direct product, abelian, monomial, 2-elementary
Aliases: C4×C112, SmallGroup(448,149)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C112 |
| C1 — C4×C112 |
| C1 — C4×C112 |
Generators and relations for C4×C112
G = < a,b | a4=b112=1, ab=ba >
Smallest permutation representation of C4×C112
►Regular action on 448 points
Generators in S448
(1 197 294 377)(2 198 295 378)(3 199 296 379)(4 200 297 380)(5 201 298 381)(6 202 299 382)(7 203 300 383)(8 204 301 384)(9 205 302 385)(10 206 303 386)(11 207 304 387)(12 208 305 388)(13 209 306 389)(14 210 307 390)(15 211 308 391)(16 212 309 392)(17 213 310 393)(18 214 311 394)(19 215 312 395)(20 216 313 396)(21 217 314 397)(22 218 315 398)(23 219 316 399)(24 220 317 400)(25 221 318 401)(26 222 319 402)(27 223 320 403)(28 224 321 404)(29 113 322 405)(30 114 323 406)(31 115 324 407)(32 116 325 408)(33 117 326 409)(34 118 327 410)(35 119 328 411)(36 120 329 412)(37 121 330 413)(38 122 331 414)(39 123 332 415)(40 124 333 416)(41 125 334 417)(42 126 335 418)(43 127 336 419)(44 128 225 420)(45 129 226 421)(46 130 227 422)(47 131 228 423)(48 132 229 424)(49 133 230 425)(50 134 231 426)(51 135 232 427)(52 136 233 428)(53 137 234 429)(54 138 235 430)(55 139 236 431)(56 140 237 432)(57 141 238 433)(58 142 239 434)(59 143 240 435)(60 144 241 436)(61 145 242 437)(62 146 243 438)(63 147 244 439)(64 148 245 440)(65 149 246 441)(66 150 247 442)(67 151 248 443)(68 152 249 444)(69 153 250 445)(70 154 251 446)(71 155 252 447)(72 156 253 448)(73 157 254 337)(74 158 255 338)(75 159 256 339)(76 160 257 340)(77 161 258 341)(78 162 259 342)(79 163 260 343)(80 164 261 344)(81 165 262 345)(82 166 263 346)(83 167 264 347)(84 168 265 348)(85 169 266 349)(86 170 267 350)(87 171 268 351)(88 172 269 352)(89 173 270 353)(90 174 271 354)(91 175 272 355)(92 176 273 356)(93 177 274 357)(94 178 275 358)(95 179 276 359)(96 180 277 360)(97 181 278 361)(98 182 279 362)(99 183 280 363)(100 184 281 364)(101 185 282 365)(102 186 283 366)(103 187 284 367)(104 188 285 368)(105 189 286 369)(106 190 287 370)(107 191 288 371)(108 192 289 372)(109 193 290 373)(110 194 291 374)(111 195 292 375)(112 196 293 376)
(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)
G:=sub<Sym(448)| (1,197,294,377)(2,198,295,378)(3,199,296,379)(4,200,297,380)(5,201,298,381)(6,202,299,382)(7,203,300,383)(8,204,301,384)(9,205,302,385)(10,206,303,386)(11,207,304,387)(12,208,305,388)(13,209,306,389)(14,210,307,390)(15,211,308,391)(16,212,309,392)(17,213,310,393)(18,214,311,394)(19,215,312,395)(20,216,313,396)(21,217,314,397)(22,218,315,398)(23,219,316,399)(24,220,317,400)(25,221,318,401)(26,222,319,402)(27,223,320,403)(28,224,321,404)(29,113,322,405)(30,114,323,406)(31,115,324,407)(32,116,325,408)(33,117,326,409)(34,118,327,410)(35,119,328,411)(36,120,329,412)(37,121,330,413)(38,122,331,414)(39,123,332,415)(40,124,333,416)(41,125,334,417)(42,126,335,418)(43,127,336,419)(44,128,225,420)(45,129,226,421)(46,130,227,422)(47,131,228,423)(48,132,229,424)(49,133,230,425)(50,134,231,426)(51,135,232,427)(52,136,233,428)(53,137,234,429)(54,138,235,430)(55,139,236