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G = C8×C56order 448 = 26·7

Abelian group of type [8,56]

direct product, abelian, monomial, 2-elementary

Aliases: C8×C56, SmallGroup(448,125)

Series: Derived Chief Lower central Upper central

C1 — C8×C56
C1C2C22C2×C4C42C4×C28C4×C56 — C8×C56
C1 — C8×C56
C1 — C8×C56

Generators and relations for C8×C56
 G = < a,b | a8=b56=1, ab=ba >

Subgroups: 74, all normal (8 characteristic)
C1, C2 [×3], C4 [×6], C22, C7, C8 [×12], C2×C4 [×3], C14 [×3], C42, C2×C8 [×6], C28 [×6], C2×C14, C4×C8 [×3], C56 [×12], C2×C28 [×3], C82, C4×C28, C2×C56 [×6], C4×C56 [×3], C8×C56
Quotients: C1, C2 [×3], C4 [×6], C22, C7, C8 [×12], C2×C4 [×3], C14 [×3], C42, C2×C8 [×6], C28 [×6], C2×C14, C4×C8 [×3], C56 [×12], C2×C28 [×3], C82, C4×C28, C2×C56 [×6], C4×C56 [×3], C8×C56

Smallest permutation representation of C8×C56
Regular action on 448 points
Generators in S448
(1 134 83 256 379 429 287 223)(2 135 84 257 380 430 288 224)(3 136 85 258 381 431 289 169)(4 137 86 259 382 432 290 170)(5 138 87 260 383 433 291 171)(6 139 88 261 384 434 292 172)(7 140 89 262 385 435 293 173)(8 141 90 263 386 436 294 174)(9 142 91 264 387 437 295 175)(10 143 92 265 388 438 296 176)(11 144 93 266 389 439 297 177)(12 145 94 267 390 440 298 178)(13 146 95 268 391 441 299 179)(14 147 96 269 392 442 300 180)(15 148 97 270 337 443 301 181)(16 149 98 271 338 444 302 182)(17 150 99 272 339 445 303 183)(18 151 100 273 340 446 304 184)(19 152 101 274 341 447 305 185)(20 153 102 275 342 448 306 186)(21 154 103 276 343 393 307 187)(22 155 104 277 344 394 308 188)(23 156 105 278 345 395 309 189)(24 157 106 279 346 396 310 190)(25 158 107 280 347 397 311 191)(26 159 108 225 348 398 312 192)(27 160 109 226 349 399 313 193)(28 161 110 227 350 400 314 194)(29 162 111 228 351 401 315 195)(30 163 112 229 352 402 316 196)(31 164 57 230 353 403 317 197)(32 165 58 231 354 404 318 198)(33 166 59 232 355 405 319 199)(34 167 60 233 356 406 320 200)(35 168 61 234 357 407 321 201)(36 113 62 235 358 408 322 202)(37 114 63 236 359 409 323 203)(38 115 64 237 360 410 324 204)(39 116 65 238 361 411 325 205)(40 117 66 239 362 412 326 206)(41 118 67 240 363 413 327 207)(42 119 68 241 364 414 328 208)(43 120 69 242 365 415 329 209)(44 121 70 243 366 416 330 210)(45 122 71 244 367 417 331 211)(46 123 72 245 368 418 332 212)(47 124 73 246 369 419 333 213)(48 125 74 247 370 420 334 214)(49 126 75 248 371 421 335 215)(50 127 76 249 372 422 336 216)(51 128 77 250 373 423 281 217)(52 129 78 251 374 424 282 218)(53 130 79 252 375 425 283 219)(54 131 80 253 376 426 284 220)(55 132 81 254 377 427 285 221)(56 133 82 255 378 428 286 222)
