direct product, abelian, monomial, 2-elementary
Aliases: C4×C116, SmallGroup(464,20)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C116 |
| C1 — C4×C116 |
| C1 — C4×C116 |
Generators and relations for C4×C116
G = < a,b | a4=b116=1, ab=ba >
Smallest permutation representation of C4×C116
►Regular action on 464 points
Generators in S464
(1 458 284 160)(2 459 285 161)(3 460 286 162)(4 461 287 163)(5 462 288 164)(6 463 289 165)(7 464 290 166)(8 349 291 167)(9 350 292 168)(10 351 293 169)(11 352 294 170)(12 353 295 171)(13 354 296 172)(14 355 297 173)(15 356 298 174)(16 357 299 175)(17 358 300 176)(18 359 301 177)(19 360 302 178)(20 361 303 179)(21 362 304 180)(22 363 305 181)(23 364 306 182)(24 365 307 183)(25 366 308 184)(26 367 309 185)(27 368 310 186)(28 369 311 187)(29 370 312 188)(30 371 313 189)(31 372 314 190)(32 373 315 191)(33 374 316 192)(34 375 317 193)(35 376 318 194)(36 377 319 195)(37 378 320 196)(38 379 321 197)(39 380 322 198)(40 381 323 199)(41 382 324 200)(42 383 325 201)(43 384 326 202)(44 385 327 203)(45 386 328 204)(46 387 329 205)(47 388 330 206)(48 389 331 207)(49 390 332 208)(50 391 333 209)(51 392 334 210)(52 393 335 211)(53 394 336 212)(54 395 337 213)(55 396 338 214)(56 397 339 215)(57 398 340 216)(58 399 341 217)(59 400 342 218)(60 401 343 219)(61 402 344 220)(62 403 345 221)(63 404 346 222)(64 405 347 223)(65 406 348 224)(66 407 233 225)(67 408 234 226)(68 409 235 227)(69 410 236 228)(70 411 237 229)(71 412 238 230)(72 413 239 231)(73 414 240 232)(74 415 241 117)(75 416 242 118)(76 417 243 119)(77 418 244 120)(78 419 245 121)(79 420 246 122)(80 421 247 123)(81 422 248 124)(82 423 249 125)(83 424 250 126)(84 425 251 127)(85 426 252 128)(86 427 253 129)(87 428 254 130)(88 429 255 131)(89 430 256 132)(90 431 257 133)(91 432 258 134)(92 433 259 135)(93 434 260 136)(94 435 261 137)(95 436 262 138)(96 437 263 139)(97 438 264 140)(98 439 265 141)(99 440 266 142)(100 441 267 143)(101 442 268 144)(102 443 269 145)(103 444 270 146)(104 445 271 147)(105 446 272 148)(106 447 273 149)(107 448 274 150)(108 449 275 151)(109 450 276 152)(110 451 277 153)(111 452 278 154)(112 453 279 155)(113 454 280 156)(114 455 281 157)(115 456 282 158)(116 457 283 159)
(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)(349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464)
G:=sub<Sym(464)| (1,458,284,160)(2,459,285,161)(3,460,286,162)(4,461,287,163)(5,462,288,164)(6,463,289,165)(7,464,290,166)(8,349,291,167)(9,350,292,168)(10,351,293,169)(11,352,294,170)(12,353,295,171)(13,354,296,172)(14,355,297,173)(15,356,298,174)(16,357,299,175)(17,358,300,176)(18,359,301,177)(19,360,302,178)(20,361,303,179)(21,362,304,180)(22,363,305,181)(23,364,306,182)(24,365,307,183)(25,366,308,184)(26,367,309,185)(27,368,310,186)(28,369,311,187)(29,370,312,188)(30,371,313,189)(31,372,314,190)(32,373,315,191)(33,374,316,192)(34,375,317,193)(35,376,318,194)(36,377,319,195)(37,378,320,196)(38,379,321,197)(39,380,322,198)(40,381,323,199)(41,382,324,200)(42,383,325,201)(43,384,326,202)(44,385,327,203)(45,386,328,204)(46,387,329,205)(47,388,330,206