metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C112⋊6C4, C16⋊4Dic7, C56.12Q8, C28.13Q16, C4.2Dic28, C14.2SD32, C22.9D56, C8.11Dic14, (C2×C16).6D7, C7⋊2(C16⋊4C4), (C2×C112).8C2, C56.70(C2×C4), (C2×C4).72D28, (C2×C14).15D8, C28.24(C4⋊C4), C56⋊1C4.3C2, (C2×C28).371D4, (C2×C8).298D14, C4.9(C4⋊Dic7), C8.15(C2×Dic7), C2.4(C56⋊1C4), C14.8(C2.D8), C2.2(C112⋊C2), (C2×C56).371C22, SmallGroup(448,62)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C112⋊6C4
G = < a,b | a112=b4=1, bab-1=a55 >
Smallest permutation representation of C112⋊6C4
►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)
(1 210 242 396)(2 153 243 339)(3 208 244 394)(4 151 245 337)(5 206 246 392)(6 149 247 447)(7 204 248 390)(8 147 249 445)(9 202 250 388)(10 145 251 443)(11 200 252 386)(12 143 253 441)(13 198 254 384)(14 141 255 439)(15 196 256 382)(16 139 257 437)(17 194 258 380)(18 137 259 435)(19 192 260 378)(20 135 261 433)(21 190 262 376)(22 133 263 431)(23 188 264 374)(24 131 265 429)(25 186 266 372)(26 129 267 427)(27 184 268 370)(28 127 269 425)(29 182 270 368)(30 125 271 423)(31 180 272 366)(32 123 273 421)(33 178 274 364)(34 121 275 419)(35 176 276 362)(36 119 277 417)(37 174 278 360)(38 117 279 415)(39 172 280 358)(40 115 281 413)(41 170 282 356)(42 113 283 411)(43 168 284 354)(44 223 285 409)(45 166 286 352)(46 221 287 407)(47 164 288 350)(48 219 289 405)(49 162 290 348)(50 217 291 403)(51 160 292 346)(52 215 293 401)(53 158 294 344)(54 213 295 399)(55 156 296 342)(56 211 297 397)(57 154 298 340)(58 209 299 395)(59 152 300 338)(60 207 301 393)(61 150 302 448)(62 205 303 391)(63 148 304 446)(64 203 305 389)(65 146 306 444)(66 201 307 387)(67 144 308 442)(68 199 309 385)(69 142 310 440)(70 197 311 383)(71 140 312 438)(72 195 313 381)(73 138 314 436)(74 193 315 379)(75 136 316 434)(76 191 317 377)(77 134 318 432)(78 189 319 375)(79 132 320 430)(80 187 321 373)(81 130 322 428)(82 185 323 371)(83 128 324 426)(84 183 325 369)(85 126 326 424)(86 181 327 367)(87 124 328 422)(88 179 329 365)(89 122 330 420)(90 177 331 363)(91 120 332 418)(92 175 333 361)(93 118 334 416)(94 173 335 359)(95 116 336 414)(96 171 225 357)(97 114 226 412)(98 169 227 355)(99 224 228 410)(100 167 229 353)(101 222 230 408)(102 165 231 351)(103 220 232 406)(104 163 233 349)(105 218 234 404)(106 161 235 347)(107 216 236 402)(108 159 237 345)(109 214 238 400)(110 157 239 343)(111 212 240 398)(112 155 241 341)
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), (1,210,242,396)(2,153,243,339)(3,208,244,394)(4,151,245,337)(5,206,246,392)(6,149,247,447)(7,204,248,390)(8,147,249,445)(9,202,250,388)(10,145,251,443)(11,200,252,386)(12,143,253,441)(13,198,254,384)(14,141,255,439)(15,196,256,382)(16,139,257,437)(17,194,258,380)(18,137,259,435)(19,192,260,378)(20,135,261,433)(21,190,262,376)(22,133,263,431)(23,188,264,374)(24,131,265,429)(25,186,266,372)(26,129,267,427)(27,184,268,370)(28,127,269,425)(29,182,270,368)(30,125,271,423)(31,180,272,366)(32,123,273,421)(33,178,274,364)(34,121,275,419)(35,176,276,362)(36,119,277,417)(37,174,278,360)(38,117,279,415)(39,172,280,358