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G = C7⋊C64order 448 = 26·7

The semidirect product of C7 and C64 acting via C64/C32=C2

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

Aliases: C7⋊C64, C14.C32, C56.2C8, C32.2D7, C112.2C4, C224.3C2, C28.2C16, C16.3Dic7, C2.(C7⋊C32), C8.4(C7⋊C8), C4.2(C7⋊C16), SmallGroup(448,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C64
C1C7C14C28C56C112C224 — C7⋊C64
C7 — C7⋊C64
C1C32

Generators and relations for C7⋊C64
 G = < a,b | a7=b64=1, bab-1=a-1 >

7C64

Smallest permutation representation of C7⋊C64
Regular action on 448 points
Generators in S448
(1 260 343 187 87 248 415)(2 416 249 88 188 344 261)(3 262 345 189 89 250 417)(4 418 251 90 190 346 263)(5 264 347 191 91 252 419)(6 420 253 92 192 348 265)(7 266 349 129 93 254 421)(8 422 255 94 130 350 267)(9 268 351 131 95 256 423)(10 424 193 96 132 352 269)(11 270 353 133 97 194 425)(12 426 195 98 134 354 271)(13 272 355 135 99 196 427)(14 428 197 100 136 356 273)(15 274 357 137 101 198 429)(16 430 199 102 138 358 275)(17 276 359 139 103 200 431)(18 432 201 104 140 360 277)(19 278 361 141 105 202 433)(20 434 203 106 142 362 279)(21 280 363 143 107 204 435)(22 436 205 108 144 364 281)(23 282 365 145 109 206 437)(24 438 207 110 146 366 283)(25 284 367 147 111 208 439)(26 440 209 112 148 368 285)(27 286 369 149 113 210 441)(28 442 211 114 150 370 287)(29 288 371 151 115 212 443)(30 444 213 116 152 372 289)(31 290 373 153 117 214 445)(32 446 215 118 154 374 291)(33 292 375 155 119 216 447)(34 448 217 120 156 376 293)(35 294 377 157 121 218 385)(36 386 219 122 158 378 295)(37 296 379 159 123 220 387)(38 388 221 124 160 380 297)(39 298 381 161 125 222 389)(40 390 223 126 162 382 299)(41 300 383 163 127 224 391)(42 392 225 128 164 384 301)(43 302 321 165 65 226 393)(44 394 227 66 166 322 303)(45 304 323 167 67 228 395)(46 396 229 68 168 324 305)(47 306 325 169 69 230 397)(48 398 231 70 170 326 307)(49 308 327 171 71 232 399)(50 400 233 72 172 328 309)(51 310 329 173 73 234 401)(52 402 235 74 174 330 311)(53 312 331 175 75 236 403)(54 404 237 76 176 332 313)(55 314 333 177 77 238 405)(56 406 239 78 178 334 315)(57 316 335 179 79 240 407)(58 408 241 80 180 336 317)(59 318 337 181 81 242 409)(60 410 243 82 182 338 319)(61 320 339 183 83 244 411)(62 412 245 84 184 340 257)(63 258 341 185 85 246 413)(64 414 247 86 186 342 259)
(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)(385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448)

G:=sub<Sym(448)| (1,260,343,187,87,248,415)(2,416,249,88,188,344,261)(3,262,345,189,89,250,417)(4,418,251,90,190,346,263)(5,264,347,191,91,252,419)(6,420,253,92,192,348,265)(7,266,349,129,93,254,421)(8,422,255,94,130,350,267)(9,268,351,131,95,256,423)(10,424,193,96,132,352,269)(11,270,353,133,97,194,425)(12,426,195,98,134,354,271)(13,272,355,135,99,196,427)(14,428,197,100,136,356,273)(15,274,357,137,101,198,429)(16,430,199,102,138,358,275)(17,276,359,139,103,200,431)(18,432,201,104,140,360,277)(19,278,361,141,105,202,433)(20,434,203,106,142,362,279)(21,280,363,143,107,204,435)(22,436,205,108,144,364,281)(23,282,365,145,109,206,437)(24,438,207,110,146,366,283)(25,284,367,147,111,208,439)(26,440,209,112,148,368,285)(27,286,369,149,113,210,441)(28,442,211,114,150,370,287)(29,288,371,151,115,212,443)(30,444,213,116,152,372,289)(31,290,373,153,117,214,445)(32,446,215,118,154,374,291)(33,292,375,155,119,216,447)(34,448,217,120,156,376,293)(35,294,377,157,121,218,385)(36,386,219,122,158,378,295)(37,296,379,159,123,220,387)(38,388,221,124,160,380,297)(39,298,381,161,125,222,389)(40,390,223,126,162,382,299)(41,300,383,163,127,224,391)(42,392,225,128,164,384,301)(43,302,321,165,65,226,393)(44,394,227,66,166,322,303)(45,304,323,167,67,228,395)(46,396,229,68,168,324,305)(47,306,325,169,69,230,397)(48,398,231,70,170,326,307)(49,308,327,171,71,232,399)(50,400,233,72,172,328,309)(51,310,329,173,73,234,401)(52,402,235,74,174,330,311)(53,312,331,175,75,236,403)(54,404,237,76,176,332,313)(55,314,333,177,77,238,405)(56,406,239,78,178,334,315)(57,316,335,179,79,240,407)(58,408,241,80,180,336,317)(59,318,337,181,81,242,409)(60,410,243,82,182,338,319)(61,320,339,183,83,244,411)(62,412,245,84,184,340,257)(63,258,341,185,85,246,413)(64,414,247,86,186,342,259), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)>;