,431)(56,140,237,432)(57,141,238,433)(58,142,239,434)(59,143,240,435)(60,144,241,436)(61,145,242,437)(62,146,243,438)(63,147,244,439)(64,148,245,440)(65,149,246,441)(66,150,247,442)(67,151,248,443)(68,152,249,444)(69,153,250,445)(70,154,251,446)(71,155,252,447)(72,156,253,448)(73,157,254,337)(74,158,255,338)(75,159,256,339)(76,160,257,340)(77,161,258,341)(78,162,259,342)(79,163,260,343)(80,164,261,344)(81,165,262,345)(82,166,263,346)(83,167,264,347)(84,168,265,348)(85,169,266,349)(86,170,267,350)(87,171,268,351)(88,172,269,352)(89,173,270,353)(90,174,271,354)(91,175,272,355)(92,176,273,356)(93,177,274,357)(94,178,275,358)(95,179,276,359)(96,180,277,360)(97,181,278,361)(98,182,279,362)(99,183,280,363)(100,184,281,364)(101,185,282,365)(102,186,283,366)(103,187,284,367)(104,188,285,368)(105,189,286,369)(106,190,287,370)(107,191,288,371)(108,192,289,372)(109,193,290,373)(110,194,291,374)(111,195,292,375)(112,196,293,376), (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)>;
G:=Group( (1,197,294,377)(2,198,295,378)(3,199,296,379)(4,200,297,380)(5,201,298,381)(6,202,299,382)(7,203,300,383)(8,204,301,384)(9,205,302,385)(10,206,303,386)(11,207,304,387)(12,208,305,388)(13,209,306,389)(14,210,307,390)(15,211,308,391)(16,212,309,392)(17,213,310,393)(18,214,311,394)(19,215,312,395)(20,216,313,396)(21,217,314,397)(22,218,315,398)(23,219,316,399)(24,220,317,400)(25,221,318,401)(26,222,319,402)(27,223,320,403)(28,224,321,404)(29,113,322,405)(30,114,323,406)(31,115,324,407)(32,116,325,408)(33,117,326,409)(34,118,327,410)(35,119,328,411)(36,120,329,412)(37,121,330,413)(38,122,331,414)(39,123,332,415)(40,124,333,416)(41,125,334,417)(42,126,335,418)(43,127,336,419)(44,128,225,420)(45,129,226,421)(46,130,227,422)(47,131,228,423)(48,132,229,424)(49,133,230,425)(50,134,231,426)(51,135,232,427)(52,136,233,428)(53,137,234,429)(54,138,235,430)(55,139,236,431)(56,140,237,432)(57,141,238,433)(58,142,239,434)(59,143,240,435)(60,144,241,436)(61,145,242,437)(62,146,243,438)(63,147,244,439)(64,148,245,440)(65,149,246,441)(66,150,247,442)(67,151,248,443)(68,152,249,444)(69,153,250,445)(70,154,251,446)(71,155,252,447)(72,156,253,448)(73,157,254,337)(74,158,255,338)(75,159,256,339)(76,160,257,340)(77,161,258,341)(78,162,259,342)(79,163,260,343)(80,164,261,344)(81,165,262,345)(82,166,263,346)(83,167,264,347)(84,168,265,348)(85,169,266,349)(86,170,267,350)(87,171,268,351)(88,172,269,352)(89,173,270,353)(90,174,271,354)(91,175,272,355)(92,176,273,356)(93,177,274,357)(94,178,275,358)(95,179,276,359)(96,180,277,360)(97,181,278,361)(98,182,279,362)(99,183,280,363)(100,184,281,364)(101,185,282,365)(102,186,283,366)(103,187,284,367)(104,188,285,368)(105,189,286,369)(106,190,287,370)(107,191,288,371)(108,192,289,372)(109,193,290,373)(110,194,291,374)(111,195,292,375)(112,196,293,376), (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) );
