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G = C53⋊C8order 424 = 23·53

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

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

Aliases: C53⋊C8, C106.C4, Dic53.2C2, C2.(C53⋊C4), SmallGroup(424,3)

Series: Derived Chief Lower central Upper central

C1C53 — C53⋊C8
C1C53C106Dic53 — C53⋊C8
C53 — C53⋊C8
C1C2

Generators and relations for C53⋊C8
 G = < a,b | a53=b8=1, bab-1=a23 >

53C4
53C8

Smallest permutation representation of C53⋊C8
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)
(1 421 160 313 69 355 107 224)(2 398 212 283 70 332 159 247)(3 375 211 306 71 362 158 217)(4 405 210 276 72 339 157 240)(5 382 209 299 73 369 156 263)(6 412 208 269 74 346 155 233)(7 389 207 292 75 323 154 256)(8 419 206 315 76 353 153 226)(9 396 205 285 77 330 152 249)(10 373 204 308 78 360 151 219)(11 403 203 278 79 337 150 242)(12 380 202 301 80 367 149 265)(13 410 201 271 81 344 148 235)(14 387 200 294 82 321 147 258)(15 417 199 317 83 351 146 228)(16 394 198 287 84 328 145 251)(17 424 197 310 85 358 144 221)(18 401 196 280 86 335 143 244)(19 378 195 303 87 365 142 214)(20 408 194 273 88 342 141 237)(21 385 193 296 89 319 140 260)(22 415 192 266 90 349 139 230)(23 392 191 289 91 326 138 253)(24 422 190 312 92 356 137 223)(25 399 189 282 93 333 136 246)(26 376 188 305 94 363 135 216)(27 406 187 275 95 340 134 239)(28 383 186 298 96 370 133 262)(29 413 185 268 97 347 132 232)(30 390 184 291 98 324 131 255)(31 420 183 314 99 354 130 225)(32 397 182 284 100 331 129 248)(33 374 181 307 101 361 128 218)(34 404 180 277 102 338 127 241)(35 381 179 300 103 368 126 264)(36 411 178 270 104 345 125 234)(37 388 177 293 105 322 124 257)(38 418 176 316 106 352 123 227)(39 395 175 286 54 329 122 250)(40 372 174 309 55 359 121 220)(41 402 173 279 56 336 120 243)(42 379 172 302 57 366 119 213)(43 409 171 272 58 343 118 236)(44 386 170 295 59 320 117 259)(45 416 169 318 60 350 116 229)(46 393 168 288 61 327 115 252)(47 423 167 311 62 357 114 222)(48 400 166 281 63 334 113 245)(49 377 165 304 64 364 112 215)(50 407 164 274 65 341 111 238)(51 384 163 297 66 371 110 261)(52 414 162 267 67 348 109 231)(53 391 161 290 68 325 108 254)

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

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

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

34 conjugacy classes

class 1  2 4A4B8A8B8C8D53A···53M106A···106M
order1244888853···53106···106
size115353535353534···44···4

34 irreducible representations

dim111144
type+++-
imageC1C2C4C8C53⋊C4C53⋊C8
kernelC53⋊C8Dic53C106C53C2C1
# reps11241313

Matrix representation of C53⋊C8 in GL4(𝔽1697) generated by

0100
0010
0001
16967101460710
,
129815281461396
2771249312316
1163134316841266
1343381110860
G:=sub<GL(4,GF(1697))| [0,0,0,1696,1,0,0,710,0,1,0,1460,0,0,1,710],[1298,277,1163,1343,1528,1249,1343,381,1461,312,1684,110,396,316,1266,860] >;

C53⋊C8 in GAP, Magma, Sage, TeX

C_{53}\rtimes C_8
% in TeX

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

G:=SmallGroup(424,3);
// by ID

G=gap.SmallGroup(424,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-53,8,21,3843,3335]);
// Polycyclic

G:=Group<a,b|a^53=b^8=1,b*a*b^-1=a^23>;
// generators/relations

Export

Subgroup lattice of C53⋊C8 in TeX

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