)(48,389,331,207)(49,390,332,208)(50,391,333,209)(51,392,334,210)(52,393,335,211)(53,394,336,212)(54,395,337,213)(55,396,338,214)(56,397,339,215)(57,398,340,216)(58,399,341,217)(59,400,342,218)(60,401,343,219)(61,402,344,220)(62,403,345,221)(63,404,346,222)(64,405,347,223)(65,406,348,224)(66,407,233,225)(67,408,234,226)(68,409,235,227)(69,410,236,228)(70,411,237,229)(71,412,238,230)(72,413,239,231)(73,414,240,232)(74,415,241,117)(75,416,242,118)(76,417,243,119)(77,418,244,120)(78,419,245,121)(79,420,246,122)(80,421,247,123)(81,422,248,124)(82,423,249,125)(83,424,250,126)(84,425,251,127)(85,426,252,128)(86,427,253,129)(87,428,254,130)(88,429,255,131)(89,430,256,132)(90,431,257,133)(91,432,258,134)(92,433,259,135)(93,434,260,136)(94,435,261,137)(95,436,262,138)(96,437,263,139)(97,438,264,140)(98,439,265,141)(99,440,266,142)(100,441,267,143)(101,442,268,144)(102,443,269,145)(103,444,270,146)(104,445,271,147)(105,446,272,148)(106,447,273,149)(107,448,274,150)(108,449,275,151)(109,450,276,152)(110,451,277,153)(111,452,278,154)(112,453,279,155)(113,454,280,156)(114,455,281,157)(115,456,282,158)(116,457,283,159), (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)(349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464)>;
G:=Group( (1,458,284,160)(2,459,285,161)(3,460,286,162)(4,461,287,163)(5,462,288,164)(6,463,289,165)(7,464,290,166)(8,349,291,167)(9,350,292,168)(10,351,293,169)(11,352,294,170)(12,353,295,171)(13,354,296,172)(14,355,297,173)(15,356,298,174)(16,357,299,175)(17,358,300,176)(18,359,301,177)(19,360,302,178)(20,361,303,179)(21,362,304,180)(22,363,305,181)(23,364,306,182)(24,365,307,183)(25,366,308,184)(26,367,309,185)(27,368,310,186)(28,369,311,187)(29,370,312,188)(30,371,313,189)(31,372,314,190)(32,373,315,191)(33,374,316,192)(34,375,317,193)(35,376,318,194)(36,377,319,195)(37,378,320,196)(38,379,321,197)(39,380,322,198)(40,381,323,199)(41,382,324,200)(42,383,325,201)(43,384,326,202)(44,385,327,203)(45,386,328,204)(46,387,329,205)(47,388,330,206)(48,389,331,207)(49,390,332,208)(50,391,333,209)(51,392,334,210)(52,393,335,211)(53,394,336,212)(54,395,337,213)(55,396,338,214)(56,397,339,215)(57,398,340,216)(58,399,341,217)(59,400,342,218)(60,401,343,219)(61,402,344,220)(62,403,345,221)(63,404,346,222)(64,405,347,223)(65,406,348,224)(66,407,233,225)(67,408,234,226)(68,409,235,227)(69,410,236,228)(70,411,237,229)(71,412,238,230)(72,413,239,231)(73,414,240,232)(74,415,241,117)(75,416,242,118)(76,417,243,119)(77,418,244,120)(78,419,245,121)(79,420,246,122)(80,421,247,123)(81,422,248,124)(82,423,249,125)(83,424,250,126)(84,425,251,127)(85,426,252,128)(86,427,253,129)(87,428,254,130)(88,429,255,131)(89,430,256,132)(90,431,257,133)(91,432,258,134)(92,433,259,135)(93,434,260,136)(94,435,261,137)(95,436,262,138)(96,437,263,139)(97,438,264,140)(98,439,265,141)(99,440,266,142)(100,441,267,143)(101,442,268,144)(102,443,269,145)(103,444,270,146)(104,445,271,147)(105,446,272,148)(106,447,273,149)(107,448,274,150)(108,449,275,151)(109,450,276,152)(110,451,277,153)(111,452,278,154)(112,453,279,155)(113,454,280,156)(114,455,281,157)(115,456,282,158)(116,457,283,159), (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)(349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464) );
