metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C55⋊3C8, C44.2D5, C4.2D55, C2.Dic55, C22.Dic5, C110.3C4, C220.2C2, C20.2D11, C10.2Dic11, C5⋊2(C11⋊C8), C11⋊(C5⋊2C8), SmallGroup(440,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C55 — C55⋊3C8 |
Generators and relations for C55⋊3C8
G = < a,b | a55=b8=1, bab-1=a-1 >
Smallest permutation representation of C55⋊3C8
►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)
(1 437 196 295 105 362 138 256)(2 436 197 294 106 361 139 255)(3 435 198 293 107 360 140 254)(4 434 199 292 108 359 141 253)(5 433 200 291 109 358 142 252)(6 432 201 290 110 357 143 251)(7 431 202 289 56 356 144 250)(8 430 203 288 57 355 145 249)(9 429 204 287 58 354 146 248)(10 428 205 286 59 353 147 247)(11 427 206 285 60 352 148 246)(12 426 207 284 61 351 149 245)(13 425 208 283 62 350 150 244)(14 424 209 282 63 349 151 243)(15 423 210 281 64 348 152 242)(16 422 211 280 65 347 153 241)(17 421 212 279 66 346 154 240)(18 420 213 278 67 345 155 239)(19 419 214 277 68 344 156 238)(20 418 215 276 69 343 157 237)(21 417 216 330 70 342 158 236)(22 416 217 329 71 341 159 235)(23 415 218 328 72 340 160 234)(24 414 219 327 73 339 161 233)(25 413 220 326 74 338 162 232)(26 412 166 325 75 337 163 231)(27 411 167 324 76 336 164 230)(28 410 168 323 77 335 165 229)(29 409 169 322 78 334 111 228)(30 408 170 321 79 333 112 227)(31 407 171 320 80 332 113 226)(32 406 172 319 81 331 114 225)(33 405 173 318 82 385 115 224)(34 404 174 317 83 384 116 223)(35 403 175 316 84 383 117 222)(36 402 176 315 85 382 118 221)(37 401 177 314 86 381 119 275)(38 400 178 313 87 380 120 274)(39 399 179 312 88 379 121 273)(40 398 180 311 89 378 122 272)(41 397 181 310 90 377 123 271)(42 396 182 309 91 376 124 270)(43 395 183 308 92 375 125 269)(44 394 184 307 93 374 126 268)(45 393 185 306 94 373 127 267)(46 392 186 305 95 372 128 266)(47 391 187 304 96 371 129 265)(48 390 188 303 97 370 130 264)(49 389 189 302 98 369 131 263)(50 388 190 301 99 368 132 262)(51 387 191 300 100 367 133 261)(52 386 192 299 101 366 134 260)(53 440 193 298 102 365 135 259)(54 439 194 297 103 364 136 258)(55 438 195 296 104 363 137 257)
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), (1,437,196,295,105,362,138,256)(2,436,197,294,106,361,139,255)(3,435,198,293,107,360,140,254)(4,434,199,292,108,359,141,253)(5,433,200,291,109,358,142,252)(6,432,201,290,110,357,143,251)(7,431,202,289,56,356,144,250)(8,430,203,288,57,355,145,249)(9,429,204,287,58,354,146,248)(10,428,205,286,59,353,147,247)(11,427,206,285,60,352,148,246)(12,426,207,284,61,351,149,245)(13,425,208,283,62,350,150,244)(14,424,209,282,63,349,151,243)(15,423,210,281,64,348,152,242)(16,422,211,280,65,347,153,241)(17,421,212,279,66,346,154,240)(18,420,213,278,67,345,155,239)(19,419,214,277,68,344,156,238)(20,418,215,276,69,343,157,237)(21,417,216,330,70,342,158,236)(22,416,217,329,71,341,159,235)(23,415,218,328,72,340,160,234)(24,414,219,327,73,339,161,233)(25,413,220,326,74,338,162,232)(26