metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C55⋊1C8, C22.F5, C110.1C4, C10.Dic11, Dic5.2D11, C11⋊(C5⋊C8), C5⋊(C11⋊C8), C2.(C11⋊F5), (C11×Dic5).2C2, SmallGroup(440,16)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — C55⋊C8 |
Generators and relations for C55⋊C8
G = < a,b | a55=b8=1, bab-1=a32 >
Smallest permutation representation of C55⋊C8
►Regular action on 440 points
Generators in S440
(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)
(1 425 171 278 86 349 123 264)(2 413 205 310 87 337 157 241)(3 401 184 287 88 380 136 273)(4 389 218 319 89 368 115 250)(5 432 197 296 90 356 149 227)(6 420 176 328 91 344 128 259)(7 408 210 305 92 332 162 236)(8 396 189 282 93 375 141 268)(9 439 168 314 94 363 120 245)(10 427 202 291 95 351 154 222)(11 415 181 323 96 339 133 254)(12 403 215 300 97 382 112 231)(13 391 194 277 98 370 146 263)(14 434 173 309 99 358 125 240)(15 422 207 286 100 346 159 272)(16 410 186 318 101 334 138 249)(17 398 220 295 102 377 117 226)(18 386 199 327 103 365 151 258)(19 429 178 304 104 353 130 235)(20 417 212 281 105 341 164 267)(21 405 191 313 106 384 143 244)(22 393 170 290 107 372 122 221)(23 436 204 322 108 360 156 253)(24 424 183 299 109 348 135 230)(25 412 217 276 110 336 114 262)(26 400 196 308 56 379 148 239)(27 388 175 285 57 367 127 271)(28 431 209 317 58 355 161 248)(29 419 188 294 59 343 140 225)(30 407 167 326 60 331 119 257)(31 395 201 303 61 374 153 234)(32 438 180 280 62 362 132 266)(33 426 214 312 63 350 111 243)(34 414 193 289 64 338 145 275)(35 402 172 321 65 381 124 252)(36 390 206 298 66 369 158 229)(37 433 185 330 67 357 137 261)(38 421 219 307 68 345 116 238)(39 409 198 284 69 333 150 270)(40 397 177 316 70 376 129 247)(41 440 211 293 71 364 163 224)(42 428 190 325 72 352 142 256)(43 416 169 302 73 340 121 233)(44 404 203 279 74 383 155 265)(45 392 182 311 75 371 134 242)(46 435 216 288 76 359 113 274)(47 423 195 320 77 347 147 251)(48 411 174 297 78 335 126 228)(49 399 208 329 79 378 160 260)(50 387 187 306 80 366 139 237)(51 430 166 283 81 354 118 269)(52 418 200 315 82 342 152 246)(53 406 179 292 83 385 131 223)(54 394 213 324 84 373 165 255)(55 437 192 301 85 361 144 232)
G:=sub<Sym(440)| (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), (1,425,171,278,86,349,123,264)(2,413,205,310,87,337,157,241)(3,401,184,287,88,380,136,273)(4,389,218,319,89,368,115,250)(5,432,197,296,90,356,149,227)(6,420,176,328,91,344,128,259)(7,408,210,305,92,332,162,236)(8,396,189,282,93,375,141,268)(9,439,168,314,94,363,120,245)(10,427,202,291,95,351,154,222)(11,415,181,323,96,339,133,254)(12,403,215,300,97,382,112,231)(13,391,194,277,98,370,146,263)(14,434,173,309,99,358,125,240)(15,422,207,286,100,346,159,272)(16,410,186,318,101,334,138,249)(17,398,220,295,102,377,117,226)(18,386,199,327,103,365,151,258)(19,429,178,304,104,353,130,235)(20,417,212,281,105,341,164,267)(21,405,191,313,106,384,143,244)(22,393,170,290,107,372,122,221)(23,436,204,322,108,360,156,253)(24,424,183,299,109,348,135,230)(25,412,217,276,110,336,114,262)(26,400,196,308,56,379,148,239)(27,388,175,285,57,