metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C112⋊9C4, C16⋊5Dic7, C28.17C42, C14.2M5(2), C7⋊C16⋊8C4, C7⋊C8.3C8, C14.8(C4×C8), C8.41(C4×D7), (C2×C16).8D7, C4.21(C8×D7), C7⋊2(C16⋊5C4), C56.58(C2×C4), C28.26(C2×C8), C2.4(C8×Dic7), (C2×C112).15C2, (C2×C8).334D14, (C2×Dic7).3C8, (C8×Dic7).9C2, C8.24(C2×Dic7), C4.16(C4×Dic7), C22.10(C8×D7), C2.2(C16⋊D7), (C4×Dic7).11C4, (C2×C56).400C22, (C2×C7⋊C8).13C4, (C2×C7⋊C16).10C2, (C2×C14).11(C2×C8), (C2×C4).168(C4×D7), (C2×C28).242(C2×C4), SmallGroup(448,59)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C112⋊9C4
G = < a,b | a112=b4=1, bab-1=a41 >
Smallest permutation representation of C112⋊9C4
►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)
(1 156 269 409)(2 197 270 338)(3 126 271 379)(4 167 272 420)(5 208 273 349)(6 137 274 390)(7 178 275 431)(8 219 276 360)(9 148 277 401)(10 189 278 442)(11 118 279 371)(12 159 280 412)(13 200 281 341)(14 129 282 382)(15 170 283 423)(16 211 284 352)(17 140 285 393)(18 181 286 434)(19 222 287 363)(20 151 288 404)(21 192 289 445)(22 121 290 374)(23 162 291 415)(24 203 292 344)(25 132 293 385)(26 173 294 426)(27 214 295 355)(28 143 296 396)(29 184 297 437)(30 113 298 366)(31 154 299 407)(32 195 300 448)(33 124 301 377)(34 165 302 418)(35 206 303 347)(36 135 304 388)(37 176 305 429)(38 217 306 358)(39 146 307 399)(40 187 308 440)(41 116 309 369)(42 157 310 410)(43 198 311 339)(44 127 312 380)(45 168 313 421)(46 209 314 350)(47 138 315 391)(48 179 316 432)(49 220 317 361)(50 149 318 402)(51 190 319 443)(52 119 320 372)(53 160 321 413)(54 201 322 342)(55 130 323 383)(56 171 324 424)(57 212 325 353)(58 141 326 394)(59 182 327 435)(60 223 328 364)(61 152 329 405)(62 193 330 446)(63 122 331 375)(64 163 332 416)(65 204 333 345)(66 133 334 386)(67 174 335 427)(68 215 336 356)(69 144 225 397)(70 185 226 438)(71 114 227 367)(72 155 228 408)(73 196 229 337)(74 125 230 378)(75 166 231 419)(76 207 232 348)(77 136 233 389)(78 177 234 430)(79 218 235 359)(80 147 236 400)(81 188 237 441)(82 117 238 370)(83 158 239 411)(84 199 240 340)(85 128 241 381)(86 169 242 422)(87 210 243 351)(88 139 244 392)(89 180 245 433)(90 221 246 362)(91 150 247 403)(92 191 248 444)(93 120 249 373)(94 161 250 414)(95 202 251 343)(96 131 252 384)(97 172 253 425)(98 213 254 354)(99 142 255 395)(100 183 256 436)(101 224 257 365)(102 153 258 406)(103 194 259 447)(104 123 260 376)(105 164 261 417)(106 205 262 346)(107 134 263 387)(108 175 264 428)(109 216 265 357)(110 145 266 398)(111 186 267 439)(112 115 268 368)
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), (1,156,269,409)(2,197,270,338)(3,126,271,379)(4,167,272,420)(5,208,273,349)(6,137,274,390)(7,178,275,431)(8,219,276,360)(9,148,277,401)(10,189,278,442)(11,118,279,371)(12,159,280,412)(13,200,281,341)(14,129,282,382)(15,170,283,423)(16,211,284,352)(17,140,285,393)(18,181,286,434)(19,222,287,363)(20,151,288,404)(21,192,289,445)(22,121,290,374)(23,162,291,415)(24,203,292,344)(25,132,293,385)(26,173,294,426)(27,214,295,355)(28,143,296,396)(29,184,297,437)(30,113,298,366)(31,154,299,407)(32,195,300,448)(33,124,301,377)(34,165,302,418)(35,206,303,347)(36,135,304,388)(37,176,305,429)(38,217,306,358)(39,146,307,399)(40,187,308,440)(41,116,309,369)(42,157