,412,166,325,75,337,163,231)(27,411,167,324,76,336,164,230)(28,410,168,323,77,335,165,229)(29,409,169,322,78,334,111,228)(30,408,170,321,79,333,112,227)(31,407,171,320,80,332,113,226)(32,406,172,319,81,331,114,225)(33,405,173,318,82,385,115,224)(34,404,174,317,83,384,116,223)(35,403,175,316,84,383,117,222)(36,402,176,315,85,382,118,221)(37,401,177,314,86,381,119,275)(38,400,178,313,87,380,120,274)(39,399,179,312,88,379,121,273)(40,398,180,311,89,378,122,272)(41,397,181,310,90,377,123,271)(42,396,182,309,91,376,124,270)(43,395,183,308,92,375,125,269)(44,394,184,307,93,374,126,268)(45,393,185,306,94,373,127,267)(46,392,186,305,95,372,128,266)(47,391,187,304,96,371,129,265)(48,390,188,303,97,370,130,264)(49,389,189,302,98,369,131,263)(50,388,190,301,99,368,132,262)(51,387,191,300,100,367,133,261)(52,386,192,299,101,366,134,260)(53,440,193,298,102,365,135,259)(54,439,194,297,103,364,136,258)(55,438,195,296,104,363,137,257)>;
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), (1,437,196,295,105,362,138,256)(2,436,197,294,106,361,139,255)(3,435,198,293,107,360,140,254)(4,434,199,292,108,359,141,253)(5,433,200,291,109,358,142,252)(6,432,201,290,110,357,143,251)(7,431,202,289,56,356,144,250)(8,430,203,288,57,355,145,249)(9,429,204,287,58,354,146,248)(10,428,205,286,59,353,147,247)(11,427,206,285,60,352,148,246)(12,426,207,284,61,351,149,245)(13,425,208,283,62,350,150,244)(14,424,209,282,63,349,151,243)(15,423,210,281,64,348,152,242)(16,422,211,280,65,347,153,241)(17,421,212,279,66,346,154,240)(18,420,213,278,67,345,155,239)(19,419,214,277,68,344,156,238)(20,418,215,276,69,343,157,237)(21,417,216,330,70,342,158,236)(22,416,217,329,71,341,159,235)(23,415,218,328,72,340,160,234)(24,414,219,327,73,339,161,233)(25,413,220,326,74,338,162,232)(26,412,166,325,75,337,163,231)(27,411,167,324,76,336,164,230)(28,410,168,323,77,335,165,229)(29,409,169,322,78,334,111,228)(30,408,170,321,79,333,112,227)(31,407,171,320,80,332,113,226)(32,406,172,319,81,331,114,225)(33,405,173,318,82,385,115,224)(34,404,174,317,83,384,116,223)(35,403,175,316,84,383,117,222)(36,402,176,315,85,382,118,221)(37,401,177,314,86,381,119,275)(38,400,178,313,87,380,120,274)(39,399,179,312,88,379,121,273)(40,398,180,311,89,378,122,272)(41,397,181,310,90,377,123,271)(42,396,182,309,91,376,124,270)(43,395,183,308,92,375,125,269)(44,394,184,307,93,374,126,268)(45,393,185,306,94,373,127,267)(46,392,186,305,95,372,128,266)(47,391,187,304,96,371,129,265)(48,390,188,303,97,370,130,264)(49,389,189,302,98,369,131,263)(50,388,190,301,99,368,132,262)(51,387,191,300,100,367,133,261)(52,386,192,299,101,366,134,260)(53,440,193,298,102,365,135,259)(54,439,194,297,103,364,136,258)(55,438,195,296,104,363,137,257) );
