direct product, abelian, monomial
Aliases: C6×C78, SmallGroup(468,55)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C6×C78 |
| C1 — C6×C78 |
| C1 — C6×C78 |
Generators and relations for C6×C78
G = < a,b | a6=b78=1, ab=ba >
Subgroups: 60, all normal (8 characteristic)
C1, C2, C3, C22, C6, C32, C2×C6, C13, C3×C6, C26, C62, C39, C2×C26, C78, C3×C39, C2×C78, C3×C78, C6×C78
Quotients: C1, C2, C3, C22, C6, C32, C2×C6, C13, C3×C6, C26, C62, C39, C2×C26, C78, C3×C39, C2×C78, C3×C78, C6×C78
►Regular action on 468 points
Generators in S468
(1 257 210 94 405 332)(2 258 211 95 406 333)(3 259 212 96 407 334)(4 260 213 97 408 335)(5 261 214 98 409 336)(6 262 215 99 410 337)(7 263 216 100 411 338)(8 264 217 101 412 339)(9 265 218 102 413 340)(10 266 219 103 414 341)(11 267 220 104 415 342)(12 268 221 105 416 343)(13 269 222 106 417 344)(14 270 223 107 418 345)(15 271 224 108 419 346)(16 272 225 109 420 347)(17 273 226 110 421 348)(18 274 227 111 422 349)(19 275 228 112 423 350)(20 276 229 113 424 351)(21 277 230 114 425 352)(22 278 231 115 426 353)(23 279 232 116 427 354)(24 280 233 117 428 355)(25 281 234 118 429 356)(26 282 157 119 430 357)(27 283 158 120 431 358)(28 284 159 121 432 359)(29 285 160 122 433 360)(30 286 161 123 434 361)(31 287 162 124 435 362)(32 288 163 125 436 363)(33 289 164 126 437 364)(34 290 165 127 438 365)(35 291 166 128 439 366)(36 292 167 129 440 367)(37 293 168 130 441 368)(38 294 169 131 442 369)(39 295 170 132 443 370)(40 296 171 133 444 371)(41 297 172 134 445 372)(42 298 173 135 446 373)(43 299 174 136 447 374)(44 300 175 137 448 375)(45 301 176 138 449 376)(46 302 177 139 450 377)(47 303 178 140 451 378)(48 304 179 141 452 379)(49 305 180 142 453 380)(50 306 181 143 454 381)(51 307 182 144 455 382)(52 308 183 145 456 383)(53 309 184 146 457 384)(54 310 185 147 458 385)(55 311 186 148 459 386)(56 312 187 149 460 387)(57 235 188 150 461 388)(58 236 189 151 462 389)(59 237 190 152 463 390)(60 238 191 153 464 313)(61 239 192 154 465 314)(62 240 193 155 466 315)(63 241 194 156 467 316)(64 242 195 79 468 317)(65 243 196 80 391 318)(66 244 197 81 392 319)(67 245 198 82 393 320)(68 246 199 83 394 321)(69 247 200 84 395 322)(70 248 201 85 396 323)(71 249 202 86 397 324)(72 250 203 87 398 325)(73 251 204 88 399 326)(74 252 205 89 400 327)(75 253 206 90 401 328)(76 254 207 91 402 329)(77 255 208 92 403 330)(78 256 209 93 404 331)
(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)(391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468)
G:=sub<Sym(468)| (1,257,210,94,405,332)(2,258,211,95,406,333)(3,259,212,96,407,334)(4,260,213,97,408,335)(5,261,214,98,409,336)(6,262,215,99,410,337)(7,263,216,100,411,338)(8,264,217,101,412,339)(9,265,218,102,413,340)(10,266,219,103,414,341)(11,267,220,104,415,342)(12,268,221,105,416,343)(13,269,222,106,417,344)(14,270,223,107,418,345)(15,271,224,108,419,346)(16,272,225,109,420,347)(17,273,226,110,421,348)(18,274,227,111,422,349)(19,275,228,112,423,350)(20,276,229,113,424,351)(21,277,230,114,425,352)(22,278,231,115,426,353)(23,279,232,116,427,354)(24,280,233,117,428,355)(25,281,234,118,429,356)(26,282,157,119,430,357)(27,283,158,120,431,358)(28,284