,310,410)(43,198,311,339)(44,127,312,380)(45,168,313,421)(46,209,314,350)(47,138,315,391)(48,179,316,432)(49,220,317,361)(50,149,318,402)(51,190,319,443)(52,119,320,372)(53,160,321,413)(54,201,322,342)(55,130,323,383)(56,171,324,424)(57,212,325,353)(58,141,326,394)(59,182,327,435)(60,223,328,364)(61,152,329,405)(62,193,330,446)(63,122,331,375)(64,163,332,416)(65,204,333,345)(66,133,334,386)(67,174,335,427)(68,215,336,356)(69,144,225,397)(70,185,226,438)(71,114,227,367)(72,155,228,408)(73,196,229,337)(74,125,230,378)(75,166,231,419)(76,207,232,348)(77,136,233,389)(78,177,234,430)(79,218,235,359)(80,147,236,400)(81,188,237,441)(82,117,238,370)(83,158,239,411)(84,199,240,340)(85,128,241,381)(86,169,242,422)(87,210,243,351)(88,139,244,392)(89,180,245,433)(90,221,246,362)(91,150,247,403)(92,191,248,444)(93,120,249,373)(94,161,250,414)(95,202,251,343)(96,131,252,384)(97,172,253,425)(98,213,254,354)(99,142,255,395)(100,183,256,436)(101,224,257,365)(102,153,258,406)(103,194,259,447)(104,123,260,376)(105,164,261,417)(106,205,262,346)(107,134,263,387)(108,175,264,428)(109,216,265,357)(110,145,266,398)(111,186,267,439)(112,115,268,368)>;
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), (1,156,269,409)(2,197,270,338)(3,126,271,379)(4,167,272,420)(5,208,273,349)(6,137,274,390)(7,178,275,431)(8,219,276,360)(9,148,277,401)(10,189,278,442)(11,118,279,371)(12,159,280,412)(13,200,281,341)(14,129,282,382)(15,170,283,423)(16,211,284,352)(17,140,285,393)(18,181,286,434)(19,222,287,363)(20,151,288,404)(21,192,289,445)(22,121,290,374)(23,162,291,415)(24,203,292,344)(25,132,293,385)(26,173,294,426)(27,214,295,355)(28,143,296,396)(29,184,297,437)(30,113,298,366)(31,154,299,407)(32,195,300,448)(33,124,301,377)(34,165,302,418)(35,206,303,347)(36,135,304,388)(37,176,305,429)(38,217,306,358)(39,146,307,399)(40,187,308,440)(41,116,309,369)(42,157,310,410)(43,198,311,339)(44,127,312,380)(45,168,313,421)(46,209,314,350)(47,138,315,391)(48,179,316,432)(49,220,317,361)(50,149,318,402)(51,190,319,443)(52,119,320,372)(53,160,321,413)(54,201,322,342)(55,130,323,383)(56,171,324,424)(57,212,325,353)(58,141,326,394)(59,182,327,435)(60,223,328,364)(61,152,329,405)(62,193,330,446)(63,122,331,375)(64,163,332,416)(65,204,333,345)(66,133,334,386)(67,174,335,427)(68,215,336,356)(69,144,225,397)(70,185,226,438)(71,114,227,367)(72,155,228,408)(73,196,229,337)(74,125,230,378)(75,166,231,419)(76,207,232,348)(77,136,233,389)(78,177,234,430)(79,218,235,359)(80,147,236,400)(81,188,237,441)(82,117,238,370)(83,158,239,411)(84,199,240,340)(85,128,241,381)(86,169,242,422)(87,210,243,351)(88,139,244,392)(89,180,245,433)(90,221,246,362)(91,150,247,403)(92,191,248,444)(93,120,249,373)(94,161,250,414)(95,202,251,343)(96,131,252,384)(97,172,253,425)(98,213,254,354)(99,142,255,395)(100,183,256,436)(101,224,257,365)(102,153,258,406)(103,194,259,447)(104,123,260,376)(105,164,261,417)(106,205,262,346)(107,134,263,387)(108,175,264,428)(109,216,265,357)(110,145,266,398)(111,186,267,439)(112,115,268,368) );