G=PermutationGroup([[(1,197,294,377),(2,198,295,378),(3,199,296,379),(4,200,297,380),(5,201,298,381),(6,202,299,382),(7,203,300,383),(8,204,301,384),(9,205,302,385),(10,206,303,386),(11,207,304,387),(12,208,305,388),(13,209,306,389),(14,210,307,390),(15,211,308,391),(16,212,309,392),(17,213,310,393),(18,214,311,394),(19,215,312,395),(20,216,313,396),(21,217,314,397),(22,218,315,398),(23,219,316,399),(24,220,317,400),(25,221,318,401),(26,222,319,402),(27,223,320,403),(28,224,321,404),(29,113,322,405),(30,114,323,406),(31,115,324,407),(32,116,325,408),(33,117,326,409),(34,118,327,410),(35,119,328,411),(36,120,329,412),(37,121,330,413),(38,122,331,414),(39,123,332,415),(40,124,333,416),(41,125,334,417),(42,126,335,418),(43,127,336,419),(44,128,225,420),(45,129,226,421),(46,130,227,422),(47,131,228,423),(48,132,229,424),(49,133,230,425),(50,134,231,426),(51,135,232,427),(52,136,233,428),(53,137,234,429),(54,138,235,430),(55,139,236,431),(56,140,237,432),(57,141,238,433),(58,142,239,434),(59,143,240,435),(60,144,241,436),(61,145,242,437),(62,146,243,438),(63,147,244,439),(64,148,245,440),(65,149,246,441),(66,150,247,442),(67,151,248,443),(68,152,249,444),(69,153,250,445),(70,154,251,446),(71,155,252,447),(72,156,253,448),(73,157,254,337),(74,158,255,338),(75,159,256,339),(76,160,257,340),(77,161,258,341),(78,162,259,342),(79,163,260,343),(80,164,261,344),(81,165,262,345),(82,166,263,346),(83,167,264,347),(84,168,265,348),(85,169,266,349),(86,170,267,350),(87,171,268,351),(88,172,269,352),(89,173,270,353),(90,174,271,354),(91,175,272,355),(92,176,273,356),(93,177,274,357),(94,178,275,358),(95,179,276,359),(96,180,277,360),(97,181,278,361),(98,182,279,362),(99,183,280,363),(100,184,281,364),(101,185,282,365),(102,186,283,366),(103,187,284,367),(104,188,285,368),(105,189,286,369),(106,190,287,370),(107,191,288,371),(108,192,289,372),(109,193,290,373),(110,194,291,374),(111,195,292,375),(112,196,293,376)], [(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)]])
448 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 7A | ··· | 7F | 8A | ··· | 8P | 14A | ··· | 14R | 16A | ··· | 16AF | 28A | ··· | 28BT | 56A | ··· | 56CR | 112A | ··· | 112GJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 7 | ··· | 7 | 8 | ··· | 8 | 14 | ··· | 14 | 16 | ··· | 16 | 28 | ··· | 28 | 56 | ··· | 56 | 112 | ··· | 112 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
448 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C7 | C8 | C8 | C14 | C14 | C16 | C28 | C28 | C28 | C56 | C56 | C112 |
| kernel | C4×C112 | C4×C56 | C2×C112 | C112 | C4×C28 | C2×C56 | C4×C16 | C56 | C2×C28 | C4×C8 | C2×C16 | C28 | C16 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 6 | 8 | 8 | 6 | 12 | 32 | 48 | 12 | 12 | 48 | 48 | 192 |
Matrix representation of C4×C112 ►in GL3(𝔽113) generated by
| 98 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 98 |
| 1 | 0 | 0 |
| 0 | 13 | 0 |
| 0 | 0 | 35 |
G:=sub<GL(3,GF(113))| [98,0,0,0,1,0,0,0,98],[1,0,0,0,13,0,0,0,35] >;
C4×C112 in GAP, Magma, Sage, TeX
C_4\times C_{112} % in TeX
G:=Group("C4xC112"); // GroupNames label
G:=SmallGroup(448,149);
// by ID
G=gap.SmallGroup(448,149);
# by ID
G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,400,136,124]);
// Polycyclic
G:=Group<a,b|a^4=b^112=1,a*b=b*a>;
// generators/relations
Export