metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C112⋊5C4, C16⋊3Dic7, C14.4D16, C2.1D112, C56.11Q8, C14.2Q32, C28.12Q16, C4.1Dic28, C2.2Dic56, C22.8D56, C8.10Dic14, (C2×C16).3D7, C7⋊2(C16⋊3C4), (C2×C112).5C2, C56.69(C2×C4), (C2×C4).71D28, (C2×C14).14D8, C28.23(C4⋊C4), C56⋊1C4.2C2, (C2×C28).370D4, (C2×C8).297D14, C4.8(C4⋊Dic7), C8.14(C2×Dic7), C2.3(C56⋊1C4), C14.7(C2.D8), (C2×C56).370C22, SmallGroup(448,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C112⋊5C4
G = < a,b | a112=b4=1, bab-1=a-1 >
Smallest permutation representation of C112⋊5C4
►Regular action on 448 points
Generators in S448
(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)(337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)
(1 168 355 235)(2 167 356 234)(3 166 357 233)(4 165 358 232)(5 164 359 231)(6 163 360 230)(7 162 361 229)(8 161 362 228)(9 160 363 227)(10 159 364 226)(11 158 365 225)(12 157 366 336)(13 156 367 335)(14 155 368 334)(15 154 369 333)(16 153 370 332)(17 152 371 331)(18 151 372 330)(19 150 373 329)(20 149 374 328)(21 148 375 327)(22 147 376 326)(23 146 377 325)(24 145 378 324)(25 144 379 323)(26 143 380 322)(27 142 381 321)(28 141 382 320)(29 140 383 319)(30 139 384 318)(31 138 385 317)(32 137 386 316)(33 136 387 315)(34 135 388 314)(35 134 389 313)(36 133 390 312)(37 132 391 311)(38 131 392 310)(39 130 393 309)(40 129 394 308)(41 128 395 307)(42 127 396 306)(43 126 397 305)(44 125 398 304)(45 124 399 303)(46 123 400 302)(47 122 401 301)(48 121 402 300)(49 120 403 299)(50 119 404 298)(51 118 405 297)(52 117 406 296)(53 116 407 295)(54 115 408 294)(55 114 409 293)(56 113 410 292)(57 224 411 291)(58 223 412 290)(59 222 413 289)(60 221 414 288)(61 220 415 287)(62 219 416 286)(63 218 417 285)(64 217 418 284)(65 216 419 283)(66 215 420 282)(67 214 421 281)(68 213 422 280)(69 212 423 279)(70 211 424 278)(71 210 425 277)(72 209 426 276)(73 208 427 275)(74 207 428 274)(75 206 429 273)(76 205 430 272)(77 204 431 271)(78 203 432 270)(79 202 433 269)(80 201 434 268)(81 200 435 267)(82 199 436 266)(83 198 437 265)(84 197 438 264)(85 196 439 263)(86 195 440 262)(87 194 441 261)(88 193 442 260)(89 192 443 259)(90 191 444 258)(91 190 445 257)(92 189 446 256)(93 188 447 255)(94 187 448 254)(95 186 337 253)(96 185 338 252)(97 184 339 251)(98 183 340 250)(99 182 341 249)(100 181 342 248)(101 180 343 247)(102 179 344 246)(103 178 345 245)(104 177 346 244)(105 176 347 243)(106 175 348 242)(107 174 349 241)(108 173 350 240)(109 172 351 239)(110 171 352 238)(111 170 353 237)(112 169 354 236)
G:=sub<Sym(448)| (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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,168,355,235)(2,167,356,234)(3,166,357,233)(4,165,358,232)(5,164,359,231)(6,163,360,230)(7,162,361,229)(8,161,362,228)(9,160,363,227)(10,159,364,226)(11,158,365,225)(12,157,366,336)(13,156,367,335)(14,155,368,334)(15,154,369,333)(16,153,370,332)(17,152,371,331)(18,151,372,330)(19,150,373,329)(20,149,374,328)(21,148,375,327)(22,147,376,326)(23,146,377,325)(24,145,378,324)(25,144,379,323)(26,143,380,322)(27,142,381,321)(28,141,382,320)(29,140,383,319)(30,139,384,318)(31,138,385,317)(32,137,386,316)(33,136,387,315)(34,135,388,314)(35,134,389,313)(36,133,390,312)(37,132,391,311