367,127,271)(28,431,209,317,58,355,161,248)(29,419,188,294,59,343,140,225)(30,407,167,326,60,331,119,257)(31,395,201,303,61,374,153,234)(32,438,180,280,62,362,132,266)(33,426,214,312,63,350,111,243)(34,414,193,289,64,338,145,275)(35,402,172,321,65,381,124,252)(36,390,206,298,66,369,158,229)(37,433,185,330,67,357,137,261)(38,421,219,307,68,345,116,238)(39,409,198,284,69,333,150,270)(40,397,177,316,70,376,129,247)(41,440,211,293,71,364,163,224)(42,428,190,325,72,352,142,256)(43,416,169,302,73,340,121,233)(44,404,203,279,74,383,155,265)(45,392,182,311,75,371,134,242)(46,435,216,288,76,359,113,274)(47,423,195,320,77,347,147,251)(48,411,174,297,78,335,126,228)(49,399,208,329,79,378,160,260)(50,387,187,306,80,366,139,237)(51,430,166,283,81,354,118,269)(52,418,200,315,82,342,152,246)(53,406,179,292,83,385,131,223)(54,394,213,324,84,373,165,255)(55,437,192,301,85,361,144,232)>;
G:=Group( (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), (1,425,171,278,86,349,123,264)(2,413,205,310,87,337,157,241)(3,401,184,287,88,380,136,273)(4,389,218,319,89,368,115,250)(5,432,197,296,90,356,149,227)(6,420,176,328,91,344,128,259)(7,408,210,305,92,332,162,236)(8,396,189,282,93,375,141,268)(9,439,168,314,94,363,120,245)(10,427,202,291,95,351,154,222)(11,415,181,323,96,339,133,254)(12,403,215,300,97,382,112,231)(13,391,194,277,98,370,146,263)(14,434,173,309,99,358,125,240)(15,422,207,286,100,346,159,272)(16,410,186,318,101,334,138,249)(17,398,220,295,102,377,117,226)(18,386,199,327,103,365,151,258)(19,429,178,304,104,353,130,235)(20,417,212,281,105,341,164,267)(21,405,191,313,106,384,143,244)(22,393,170,290,107,372,122,221)(23,436,204,322,108,360,156,253)(24,424,183,299,109,348,135,230)(25,412,217,276,110,336,114,262)(26,400,196,308,56,379,148,239)(27,388,175,285,57,367,127,271)(28,431,209,317,58,355,161,248)(29,419,188,294,59,343,140,225)(30,407,167,326,60,331,119,257)(31,395,201,303,61,374,153,234)(32,438,180,280,62,362,132,266)(33,426,214,312,63,350,111,243)(34,414,193,289,64,338,145,275)(35,402,172,321,65,381,124,252)(36,390,206,298,66,369,158,229)(37,433,185,330,67,357,137,261)(38,421,219,307,68,345,116,238)(39,409,198,284,69,333,150,270)(40,397,177,316,70,376,129,247)(41,440,211,293,71,364,163,224)(42,428,190,325,72,352,142,256)(43,416,169,302,73,340,121,233)(44,404,203,279,74,383,155,265)(45,392,182,311,75,371,134,242)(46,435,216,288,76,359,113,274)(47,423,195,320,77,347,147,251)(48,411,174,297,78,335,126,228)(49,399,208,329,79,378,160,260)(50,387,187,306,80,366,139,237)(51,430,166,283,81,354,118,269)(52,418,200,315,82,342,152,246)(53,406,179,292,83,385,131,223)(54,394,213,324,84,373,165,255)(55,437,192,301,85,361,144,232) );