(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)| (1,134,83,256,379,429,287,223)(2,135,84,257,380,430,288,224)(3,136,85,258,381,431,289,169)(4,137,86,259,382,432,290,170)(5,138,87,260,383,433,291,171)(6,139,88,261,384,434,292,172)(7,140,89,262,385,435,293,173)(8,141,90,263,386,436,294,174)(9,142,91,264,387,437,295,175)(10,143,92,265,388,438,296,176)(11,144,93,266,389,439,297,177)(12,145,94,267,390,440,298,178)(13,146,95,268,391,441,299,179)(14,147,96,269,392,442,300,180)(15,148,97,270,337,443,301,181)(16,149,98,271,338,444,302,182)(17,150,99,272,339,445,303,183)(18,151,100,273,340,446,304,184)(19,152,101,274,341,447,305,185)(20,153,102,275,342,448,306,186)(21,154,103,276,343,393,307,187)(22,155,104,277,344,394,308,188)(23,156,105,278,345,395,309,189)(24,157,106,279,346,396,310,190)(25,158,107,280,347,397,311,191)(26,159,108,225,348,398,312,192)(27,160,109,226,349,399,313,193)(28,161,110,227,350,400,314,194)(29,162,111,228,351,401,315,195)(30,163,112,229,352,402,316,196)(31,164,57,230,353,403,317,197)(32,165,58,231,354,404,318,198)(33,166,59,232,355,405,319,199)(34,167,60,233,356,406,320,200)(35,168,61,234,357,407,321,201)(36,113,62,235,358,408,322,202)(37,114,63,236,359,409,323,203)(38,115,64,237,360,410,324,204)(39,116,65,238,361,411,325,205)(40,117,66,239,362,412,326,206)(41,118,67,240,363,413,327,207)(42,119,68,241,364,414,328,208)(43,120,69,242,365,415,329,209)(44,121,70,243,366,416,330,210)(45,122,71,244,367,417,331,211)(46,123,72,245,368,418,332,212)(47,124,73,246,369,419,333,213)(48,125,74,247,370,420,334,214)(49,126,75,248,371,421,335,215)(50,127,76,249,372,422,336,216)(51,128,77,250,373,423,281,217)(52,129,78,251,374,424,282,218)(53,130,79,252,375,425,283,219)(54,131,80,253,376,426,284,220)(55,132,81,254,377,427,285,221)(56,133,82,255,378,428,286,222), (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( (1,134,83,256,379,429,287,223)(2,135,84,257,380,430,288,224)(3,136,85,258,381,431,289,169)(4,137,86,259,382,432,290,170)(5,138,87,260,383,433,291,171)(6,139,88,261,384,434,292,172)(7,140,89,262,385,435,293,173)(8,141,90,263,386,436,294,174)(9,142,91,264,387,437,295,175)(10,143,92,265,388,438,296,176)(11,144,93,266,389,439,297,177)(12,145,94,267,390,440,298,178)(13,146,95,268,391,441,299,179)(14,147,96,269,392,442,300,180)(15,148,97,270,337,443,301,181)(16,149,98,271,338,444,302,182)(17,150,99,272,339,445,303,183)(18,151,100,273,340,446,304,184)(19,152,101,274,341,447,305,185)(20,153,102,275,342,448,306,186)(21,154,103,276,343,393,307,187)(22,155,104,277,344,394,308,188)(23,156,105,278,345,395,309,189)(24,157,106,279,346,396,310,190)(25,158,107,280,347,397,311,191)(26,159,108,225,348,398,312,192)(27,160,109,226,349,399,313,193)(28,161,110,227,350,400,314,194)(29,162,111,228,351,401,315,195)(30,163,112,229,352,402,316,196)(31,164,57,230,353,403,317,197)(32,165,58,231,354,404,318,198)(33,166,59,232,355,405,319,199)(34,167,60,233,356,406,320,200)(35,168,61,234,357,407,321,201)(36,113,62,235,358,408,322,202)(37,114,63,236,359,409,323,203)(38,115,64,237,360,410,324,204)(39,116,65,238,361,411,325,205)(40,117,66,239,362,412,326,206)(41,118,67,240,363,413,327,207)(42,119,68,241,364,414,328,208)(43,120,69,242,365,415,329,209)(44,121,70,243,366,416,330,210)(45,122,71,244,367,417,331,211)(46,123,72,245,368,