)(40,115,281,413)(41,170,282,356)(42,113,283,411)(43,168,284,354)(44,223,285,409)(45,166,286,352)(46,221,287,407)(47,164,288,350)(48,219,289,405)(49,162,290,348)(50,217,291,403)(51,160,292,346)(52,215,293,401)(53,158,294,344)(54,213,295,399)(55,156,296,342)(56,211,297,397)(57,154,298,340)(58,209,299,395)(59,152,300,338)(60,207,301,393)(61,150,302,448)(62,205,303,391)(63,148,304,446)(64,203,305,389)(65,146,306,444)(66,201,307,387)(67,144,308,442)(68,199,309,385)(69,142,310,440)(70,197,311,383)(71,140,312,438)(72,195,313,381)(73,138,314,436)(74,193,315,379)(75,136,316,434)(76,191,317,377)(77,134,318,432)(78,189,319,375)(79,132,320,430)(80,187,321,373)(81,130,322,428)(82,185,323,371)(83,128,324,426)(84,183,325,369)(85,126,326,424)(86,181,327,367)(87,124,328,422)(88,179,329,365)(89,122,330,420)(90,177,331,363)(91,120,332,418)(92,175,333,361)(93,118,334,416)(94,173,335,359)(95,116,336,414)(96,171,225,357)(97,114,226,412)(98,169,227,355)(99,224,228,410)(100,167,229,353)(101,222,230,408)(102,165,231,351)(103,220,232,406)(104,163,233,349)(105,218,234,404)(106,161,235,347)(107,216,236,402)(108,159,237,345)(109,214,238,400)(110,157,239,343)(111,212,240,398)(112,155,241,341)>;
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), (1,210,242,396)(2,153,243,339)(3,208,244,394)(4,151,245,337)(5,206,246,392)(6,149,247,447)(7,204,248,390)(8,147,249,445)(9,202,250,388)(10,145,251,443)(11,200,252,386)(12,143,253,441)(13,198,254,384)(14,141,255,439)(15,196,256,382)(16,139,257,437)(17,194,258,380)(18,137,259,435)(19,192,260,378)(20,135,261,433)(21,190,262,376)(22,133,263,431)(23,188,264,374)(24,131,265,429)(25,186,266,372)(26,129,267,427)(27,184,268,370)(28,127,269,425)(29,182,270,368)(30,125,271,423)(31,180,272,366)(32,123,273,421)(33,178,274,364)(34,121,275,419)(35,176,276,362)(36,119,277,417)(37,174,278,360)(38,117,279,415)(39,172,280,358)(40,115,281,413)(41,170,282,356)(42,113,283,411)(43,168,284,354)(44,223,285,409)(45,166,286,352)(46,221,287,407)(47,164,288,350)(48,219,289,405)(49,162,290,348)(50,217,291,403)(51,160,292,346)(52,215,293,401)(53,158,294,344)(54,213,295,399)(55,156,296,342)(56,211,297,397)(57,154,298,340)(58,209,299,395)(59,152,300,338)(60,207,301,393)(61,150,302,448)(62,205,303,391)(63,148,304,446)(64,203,305,389)(65,146,306,444)(66,201,307,387)(67,144,308,442)(68,199,309,385)(69,142,310,440)(70,197,311,383)(71,140,312,438)(72,195,313,381)(73,138,314,436)(74,193,315,379)(75,136,316,434)(76,191,317,377)(77,134,318,432)(78,189,319,375)(79,132,320,430)(80,187,321,373)(81,130,322,428)(82,185,323,371)(83,128,324,426)(84,183,325,369)(85,126,326,424)(86,181,327,367)(87,124,328,422)(88,179,329,365)(89,122,330,420)(90,177,331,363)(91,120,332,418)(92,175,333,361)(93,118,334,416)(94,173,335,359)(95,116,336,414)(96,171,225,357)(97,114,226,412)(98,169,227,355)(99,224,228,410)(100,167,229,353)(101,222,230,408)(102,165,231,351)(103,220,232,406)(104,163,233,349)(105,218,234,404)(106,161,235,347)(107,216,236,402)(108,159,237,345)(109,214,238,400)(110,157,239,343)(111,212,240,398)(112,155,241,341) );