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)], [(1,437,196,295,105,362,138,256),(2,436,197,294,106,361,139,255),(3,435,198,293,107,360,140,254),(4,434,199,292,108,359,141,253),(5,433,200,291,109,358,142,252),(6,432,201,290,110,357,143,251),(7,431,202,289,56,356,144,250),(8,430,203,288,57,355,145,249),(9,429,204,287,58,354,146,248),(10,428,205,286,59,353,147,247),(11,427,206,285,60,352,148,246),(12,426,207,284,61,351,149,245),(13,425,208,283,62,350,150,244),(14,424,209,282,63,349,151,243),(15,423,210,281,64,348,152,242),(16,422,211,280,65,347,153,241),(17,421,212,279,66,346,154,240),(18,420,213,278,67,345,155,239),(19,419,214,277,68,344,156,238),(20,418,215,276,69,343,157,237),(21,417,216,330,70,342,158,236),(22,416,217,329,71,341,159,235),(23,415,218,328,72,340,160,234),(24,414,219,327,73,339,161,233),(25,413,220,326,74,338,162,232),(26,412,166,325,75,337,163,231),(27,411,167,324,76,336,164,230),(28,410,168,323,77,335,165,229),(29,409,169,322,78,334,111,228),(30,408,170,321,79,333,112,227),(31,407,171,320,80,332,113,226),(32,406,172,319,81,331,114,225),(33,405,173,318,82,385,115,224),(34,404,174,317,83,384,116,223),(35,403,175,316,84,383,117,222),(36,402,176,315,85,382,118,221),(37,401,177,314,86,381,119,275),(38,400,178,313,87,380,120,274),(39,399,179,312,88,379,121,273),(40,398,180,311,89,378,122,272),(41,397,181,310,90,377,123,271),(42,396,182,309,91,376,124,270),(43,395,183,308,92,375,125,269),(44,394,184,307,93,374,126,268),(45,393,185,306,94,373,127,267),(46,392,186,305,95,372,128,266),(47,391,187,304,96,371,129,265),(48,390,188,303,97,370,130,264),(49,389,189,302,98,369,131,263),(50,388,190,301,99,368,132,262),(51,387,191,300,100,367,133,261),(52,386,192,299,101,366,134,260),(53,440,193,298,102,365,135,259),(54,439,194,297,103,364,136,258),(55,438,195,296,104,363,137,257)]])
116 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 8A | 8B | 8C | 8D | 10A | 10B | 11A | ··· | 11E | 20A | 20B | 20C | 20D | 22A | ··· | 22E | 44A | ··· | 44J | 55A | ··· | 55T | 110A | ··· | 110T | 220A | ··· | 220AN |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | 10 | 11 | ··· | 11 | 20 | 20 | 20 | 20 | 22 | ··· | 22 | 44 | ··· | 44 | 55 | ··· | 55 | 110 | ··· | 110 | 220 | ··· | 220 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 55 | 55 | 55 | 55 | 2 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
116 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||
| image | C1 | C2 | C4 | C8 | D5 | Dic5 | D11 | C5⋊2C8 | Dic11 | C11⋊C8 | D55 | Dic55 | C55⋊3C8 |
| kernel | C55⋊3C8 | C220 | C110 | C55 | C44 | C22 | C20 | C11 | C10 | C5 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 2 | 2 | 5 | 4 | 5 | 10 | 20 | 20 | 40 |
Matrix representation of C55⋊3C8 ►in GL3(𝔽881) generated by
| 1 | 0 | 0 |
| 0 | 155 | 116 |
| 0 | 765 | 510 |
| 219 | 0 | 0 |
| 0 | 515 | 428 |
| 0 | 356 | 366 |
G:=sub<GL(3,GF(881))| [1,0,0,0,155,765,0,116,510],[219,0,0,0,515,356,0,428,366] >;
C55⋊3C8 in GAP, Magma, Sage, TeX
C_{55}\rtimes_3C_8 % in TeX
G:=Group("C55:3C8"); // GroupNames label
G:=SmallGroup(440,5);
// by ID
G=gap.SmallGroup(440,5);
# by ID
G:=PCGroup([5,-2,-2,-2,-5,-11,10,26,643,10004]);
// Polycyclic
G:=Group<a,b|a^55=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
Export