)(38,131,392,310)(39,130,393,309)(40,129,394,308)(41,128,395,307)(42,127,396,306)(43,126,397,305)(44,125,398,304)(45,124,399,303)(46,123,400,302)(47,122,401,301)(48,121,402,300)(49,120,403,299)(50,119,404,298)(51,118,405,297)(52,117,406,296)(53,116,407,295)(54,115,408,294)(55,114,409,293)(56,113,410,292)(57,224,411,291)(58,223,412,290)(59,222,413,289)(60,221,414,288)(61,220,415,287)(62,219,416,286)(63,218,417,285)(64,217,418,284)(65,216,419,283)(66,215,420,282)(67,214,421,281)(68,213,422,280)(69,212,423,279)(70,211,424,278)(71,210,425,277)(72,209,426,276)(73,208,427,275)(74,207,428,274)(75,206,429,273)(76,205,430,272)(77,204,431,271)(78,203,432,270)(79,202,433,269)(80,201,434,268)(81,200,435,267)(82,199,436,266)(83,198,437,265)(84,197,438,264)(85,196,439,263)(86,195,440,262)(87,194,441,261)(88,193,442,260)(89,192,443,259)(90,191,444,258)(91,190,445,257)(92,189,446,256)(93,188,447,255)(94,187,448,254)(95,186,337,253)(96,185,338,252)(97,184,339,251)(98,183,340,250)(99,182,341,249)(100,181,342,248)(101,180,343,247)(102,179,344,246)(103,178,345,245)(104,177,346,244)(105,176,347,243)(106,175,348,242)(107,174,349,241)(108,173,350,240)(109,172,351,239)(110,171,352,238)(111,170,353,237)(112,169,354,236)>;
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)(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448), (1,168,355,235)(2,167,356,234)(3,166,357,233)(4,165,358,232)(5,164,359,231)(6,163,360,230)(7,162,361,229)(8,161,362,228)(9,160,363,227)(10,159,364,226)(11,158,365,225)(12,157,366,336)(13,156,367,335)(14,155,368,334)(15,154,369,333)(16,153,370,332)(17,152,371,331)(18,151,372,330)(19,150,373,329)(20,149,374,328)(21,148,375,327)(22,147,376,326)(23,146,377,325)(24,145,378,324)(25,144,379,323)(26,143,380,322)(27,142,381,321)(28,141,382,320)(29,140,383,319)(30,139,384,318)(31,138,385,317)(32,137,386,316)(33,136,387,315)(34,135,388,314)(35,134,389,313)(36,133,390,312)(37,132,391,311)(38,131,392,310)(39,130,393,309)(40,129,394,308)(41,128,395,307)(42,127,396,306)(43,126,397,305)(44,125,398,304)(45,124,399,303)(46,123,400,302)(47,122,401,301)(48,121,402,300)(49,120,403,299)(50,119,404,298)(51,118,405,297)(52,117,406,296)(53,116,407,295)(54,115,408,294)(55,114,409,293)(56,113,410,292)(57,224,411,291)(58,223,412,290)(59,222,413,289)(60,221,414,288)(61,220,415,287)(62,219,416,286)(63,218,417,285)(64,217,418,284)(65,216,419,283)(66,215,420,282)(67,214,421,281)(68,213,422,280)(69,212,423,279)(70,211,424,278)(71,210,425,277)(72,209,426,276)(73,208,427,275)(74,207,428,274)(75,206,429,273)(76,205,430,272)(77,204,431,271)(78,203,432,270)(79,202,433,269)(80,201,434,268)(81,200,435,267)(82,199,436,266)(83,198,437,265)(84,197,438,264)(85,196,439,263)(86,195,440,262)(87,194,441,261)(88,193,442,260)(89,192,443,259)(90,191,444,258)(91,190,445,257)(92,189,446,256)(93,188,447,255)(94,187,448,254)(95,186,337,253)(96,185,338,252)(97,184,339,251)(98,183,340,250)(99,182,341,249)(100,181,342,248)(101,180,343,247)(102,179,344,246)(103,178,345,245)(104,177,346,244)(105,176,347,243)(106,175,348,242)(107,174,349,241)(108,173,350,240)(109,172,351,239)(110,171,352,238)(111,170,353,237)(112,169,354,236) );