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)], [(1,210,242,396),(2,153,243,339),(3,208,244,394),(4,151,245,337),(5,206,246,392),(6,149,247,447),(7,204,248,390),(8,147,249,445),(9,202,250,388),(10,145,251,443),(11,200,252,386),(12,143,253,441),(13,198,254,384),(14,141,255,439),(15,196,256,382),(16,139,257,437),(17,194,258,380),(18,137,259,435),(19,192,260,378),(20,135,261,433),(21,190,262,376),(22,133,263,431),(23,188,264,374),(24,131,265,429),(25,186,266,372),(26,129,267,427),(27,184,268,370),(28,127,269,425),(29,182,270,368),(30,125,271,423),(31,180,272,366),(32,123,273,421),(33,178,274,364),(34,121,275,419),(35,176,276,362),(36,119,277,417),(37,174,278,360),(38,117,279,415),(39,172,280,358),(40,115,281,413),(41,170,282,356),(42,113,283,411),(43,168,284,354),(44,223,285,409),(45,166,286,352),(46,221,287,407),(47,164,288,350),(48,219,289,405),(49,162,290,348),(50,217,291,403),(51,160,292,346),(52,215,293,401),(53,158,294,344),(54,213,295,399),(55,156,296,342),(56,211,297,397),(57,154,298,340),(58,209,299,395),(59,152,300,338),(60,207,301,393),(61,150,302,448),(62,205,303,391),(63,148,304,446),(64,203,305,389),(65,146,306,444),(66,201,307,387),(67,144,308,442),(68,199,309,385),(69,142,310,440),(70,197,311,383),(71,140,312,438),(72,195,313,381),(73,138,314,436),(74,193,315,379),(75,136,316,434),(76,191,317,377),(77,134,318,432),(78,189,319,375),(79,132,320,430),(80,187,321,373),(81,130,322,428),(82,185,323,371),(83,128,324,426),(84,183,325,369),(85,126,326,424),(86,181,327,367),(87,124,328,422),(88,179,329,365),(89,122,330,420),(90,177,331,363),(91,120,332,418),(92,175,333,361),(93,118,334,416),(94,173,335,359),(95,116,336,414),(96,171,225,357),(97,114,226,412),(98,169,227,355),(99,224,228,410),(100,167,229,353),(101,222,230,408),(102,165,231,351),(103,220,232,406),(104,163,233,349),(105,218,234,404),(106,161,235,347),(107,216,236,402),(108,159,237,345),(109,214,238,400),(110,157,239,343),(111,212,240,398),(112,155,241,341)]])
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 |
| type | + | + | + | - | + | + | - | + | - | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C4 | Q8 | D4 | D7 | Q16 | D8 | Dic7 | D14 | SD32 | Dic14 | D28 | Dic28 | D56 | C112⋊C2 |
| kernel | C112⋊6C4 | C56⋊1C4 | C2×C112 | C112 | C56 | C2×C28 | C2×C16 | C28 | C2×C14 | C16 | C2×C8 | C14 | C8 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 3 | 2 | 2 | 6 | 3 | 8 | 6 | 6 | 12 | 12 | 48 |
Matrix representation of C112⋊6C4 ►in GL4(𝔽113) generated by
| 100 | 109 | 0 | 0 |
| 8 | 98 | 0 | 0 |
| 0 | 0 | 109 | 9 |
| 0 | 0 | 104 | 77 |
| 9 | 57 | 0 | 0 |
| 62 | 104 | 0 | 0 |
| 0 | 0 | 22 | 43 |
| 0 | 0 | 15 | 91 |
G:=sub<GL(4,GF(113))| [100,8,0,0,109,98,0,0,0,0,109,104,0,0,9,77],[9,62,0,0,57,104,0,0,0,0,22,15,0,0,43,91] >;
C112⋊6C4 in GAP, Magma, Sage, TeX
C_{112}\rtimes_6C_4 % in TeX
G:=Group("C112:6C4"); // GroupNames label
G:=SmallGroup(448,62);
// by ID
G=gap.SmallGroup(448,62);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,141,176,1571,80,1684,102,18822]);
// Polycyclic
G:=Group<a,b|a^112=b^4=1,b*a*b^-1=a^55>;
// generators/relations
Export