,118,330)(11,221,119,329)(12,220,120,328)(13,219,121,327)(14,218,122,326)(15,217,123,325)(16,432,124,324)(17,431,125,323)(18,430,126,322)(19,429,127,321)(20,428,128,320)(21,427,129,319)(22,426,130,318)(23,425,131,317)(24,424,132,316)(25,423,133,315)(26,422,134,314)(27,421,135,313)(28,420,136,312)(29,419,137,311)(30,418,138,310)(31,417,139,309)(32,416,140,308)(33,415,141,307)(34,414,142,306)(35,413,143,305)(36,412,144,304)(37,411,145,303)(38,410,146,302)(39,409,147,301)(40,408,148,300)(41,407,149,299)(42,406,150,298)(43,405,151,297)(44,404,152,296)(45,403,153,295)(46,402,154,294)(47,401,155,293)(48,400,156,292)(49,399,157,291)(50,398,158,290)(51,397,159,289)(52,396,160,288)(53,395,161,287)(54,394,162,286)(55,393,163,285)(56,392,164,284)(57,391,165,283)(58,390,166,282)(59,389,167,281)(60,388,168,280)(61,387,169,279)(62,386,170,278)(63,385,171,277)(64,384,172,276)(65,383,173,275)(66,382,174,274)(67,381,175,273)(68,380,176,272)(69,379,177,271)(70,378,178,270)(71,377,179,269)(72,376,180,268)(73,375,181,267)(74,374,182,266)(75,373,183,265)(76,372,184,264)(77,371,185,263)(78,370,186,262)(79,369,187,261)(80,368,188,260)(81,367,189,259)(82,366,190,258)(83,365,191,257)(84,364,192,256)(85,363,193,255)(86,362,194,254)(87,361,195,253)(88,360,196,252)(89,359,197,251)(90,358,198,250)(91,357,199,249)(92,356,200,248)(93,355,201,247)(94,354,202,246)(95,353,203,245)(96,352,204,244)(97,351,205,243)(98,350,206,242)(99,349,207,241)(100,348,208,240)(101,347,209,239)(102,346,210,238)(103,345,211,237)(104,344,212,236)(105,343,213,235)(106,342,214,234)(107,341,215,233)(108,340,216,232)>;
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)(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432), (1,231,109,339)(2,230,110,338)(3,229,111,337)(4,228,112,336)(5,227,113,335)(6,226,114,334)(7,225,115,333)(8,224,116,332)(9,223,117,331)(10,222,118,330)(11,221,119,329)(12,220,120,328)(13,219,121,327)(14,218,122,326)(15,217,123,325)(16,432,124,324)(17,431,125,323)(18,430,126,322)(19,429,127,321)(20,428,128,320)(21,427,129,319)(22,426,130,318)(23,425,131,317)(24,424,132,316)(25,423,133,315)(26,422,134,314)(27,421,135,313)(28,420,136,312)(29,419,137,311)(30,418,138,310)(31,417,139,309)(32,416,140,308)(33,415,141,307)(34,414,142,306)(35,413,143,305)(36,412,144,304)(37,411,145,303)(38,410,146,302)(39,409,147,301)(40,408,148,300)(41,407,149,299)(42,406,150,298)(43,405,151,297)(44,404,152,296)(45,403,153,295)(46,402,154,294)(47,401,155,293)(48,400,156,292)(49,399,157,291)(50,398,158,290)(51,397,159,289)(52,396,160,288)(53,395,161,287)(54,394,162,286)(55,393,163,285)(56,392,164,284)(57,391,165,283)(58,390,166,282)(59,389,167,281)(60,388,168,280)(61,387,169,279)(62,386,170,278)(63,385,171,277)(64,384,172,276)(65,383,173,275)(66,382,174,274)(67,381,175,273)(68,380,176,272)(69,379,177,271)(70,378,178,270)(71,377,179,269)(72,376,180,268)(73,375,181,267)(74,374,182,266)(75,373,183,265)(76,372,184,264)(77,371,185,263)(78,370,186,262)(79,369,187,261)(80,368,188,260)(81,367,189,259)(82,366,190,258)(83,365,191,257)(84,364,192,256)(85,363,193,255)(86,362,194,254)(87,361,195,253)(88,360,196,252)(89,359,197,251)(90,358,198,250)(91,357,199,249)(92,356,200,248)(93,355,201,247)(94,354,202,246)(95,353,203,245)(96,352,204,244)(97,351,205,243)(98,350,206,242)(99,349,207,241)(100,348,208,240)(101,347,209,239)(102,346,210,238)(103,345,211,237)(104,344,212,236)(105,343,213,235)(106,342,214,234)(107,341,215,233)(108,340,216,232) );
