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G = C553C8order 440 = 23·5·11

1st semidirect product of C55 and C8 acting via C8/C4=C2

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C553C8, C44.2D5, C4.2D55, C2.Dic55, C22.Dic5, C110.3C4, C220.2C2, C20.2D11, C10.2Dic11, C52(C11⋊C8), C11⋊(C52C8), SmallGroup(440,5)

Series: Derived Chief Lower central Upper central

C1C55 — C553C8
C1C11C55C110C220 — C553C8
C55 — C553C8
C1C4

Generators and relations for C553C8
 G = < a,b | a55=b8=1, bab-1=a-1 >

55C8
11C52C8
5C11⋊C8

Smallest permutation representation of C553C8
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)
(1 401 212 307 87 331 124 264)(2 400 213 306 88 385 125 263)(3 399 214 305 89 384 126 262)(4 398 215 304 90 383 127 261)(5 397 216 303 91 382 128 260)(6 396 217 302 92 381 129 259)(7 395 218 301 93 380 130 258)(8 394 219 300 94 379 131 257)(9 393 220 299 95 378 132 256)(10 392 166 298 96 377 133 255)(11 391 167 297 97 376 134 254)(12 390 168 296 98 375 135 253)(13 389 169 295 99 374 136 252)(14 388 170 294 100 373 137 251)(15 387 171 293 101 372 138 250)(16 386 172 292 102 371 139 249)(17 440 173 291 103 370 140 248)(18 439 174 290 104 369 141 247)(19 438 175 289 105 368 142 246)(20 437 176 288 106 367 143 245)(21 436 177 287 107 366 144 244)(22 435 178 286 108 365 145 243)(23 434 179 285 109 364 146 242)(24 433 180 284 110 363 147 241)(25 432 181 283 56 362 148 240)(26 431 182 282 57 361 149 239)(27 430 183 281 58 360 150 238)(28 429 184 280 59 359 151 237)(29 428 185 279 60 358 152 236)(30 427 186 278 61 357 153 235)(31 426 187 277 62 356 154 234)(32 425 188 276 63 355 155 233)(33 424 189 330 64 354 156 232)(34 423 190 329 65 353 157 231)(35 422 191 328 66 352 158 230)(36 421 192 327 67 351 159 229)(37 420 193 326 68 350 160 228)(38 419 194 325 69 349 161 227)(39 418 195 324 70 348 162 226)(40 417 196 323 71 347 163 225)(41 416 197 322 72 346 164 224)(42 415 198 321 73 345 165 223)(43 414 199 320 74 344 111 222)(44 413 200 319 75 343 112 221)(45 412 201 318 76 342 113 275)(46 411 202 317 77 341 114 274)(47 410 203 316 78 340 115 273)(48 409 204 315 79 339 116 272)(49 408 205 314 80 338 117 271)(50 407 206 313 81 337 118 270)(51 406 207 312 82 336 119 269)(52 405 208 311 83 335 120 268)(53 404 209 310 84 334 121 267)(54 403 210 309 85 333 122 266)(55 402 211 308 86 332 123 265)

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

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

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

116 conjugacy classes

class 1  2 4A4B5A5B8A8B8C8D10A10B11A···11E20A20B20C20D22A···22E44A···44J55A···55T110A···110T220A···220AN
order1244558888101011···112020202022···2244···4455···55110···110220···220
size11112255555555222···222222···22···22···22···22···2

116 irreducible representations

dim1111222222222
type+++-+-+-
imageC1C2C4C8D5Dic5D11C52C8Dic11C11⋊C8D55Dic55C553C8
kernelC553C8C220C110C55C44C22C20C11C10C5C4C2C1
# reps11242254510202040

Matrix representation of C553C8 in GL3(𝔽881) generated by

100
0155116
0765510
,
21900
0515428
0356366
G:=sub<GL(3,GF(881))| [1,0,0,0,155,765,0,116,510],[219,0,0,0,515,356,0,428,366] >;

C553C8 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

Subgroup lattice of C553C8 in TeX

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