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)], [(1,156,269,409),(2,197,270,338),(3,126,271,379),(4,167,272,420),(5,208,273,349),(6,137,274,390),(7,178,275,431),(8,219,276,360),(9,148,277,401),(10,189,278,442),(11,118,279,371),(12,159,280,412),(13,200,281,341),(14,129,282,382),(15,170,283,423),(16,211,284,352),(17,140,285,393),(18,181,286,434),(19,222,287,363),(20,151,288,404),(21,192,289,445),(22,121,290,374),(23,162,291,415),(24,203,292,344),(25,132,293,385),(26,173,294,426),(27,214,295,355),(28,143,296,396),(29,184,297,437),(30,113,298,366),(31,154,299,407),(32,195,300,448),(33,124,301,377),(34,165,302,418),(35,206,303,347),(36,135,304,388),(37,176,305,429),(38,217,306,358),(39,146,307,399),(40,187,308,440),(41,116,309,369),(42,157,310,410),(43,198,311,339),(44,127,312,380),(45,168,313,421),(46,209,314,350),(47,138,315,391),(48,179,316,432),(49,220,317,361),(50,149,318,402),(51,190,319,443),(52,119,320,372),(53,160,321,413),(54,201,322,342),(55,130,323,383),(56,171,324,424),(57,212,325,353),(58,141,326,394),(59,182,327,435),(60,223,328,364),(61,152,329,405),(62,193,330,446),(63,122,331,375),(64,163,332,416),(65,204,333,345),(66,133,334,386),(67,174,335,427),(68,215,336,356),(69,144,225,397),(70,185,226,438),(71,114,227,367),(72,155,228,408),(73,196,229,337),(74,125,230,378),(75,166,231,419),(76,207,232,348),(77,136,233,389),(78,177,234,430),(79,218,235,359),(80,147,236,400),(81,188,237,441),(82,117,238,370),(83,158,239,411),(84,199,240,340),(85,128,241,381),(86,169,242,422),(87,210,243,351),(88,139,244,392),(89,180,245,433),(90,221,246,362),(91,150,247,403),(92,191,248,444),(93,120,249,373),(94,161,250,414),(95,202,251,343),(96,131,252,384),(97,172,253,425),(98,213,254,354),(99,142,255,395),(100,183,256,436),(101,224,257,365),(102,153,258,406),(103,194,259,447),(104,123,260,376),(105,164,261,417),(106,205,262,346),(107,134,263,387),(108,175,264,428),(109,216,265,357),(110,145,266,398),(111,186,267,439),(112,115,268,368)]])
136 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 7A | 7B | 7C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 14A | ··· | 14I | 16A | ··· | 16H | 16I | ··· | 16P | 28A | ··· | 28L | 56A | ··· | 56X | 112A | ··· | 112AV |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 14 | 14 | 14 | 14 | 2 | 2 | 2 | 1 | ··· | 1 | 14 | 14 | 14 | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 14 | ··· | 14 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
136 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | D7 | Dic7 | D14 | M5(2) | C4×D7 | C4×D7 | C8×D7 | C8×D7 | C16⋊D7 |
| kernel | C112⋊9C4 | C2×C7⋊C16 | C8×Dic7 | C2×C112 | C7⋊C16 | C112 | C2×C7⋊C8 | C4×Dic7 | C7⋊C8 | C2×Dic7 | C2×C16 | C16 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 3 | 6 | 3 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C112⋊9C4 ►in GL4(𝔽113) generated by
| 103 | 103 | 0 | 0 |
| 10 | 89 | 0 | 0 |
| 0 | 0 | 99 | 14 |
| 0 | 0 | 99 | 96 |
| 0 | 15 | 0 | 0 |
| 15 | 0 | 0 | 0 |
| 0 | 0 | 76 | 21 |
| 0 | 0 | 37 | 37 |
G:=sub<GL(4,GF(113))| [103,10,0,0,103,89,0,0,0,0,99,99,0,0,14,96],[0,15,0,0,15,0,0,0,0,0,76,37,0,0,21,37] >;
C112⋊9C4 in GAP, Magma, Sage, TeX
C_{112}\rtimes_9C_4 % in TeX
G:=Group("C112:9C4"); // GroupNames label
G:=SmallGroup(448,59);
// by ID
G=gap.SmallGroup(448,59);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,477,64,80,102,18822]);
// Polycyclic
G:=Group<a,b|a^112=b^4=1,b*a*b^-1=a^41>;
// generators/relations
Export