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),(217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432)], [(1,231,109,339),(2,230,110,338),(3,229,111,337),(4,228,112,336),(5,227,113,335),(6,226,114,334),(7,225,115,333),(8,224,116,332),(9,223,117,331),(10,222,118,330),(11,221,119,329),(12,220,120,328),(13,219,121,327),(14,218,122,326),(15,217,123,325),(16,432,124,324),(17,431,125,323),(18,430,126,322),(19,429,127,321),(20,428,128,320),(21,427,129,319),(22,426,130,318),(23,425,131,317),(24,424,132,316),(25,423,133,315),(26,422,134,314),(27,421,135,313),(28,420,136,312),(29,419,137,311),(30,418,138,310),(31,417,139,309),(32,416,140,308),(33,415,141,307),(34,414,142,306),(35,413,143,305),(36,412,144,304),(37,411,145,303),(38,410,146,302),(39,409,147,301),(40,408,148,300),(41,407,149,299),(42,406,150,298),(43,405,151,297),(44,404,152,296),(45,403,153,295),(46,402,154,294),(47,401,155,293),(48,400,156,292),(49,399,157,291),(50,398,158,290),(51,397,159,289),(52,396,160,288),(53,395,161,287),(54,394,162,286),(55,393,163,285),(56,392,164,284),(57,391,165,283),(58,390,166,282),(59,389,167,281),(60,388,168,280),(61,387,169,279),(62,386,170,278),(63,385,171,277),(64,384,172,276),(65,383,173,275),(66,382,174,274),(67,381,175,273),(68,380,176,272),(69,379,177,271),(70,378,178,270),(71,377,179,269),(72,376,180,268),(73,375,181,267),(74,374,182,266),(75,373,183,265),(76,372,184,264),(77,371,185,263),(78,370,186,262),(79,369,187,261),(80,368,188,260),(81,367,189,259),(82,366,190,258),(83,365,191,257),(84,364,192,256),(85,363,193,255),(86,362,194,254),(87,361,195,253),(88,360,196,252),(89,359,197,251),(90,358,198,250),(91,357,199,249),(92,356,200,248),(93,355,201,247),(94,354,202,246),(95,353,203,245),(96,352,204,244),(97,351,205,243),(98,350,206,242),(99,349,207,241),(100,348,208,240),(101,347,209,239),(102,346,210,238),(103,345,211,237),(104,344,212,236),(105,343,213,235),(106,342,214,234),(107,341,215,233),(108,340,216,232)]])
111 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 18A | 18B | 18C | 24A | 24B | 24C | 24D | 27A | ··· | 27I | 36A | ··· | 36F | 54A | ··· | 54I | 72A | ··· | 72L | 108A | ··· | 108R | 216A | ··· | 216AJ |
| order | 1 | 2 | 3 | 4 | 4 | 4 | 6 | 8 | 8 | 9 | 9 | 9 | 12 | 12 | 18 | 18 | 18 | 24 | 24 | 24 | 24 | 27 | ··· | 27 | 36 | ··· | 36 | 54 | ··· | 54 | 72 | ··· | 72 | 108 | ··· | 108 | 216 | ··· | 216 |
| size | 1 | 1 | 2 | 2 | 108 | 108 | 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 |
111 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | + | + | - | + | - |
| image | C1 | C2 | C2 | S3 | D4 | D6 | Q16 | D9 | D12 | D18 | Dic12 | D27 | D36 | D54 | Dic36 | D108 | Dic108 |
| kernel | Dic108 | C216 | Dic54 | C72 | C54 | C36 | C27 | C24 | C18 | C12 | C9 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 3 | 2 | 3 | 4 | 9 | 6 | 9 | 12 | 18 | 36 |
Matrix representation of Dic108 ►in GL2(𝔽433) generated by
| 346 | 102 |
| 331 | 244 |
| 168 | 288 |
| 120 | 265 |
G:=sub<GL(2,GF(433))| [346,331,102,244],[168,120,288,265] >;
Dic108 in GAP, Magma, Sage, TeX
{\rm Dic}_{108} % in TeX
G:=Group("Dic108"); // GroupNames label
G:=SmallGroup(432,4);
// by ID
G=gap.SmallGroup(432,4);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,2804,557,10085,292,14118]);
// Polycyclic
G:=Group<a,b|a^216=1,b^2=a^108,b*a*b^-1=a^-1>;
// generators/relations
Export