,159,121,432,359)(29,285,160,122,433,360)(30,286,161,123,434,361)(31,287,162,124,435,362)(32,288,163,125,436,363)(33,289,164,126,437,364)(34,290,165,127,438,365)(35,291,166,128,439,366)(36,292,167,129,440,367)(37,293,168,130,441,368)(38,294,169,131,442,369)(39,295,170,132,443,370)(40,296,171,133,444,371)(41,297,172,134,445,372)(42,298,173,135,446,373)(43,299,174,136,447,374)(44,300,175,137,448,375)(45,301,176,138,449,376)(46,302,177,139,450,377)(47,303,178,140,451,378)(48,304,179,141,452,379)(49,305,180,142,453,380)(50,306,181,143,454,381)(51,307,182,144,455,382)(52,308,183,145,456,383)(53,309,184,146,457,384)(54,310,185,147,458,385)(55,311,186,148,459,386)(56,312,187,149,460,387)(57,235,188,150,461,388)(58,236,189,151,462,389)(59,237,190,152,463,390)(60,238,191,153,464,313)(61,239,192,154,465,314)(62,240,193,155,466,315)(63,241,194,156,467,316)(64,242,195,79,468,317)(65,243,196,80,391,318)(66,244,197,81,392,319)(67,245,198,82,393,320)(68,246,199,83,394,321)(69,247,200,84,395,322)(70,248,201,85,396,323)(71,249,202,86,397,324)(72,250,203,87,398,325)(73,251,204,88,399,326)(74,252,205,89,400,327)(75,253,206,90,401,328)(76,254,207,91,402,329)(77,255,208,92,403,330)(78,256,209,93,404,331), (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)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468)>;
G:=Group( (1,257,210,94,405,332)(2,258,211,95,406,333)(3,259,212,96,407,334)(4,260,213,97,408,335)(5,261,214,98,409,336)(6,262,215,99,410,337)(7,263,216,100,411,338)(8,264,217,101,412,339)(9,265,218,102,413,340)(10,266,219,103,414,341)(11,267,220,104,415,342)(12,268,221,105,416,343)(13,269,222,106,417,344)(14,270,223,107,418,345)(15,271,224,108,419,346)(16,272,225,109,420,347)(17,273,226,110,421,348)(18,274,227,111,422,349)(19,275,228,112,423,350)(20,276,229,113,424,351)(21,277,230,114,425,352)(22,278,231,115,426,353)(23,279,232,116,427,354)(24,280,233,117,428,355)(25,281,234,118,429,356)(26,282,157,119,430,357)(27,283,158,120,431,358)(28,284,159,121,432,359)(29,285,160,122,433,360)(30,286,161,123,434,361)(31,287,162,124,435,362)(32,288,163,125,436,363)(33,289,164,126,437,364)(34,290,165,127,438,365)(35,291,166,128,439,366)(36,292,167,129,440,367)(37,293,168,130,441,368)(38,294,169,131,442,369)(39,295,170,132,443,370)(40,296,171,133,444,371)(41,297,172,134,445,372)(42,298,173,135,446,373)(43,299,174,136,447,374)(44,300,175,137,448,375)(45,301,176,138,449,376)(46,302,177,139,450,377)(47,303,178,140,451,378)(48,304,179,141,452,379)(49,305,180,142,453,380)(50,306,181,143,454,381)(51,307,182,144,455,382)(52,308,183,145,456,383)(53,309,184,146,457,384)(54,310,185,147,458,385)(55,311,186,148,459,386)(56,312,187,149,460,387)(57,235,188,150,461,388)(58,236,189,151,462,389)(59,237,190,152,463,390)(60,238,191,153,464,313)(61,239,192,154,465,314)(62,240,193,155,466,315)(63,241,194,156,467,316)(64,242,195,79,468,317)(65,243,196,80,391,318)(66,244,197,81,392,319)(67,245,198,82,393,320)(68,246,199,83,394,321)(69,247,200,84,395,322)(70,248,201,85,396,323)(71,249,202,86,397,324)(72,250,203,87,398,325)(73,251,204,88,399,326)(74,252,205,89,400,327)(75,253,206,90,401,328)(76,254,207,91,402,329)(77,255,208,92,403,330)(78,256,209,93,404,331), (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)(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468) );