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),(337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)], [(1,168,355,235),(2,167,356,234),(3,166,357,233),(4,165,358,232),(5,164,359,231),(6,163,360,230),(7,162,361,229),(8,161,362,228),(9,160,363,227),(10,159,364,226),(11,158,365,225),(12,157,366,336),(13,156,367,335),(14,155,368,334),(15,154,369,333),(16,153,370,332),(17,152,371,331),(18,151,372,330),(19,150,373,329),(20,149,374,328),(21,148,375,327),(22,147,376,326),(23,146,377,325),(24,145,378,324),(25,144,379,323),(26,143,380,322),(27,142,381,321),(28,141,382,320),(29,140,383,319),(30,139,384,318),(31,138,385,317),(32,137,386,316),(33,136,387,315),(34,135,388,314),(35,134,389,313),(36,133,390,312),(37,132,391,311),(38,131,392,310),(39,130,393,309),(40,129,394,308),(41,128,395,307),(42,127,396,306),(43,126,397,305),(44,125,398,304),(45,124,399,303),(46,123,400,302),(47,122,401,301),(48,121,402,300),(49,120,403,299),(50,119,404,298),(51,118,405,297),(52,117,406,296),(53,116,407,295),(54,115,408,294),(55,114,409,293),(56,113,410,292),(57,224,411,291),(58,223,412,290),(59,222,413,289),(60,221,414,288),(61,220,415,287),(62,219,416,286),(63,218,417,285),(64,217,418,284),(65,216,419,283),(66,215,420,282),(67,214,421,281),(68,213,422,280),(69,212,423,279),(70,211,424,278),(71,210,425,277),(72,209,426,276),(73,208,427,275),(74,207,428,274),(75,206,429,273),(76,205,430,272),(77,204,431,271),(78,203,432,270),(79,202,433,269),(80,201,434,268),(81,200,435,267),(82,199,436,266),(83,198,437,265),(84,197,438,264),(85,196,439,263),(86,195,440,262),(87,194,441,261),(88,193,442,260),(89,192,443,259),(90,191,444,258),(91,190,445,257),(92,189,446,256),(93,188,447,255),(94,187,448,254),(95,186,337,253),(96,185,338,252),(97,184,339,251),(98,183,340,250),(99,182,341,249),(100,181,342,248),(101,180,343,247),(102,179,344,246),(103,178,345,245),(104,177,346,244),(105,176,347,243),(106,175,348,242),(107,174,349,241),(108,173,350,240),(109,172,351,239),(110,171,352,238),(111,170,353,237),(112,169,354,236)]])
118 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | ··· | 14I | 16A | ··· | 16H | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 56 | 56 | 56 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
118 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | + | - | + | + | - | - | + | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D7 | Q16 | D8 | Dic7 | D14 | D16 | Q32 | Dic14 | D28 | Dic28 | D56 | D112 | Dic56 |
| kernel | C112⋊5C4 | C56⋊1C4 | C2×C112 | C112 | C56 | C2×C28 | C2×C16 | C28 | C2×C14 | C16 | C2×C8 | C14 | C14 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 24 | 24 |
Matrix representation of C112⋊5C4 ►in GL4(𝔽113) generated by
| 103 | 103 | 0 | 0 |
| 10 | 89 | 0 | 0 |
| 0 | 0 | 25 | 67 |
| 0 | 0 | 46 | 51 |
| 15 | 92 | 0 | 0 |
| 0 | 98 | 0 | 0 |
| 0 | 0 | 106 | 33 |
| 0 | 0 | 91 | 7 |
G:=sub<GL(4,GF(113))| [103,10,0,0,103,89,0,0,0,0,25,46,0,0,67,51],[15,0,0,0,92,98,0,0,0,0,106,91,0,0,33,7] >;
C112⋊5C4 in GAP, Magma, Sage, TeX
C_{112}\rtimes_5C_4 % in TeX
G:=Group("C112:5C4"); // GroupNames label
G:=SmallGroup(448,61);
// by ID
G=gap.SmallGroup(448,61);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,176,675,192,1684,102,18822]);
// Polycyclic
G:=Group<a,b|a^112=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
Export