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G = C6×C78order 468 = 22·32·13

Abelian group of type [6,78]

Aliases: C6×C78, SmallGroup(468,55)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6×C78
 Chief series C1 — C13 — C39 — C3×C39 — C3×C78 — C6×C78
 Lower central C1 — C6×C78
 Upper central 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 [×3], C3 [×4], C22, C6 [×12], C32, C2×C6 [×4], C13, C3×C6 [×3], C26 [×3], C62, C39 [×4], C2×C26, C78 [×12], C3×C39, C2×C78 [×4], C3×C78 [×3], C6×C78
Quotients: C1, C2 [×3], C3 [×4], C22, C6 [×12], C32, C2×C6 [×4], C13, C3×C6 [×3], C26 [×3], C62, C39 [×4], C2×C26, C78 [×12], C3×C39, C2×C78 [×4], C3×C78 [×3], C6×C78

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

׿
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