G=PermutationGroup([[(1,257,210,94,405,332),(2,258,211,95,406,333),(3,259,212,96,407,334),(4,260,213,97,408,335),(5,261,214,98,409,336),(6,262,215,99,410,337),(7,263,216,100,411,338),(8,264,217,101,412,339),(9,265,218,102,413,340),(10,266,219,103,414,341),(11,267,220,104,415,342),(12,268,221,105,416,343),(13,269,222,106,417,344),(14,270,223,107,418,345),(15,271,224,108,419,346),(16,272,225,109,420,347),(17,273,226,110,421,348),(18,274,227,111,422,349),(19,275,228,112,423,350),(20,276,229,113,424,351),(21,277,230,114,425,352),(22,278,231,115,426,353),(23,279,232,116,427,354),(24,280,233,117,428,355),(25,281,234,118,429,356),(26,282,157,119,430,357),(27,283,158,120,431,358),(28,284,159,121,432,359),(29,285,160,122,433,360),(30,286,161,123,434,361),(31,287,162,124,435,362),(32,288,163,125,436,363),(33,289,164,126,437,364),(34,290,165,127,438,365),(35,291,166,128,439,366),(36,292,167,129,440,367),(37,293,168,130,441,368),(38,294,169,131,442,369),(39,295,170,132,443,370),(40,296,171,133,444,371),(41,297,172,134,445,372),(42,298,173,135,446,373),(43,299,174,136,447,374),(44,300,175,137,448,375),(45,301,176,138,449,376),(46,302,177,139,450,377),(47,303,178,140,451,378),(48,304,179,141,452,379),(49,305,180,142,453,380),(50,306,181,143,454,381),(51,307,182,144,455,382),(52,308,183,145,456,383),(53,309,184,146,457,384),(54,310,185,147,458,385),(55,311,186,148,459,386),(56,312,187,149,460,387),(57,235,188,150,461,388),(58,236,189,151,462,389),(59,237,190,152,463,390),(60,238,191,153,464,313),(61,239,192,154,465,314),(62,240,193,155,466,315),(63,241,194,156,467,316),(64,242,195,79,468,317),(65,243,196,80,391,318),(66,244,197,81,392,319),(67,245,198,82,393,320),(68,246,199,83,394,321),(69,247,200,84,395,322),(70,248,201,85,396,323),(71,249,202,86,397,324),(72,250,203,87,398,325),(73,251,204,88,399,326),(74,252,205,89,400,327),(75,253,206,90,401,328),(76,254,207,91,402,329),(77,255,208,92,403,330),(78,256,209,93,404,331)], [(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),(391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468)]])
468 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 6A | ··· | 6X | 13A | ··· | 13L | 26A | ··· | 26AJ | 39A | ··· | 39CR | 78A | ··· | 78KB |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 13 | ··· | 13 | 26 | ··· | 26 | 39 | ··· | 39 | 78 | ··· | 78 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
468 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C3 | C6 | C13 | C26 | C39 | C78 |
| kernel | C6×C78 | C3×C78 | C2×C78 | C78 | C62 | C3×C6 | C2×C6 | C6 |
| # reps | 1 | 3 | 8 | 24 | 12 | 36 | 96 | 288 |
Matrix representation of C6×C78 ►in GL2(𝔽79) generated by
| 24 | 0 |
| 0 | 56 |
| 33 | 0 |
| 0 | 4 |
G:=sub<GL(2,GF(79))| [24,0,0,56],[33,0,0,4] >;
C6×C78 in GAP, Magma, Sage, TeX
C_6\times C_{78} % in TeX
G:=Group("C6xC78"); // GroupNames label
G:=SmallGroup(468,55);
// by ID
G=gap.SmallGroup(468,55);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-13]);
// Polycyclic
G:=Group<a,b|a^6=b^78=1,a*b=b*a>;
// generators/relations