418,332,212)(47,124,73,246,369,419,333,213)(48,125,74,247,370,420,334,214)(49,126,75,248,371,421,335,215)(50,127,76,249,372,422,336,216)(51,128,77,250,373,423,281,217)(52,129,78,251,374,424,282,218)(53,130,79,252,375,425,283,219)(54,131,80,253,376,426,284,220)(55,132,81,254,377,427,285,221)(56,133,82,255,378,428,286,222), (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([(1,134,83,256,379,429,287,223),(2,135,84,257,380,430,288,224),(3,136,85,258,381,431,289,169),(4,137,86,259,382,432,290,170),(5,138,87,260,383,433,291,171),(6,139,88,261,384,434,292,172),(7,140,89,262,385,435,293,173),(8,141,90,263,386,436,294,174),(9,142,91,264,387,437,295,175),(10,143,92,265,388,438,296,176),(11,144,93,266,389,439,297,177),(12,145,94,267,390,440,298,178),(13,146,95,268,391,441,299,179),(14,147,96,269,392,442,300,180),(15,148,97,270,337,443,301,181),(16,149,98,271,338,444,302,182),(17,150,99,272,339,445,303,183),(18,151,100,273,340,446,304,184),(19,152,101,274,341,447,305,185),(20,153,102,275,342,448,306,186),(21,154,103,276,343,393,307,187),(22,155,104,277,344,394,308,188),(23,156,105,278,345,395,309,189),(24,157,106,279,346,396,310,190),(25,158,107,280,347,397,311,191),(26,159,108,225,348,398,312,192),(27,160,109,226,349,399,313,193),(28,161,110,227,350,400,314,194),(29,162,111,228,351,401,315,195),(30,163,112,229,352,402,316,196),(31,164,57,230,353,403,317,197),(32,165,58,231,354,404,318,198),(33,166,59,232,355,405,319,199),(34,167,60,233,356,406,320,200),(35,168,61,234,357,407,321,201),(36,113,62,235,358,408,322,202),(37,114,63,236,359,409,323,203),(38,115,64,237,360,410,324,204),(39,116,65,238,361,411,325,205),(40,117,66,239,362,412,326,206),(41,118,67,240,363,413,327,207),(42,119,68,241,364,414,328,208),(43,120,69,242,365,415,329,209),(44,121,70,243,366,416,330,210),(45,122,71,244,367,417,331,211),(46,123,72,245,368,418,332,212),(47,124,73,246,369,419,333,213),(48,125,74,247,370,420,334,214),(49,126,75,248,371,421,335,215),(50,127,76,249,372,422,336,216),(51,128,77,250,373,423,281,217),(52,129,78,251,374,424,282,218),(53,130,79,252,375,425,283,219),(54,131,80,253,376,426,284,220),(55,132,81,254,377,427,285,221),(56,133,82,255,378,428,286,222)], [(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 2A2B2C4A···4L7A···7F8A···8AV14A···14R28A···28BT56A···56KB
order12224···47···78···814···1428···2856···56
size11111···11···11···11···11···11···1

448 irreducible representations

dim11111111
type++
imageC1C2C4C7C8C14C28C56
kernelC8×C56C4×C56C2×C56C82C56C4×C8C2×C8C8
# reps13126481872288

Matrix representation of C8×C56 in GL2(𝔽113) generated by

950
01
,
720
069
G:=sub<GL(2,GF(113))| [95,0,0,1],[72,0,0,69] >;

C8×C56 in GAP, Magma, Sage, TeX

C_8\times C_{56}
% in TeX

G:=Group("C8xC56");
// GroupNames label

G:=SmallGroup(448,125);
// by ID

G=gap.SmallGroup(448,125);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,196,400,136,172]);
// Polycyclic

G:=Group<a,b|a^8=b^56=1,a*b=b*a>;
// generators/relations

׿
×
𝔽