G=PermutationGroup([[(1,458,284,160),(2,459,285,161),(3,460,286,162),(4,461,287,163),(5,462,288,164),(6,463,289,165),(7,464,290,166),(8,349,291,167),(9,350,292,168),(10,351,293,169),(11,352,294,170),(12,353,295,171),(13,354,296,172),(14,355,297,173),(15,356,298,174),(16,357,299,175),(17,358,300,176),(18,359,301,177),(19,360,302,178),(20,361,303,179),(21,362,304,180),(22,363,305,181),(23,364,306,182),(24,365,307,183),(25,366,308,184),(26,367,309,185),(27,368,310,186),(28,369,311,187),(29,370,312,188),(30,371,313,189),(31,372,314,190),(32,373,315,191),(33,374,316,192),(34,375,317,193),(35,376,318,194),(36,377,319,195),(37,378,320,196),(38,379,321,197),(39,380,322,198),(40,381,323,199),(41,382,324,200),(42,383,325,201),(43,384,326,202),(44,385,327,203),(45,386,328,204),(46,387,329,205),(47,388,330,206),(48,389,331,207),(49,390,332,208),(50,391,333,209),(51,392,334,210),(52,393,335,211),(53,394,336,212),(54,395,337,213),(55,396,338,214),(56,397,339,215),(57,398,340,216),(58,399,341,217),(59,400,342,218),(60,401,343,219),(61,402,344,220),(62,403,345,221),(63,404,346,222),(64,405,347,223),(65,406,348,224),(66,407,233,225),(67,408,234,226),(68,409,235,227),(69,410,236,228),(70,411,237,229),(71,412,238,230),(72,413,239,231),(73,414,240,232),(74,415,241,117),(75,416,242,118),(76,417,243,119),(77,418,244,120),(78,419,245,121),(79,420,246,122),(80,421,247,123),(81,422,248,124),(82,423,249,125),(83,424,250,126),(84,425,251,127),(85,426,252,128),(86,427,253,129),(87,428,254,130),(88,429,255,131),(89,430,256,132),(90,431,257,133),(91,432,258,134),(92,433,259,135),(93,434,260,136),(94,435,261,137),(95,436,262,138),(96,437,263,139),(97,438,264,140),(98,439,265,141),(99,440,266,142),(100,441,267,143),(101,442,268,144),(102,443,269,145),(103,444,270,146),(104,445,271,147),(105,446,272,148),(106,447,273,149),(107,448,274,150),(108,449,275,151),(109,450,276,152),(110,451,277,153),(111,452,278,154),(112,453,279,155),(113,454,280,156),(114,455,281,157),(115,456,282,158),(116,457,283,159)], [(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),(349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464)]])
464 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 29A | ··· | 29AB | 58A | ··· | 58CF | 116A | ··· | 116LX |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
464 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C29 | C58 | C116 |
| kernel | C4×C116 | C2×C116 | C116 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 28 | 84 | 336 |
Matrix representation of C4×C116 ►in GL2(𝔽233) generated by
| 232 | 0 |
| 0 | 89 |
| 219 | 0 |
| 0 | 30 |
G:=sub<GL(2,GF(233))| [232,0,0,89],[219,0,0,30] >;
C4×C116 in GAP, Magma, Sage, TeX
C_4\times C_{116} % in TeX
G:=Group("C4xC116"); // GroupNames label
G:=SmallGroup(464,20);
// by ID
G=gap.SmallGroup(464,20);
# by ID
G:=PCGroup([5,-2,-2,-29,-2,-2,580,1166]);
// Polycyclic
G:=Group<a,b|a^4=b^116=1,a*b=b*a>;
// generators/relations
Export