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G = C532C8order 424 = 23·53

The semidirect product of C53 and C8 acting via C8/C4=C2

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

Aliases: C532C8, C4.2D53, C2.Dic53, C106.2C4, C212.2C2, SmallGroup(424,1)

Series: Derived Chief Lower central Upper central

C1C53 — C532C8
C1C53C106C212 — C532C8
C53 — C532C8
C1C4

Generators and relations for C532C8
 G = < a,b | a53=b8=1, bab-1=a-1 >

53C8

Smallest permutation representation of C532C8
Regular action on 424 points
Generators in S424
(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)
(1 372 172 266 92 319 159 213)(2 424 173 318 93 371 107 265)(3 423 174 317 94 370 108 264)(4 422 175 316 95 369 109 263)(5 421 176 315 96 368 110 262)(6 420 177 314 97 367 111 261)(7 419 178 313 98 366 112 260)(8 418 179 312 99 365 113 259)(9 417 180 311 100 364 114 258)(10 416 181 310 101 363 115 257)(11 415 182 309 102 362 116 256)(12 414 183 308 103 361 117 255)(13 413 184 307 104 360 118 254)(14 412 185 306 105 359 119 253)(15 411 186 305 106 358 120 252)(16 410 187 304 54 357 121 251)(17 409 188 303 55 356 122 250)(18 408 189 302 56 355 123 249)(19 407 190 301 57 354 124 248)(20 406 191 300 58 353 125 247)(21 405 192 299 59 352 126 246)(22 404 193 298 60 351 127 245)(23 403 194 297 61 350 128 244)(24 402 195 296 62 349 129 243)(25 401 196 295 63 348 130 242)(26 400 197 294 64 347 131 241)(27 399 198 293 65 346 132 240)(28 398 199 292 66 345 133 239)(29 397 200 291 67 344 134 238)(30 396 201 290 68 343 135 237)(31 395 202 289 69 342 136 236)(32 394 203 288 70 341 137 235)(33 393 204 287 71 340 138 234)(34 392 205 286 72 339 139 233)(35 391 206 285 73 338 140 232)(36 390 207 284 74 337 141 231)(37 389 208 283 75 336 142 230)(38 388 209 282 76 335 143 229)(39 387 210 281 77 334 144 228)(40 386 211 280 78 333 145 227)(41 385 212 279 79 332 146 226)(42 384 160 278 80 331 147 225)(43 383 161 277 81 330 148 224)(44 382 162 276 82 329 149 223)(45 381 163 275 83 328 150 222)(46 380 164 274 84 327 151 221)(47 379 165 273 85 326 152 220)(48 378 166 272 86 325 153 219)(49 377 167 271 87 324 154 218)(50 376 168 270 88 323 155 217)(51 375 169 269 89 322 156 216)(52 374 170 268 90 321 157 215)(53 373 171 267 91 320 158 214)

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

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

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

112 conjugacy classes

class 1  2 4A4B8A8B8C8D53A···53Z106A···106Z212A···212AZ
order1244888853···53106···106212···212
size1111535353532···22···22···2

112 irreducible representations

dim1111222
type+++-
imageC1C2C4C8D53Dic53C532C8
kernelC532C8C212C106C53C4C2C1
# reps1124262652

Matrix representation of C532C8 in GL2(𝔽1697) generated by

01
1696104
,
4241437
14111273
G:=sub<GL(2,GF(1697))| [0,1696,1,104],[424,1411,1437,1273] >;

C532C8 in GAP, Magma, Sage, TeX

C_{53}\rtimes_2C_8
% in TeX

G:=Group("C53:2C8");
// GroupNames label

G:=SmallGroup(424,1);
// by ID

G=gap.SmallGroup(424,1);
# by ID

G:=PCGroup([4,-2,-2,-2,-53,8,21,6659]);
// Polycyclic

G:=Group<a,b|a^53=b^8=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C532C8 in TeX

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