G=PermutationGroup([[(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)], [(1,425,171,278,86,349,123,264),(2,413,205,310,87,337,157,241),(3,401,184,287,88,380,136,273),(4,389,218,319,89,368,115,250),(5,432,197,296,90,356,149,227),(6,420,176,328,91,344,128,259),(7,408,210,305,92,332,162,236),(8,396,189,282,93,375,141,268),(9,439,168,314,94,363,120,245),(10,427,202,291,95,351,154,222),(11,415,181,323,96,339,133,254),(12,403,215,300,97,382,112,231),(13,391,194,277,98,370,146,263),(14,434,173,309,99,358,125,240),(15,422,207,286,100,346,159,272),(16,410,186,318,101,334,138,249),(17,398,220,295,102,377,117,226),(18,386,199,327,103,365,151,258),(19,429,178,304,104,353,130,235),(20,417,212,281,105,341,164,267),(21,405,191,313,106,384,143,244),(22,393,170,290,107,372,122,221),(23,436,204,322,108,360,156,253),(24,424,183,299,109,348,135,230),(25,412,217,276,110,336,114,262),(26,400,196,308,56,379,148,239),(27,388,175,285,57,367,127,271),(28,431,209,317,58,355,161,248),(29,419,188,294,59,343,140,225),(30,407,167,326,60,331,119,257),(31,395,201,303,61,374,153,234),(32,438,180,280,62,362,132,266),(33,426,214,312,63,350,111,243),(34,414,193,289,64,338,145,275),(35,402,172,321,65,381,124,252),(36,390,206,298,66,369,158,229),(37,433,185,330,67,357,137,261),(38,421,219,307,68,345,116,238),(39,409,198,284,69,333,150,270),(40,397,177,316,70,376,129,247),(41,440,211,293,71,364,163,224),(42,428,190,325,72,352,142,256),(43,416,169,302,73,340,121,233),(44,404,203,279,74,383,155,265),(45,392,182,311,75,371,134,242),(46,435,216,288,76,359,113,274),(47,423,195,320,77,347,147,251),(48,411,174,297,78,335,126,228),(49,399,208,329,79,378,160,260),(50,387,187,306,80,366,139,237),(51,430,166,283,81,354,118,269),(52,418,200,315,82,342,152,246),(53,406,179,292,83,385,131,223),(54,394,213,324,84,373,165,255),(55,437,192,301,85,361,144,232)]])
50 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5 | 8A | 8B | 8C | 8D | 10 | 11A | ··· | 11E | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55J | 110A | ··· | 110J |
| order | 1 | 2 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 10 | 11 | ··· | 11 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 |
| size | 1 | 1 | 5 | 5 | 4 | 55 | 55 | 55 | 55 | 4 | 2 | ··· | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | D11 | Dic11 | C11⋊C8 | F5 | C5⋊C8 | C11⋊F5 | C55⋊C8 |
| kernel | C55⋊C8 | C11×Dic5 | C110 | C55 | Dic5 | C10 | C5 | C22 | C11 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 5 | 5 | 10 | 1 | 1 | 10 | 10 |
Matrix representation of C55⋊C8 ►in GL6(𝔽881)
| 75 | 202 | 0 | 0 | 0 | 0 |
| 755 | 201 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 49 | 0 | 0 |
| 0 | 0 | 0 | 880 | 1 | 0 |
| 0 | 0 | 0 | 880 | 0 | 1 |
| 0 | 0 | 791 | 879 | 880 | 880 |
| 402 | 694 | 0 | 0 | 0 | 0 |
| 169 | 479 | 0 | 0 | 0 | 0 |
| 0 | 0 | 231 | 64 | 242 | 523 |
| 0 | 0 | 5 | 659 | 384 | 597 |
| 0 | 0 | 690 | 275 | 367 | 872 |
| 0 | 0 | 262 | 292 | 597 | 505 |
G:=sub<GL(6,GF(881))| [75,755,0,0,0,0,202,201,0,0,0,0,0,0,1,0,0,791,0,0,49,880,880,879,0,0,0,1,0,880,0,0,0,0,1,880],[402,169,0,0,0,0,694,479,0,0,0,0,0,0,231,5,690,262,0,0,64,659,275,292,0,0,242,384,367,597,0,0,523,597,872,505] >;
C55⋊C8 in GAP, Magma, Sage, TeX
C_{55}\rtimes C_8 % in TeX
G:=Group("C55:C8"); // GroupNames label
G:=SmallGroup(440,16);
// by ID
G=gap.SmallGroup(440,16);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,10,26,483,328,10004]);
// Polycyclic
G:=Group<a,b|a^55=b^8=1,b*a*b^-1=a^32>;
// generators/relations
Export