G:=Group( (1,260,343,187,87,248,415)(2,416,249,88,188,344,261)(3,262,345,189,89,250,417)(4,418,251,90,190,346,263)(5,264,347,191,91,252,419)(6,420,253,92,192,348,265)(7,266,349,129,93,254,421)(8,422,255,94,130,350,267)(9,268,351,131,95,256,423)(10,424,193,96,132,352,269)(11,270,353,133,97,194,425)(12,426,195,98,134,354,271)(13,272,355,135,99,196,427)(14,428,197,100,136,356,273)(15,274,357,137,101,198,429)(16,430,199,102,138,358,275)(17,276,359,139,103,200,431)(18,432,201,104,140,360,277)(19,278,361,141,105,202,433)(20,434,203,106,142,362,279)(21,280,363,143,107,204,435)(22,436,205,108,144,364,281)(23,282,365,145,109,206,437)(24,438,207,110,146,366,283)(25,284,367,147,111,208,439)(26,440,209,112,148,368,285)(27,286,369,149,113,210,441)(28,442,211,114,150,370,287)(29,288,371,151,115,212,443)(30,444,213,116,152,372,289)(31,290,373,153,117,214,445)(32,446,215,118,154,374,291)(33,292,375,155,119,216,447)(34,448,217,120,156,376,293)(35,294,377,157,121,218,385)(36,386,219,122,158,378,295)(37,296,379,159,123,220,387)(38,388,221,124,160,380,297)(39,298,381,161,125,222,389)(40,390,223,126,162,382,299)(41,300,383,163,127,224,391)(42,392,225,128,164,384,301)(43,302,321,165,65,226,393)(44,394,227,66,166,322,303)(45,304,323,167,67,228,395)(46,396,229,68,168,324,305)(47,306,325,169,69,230,397)(48,398,231,70,170,326,307)(49,308,327,171,71,232,399)(50,400,233,72,172,328,309)(51,310,329,173,73,234,401)(52,402,235,74,174,330,311)(53,312,331,175,75,236,403)(54,404,237,76,176,332,313)(55,314,333,177,77,238,405)(56,406,239,78,178,334,315)(57,316,335,179,79,240,407)(58,408,241,80,180,336,317)(59,318,337,181,81,242,409)(60,410,243,82,182,338,319)(61,320,339,183,83,244,411)(62,412,245,84,184,340,257)(63,258,341,185,85,246,413)(64,414,247,86,186,342,259), (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)(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448) );

G=PermutationGroup([(1,260,343,187,87,248,415),(2,416,249,88,188,344,261),(3,262,345,189,89,250,417),(4,418,251,90,190,346,263),(5,264,347,191,91,252,419),(6,420,253,92,192,348,265),(7,266,349,129,93,254,421),(8,422,255,94,130,350,267),(9,268,351,131,95,256,423),(10,424,193,96,132,352,269),(11,270,353,133,97,194,425),(12,426,195,98,134,354,271),(13,272,355,135,99,196,427),(14,428,197,100,136,356,273),(15,274,357,137,101,198,429),(16,430,199,102,138,358,275),(17,276,359,139,103,200,431),(18,432,201,104,140,360,277),(19,278,361,141,105,202,433),(20,434,203,106,142,362,279),(21,280,363,143,107,204,435),(22,436,205,108,144,364,281),(23,282,365,145,109,206,437),(24,438,207,110,146,366,283),(25,284,367,147,111,208,439),(26,440,209,112,148,368,285),(27,286,369,149,113,210,441),(28,442,211,114,150,370,287),(29,288,371,151,115,212,443),(30,444,213,116,152,372,289),(31,290,373,153,117,214,445),(32,446,215,118,154,374,291),(33,292,375,155,119,216,447),(34,448,217,120,156,376,293),(35,294,377,157,121,218,385),(36,386,219,122,158,378,295),(37,296,379,159,123,220,387),(38,388,221,124,160,380,297),(39,298,381,161,125,222,389),(40,390,223,126,162,382,299),(41,300,383,163,127,224,391),(42,392,225,128,164,384,301),(43,302,321,165,65,226,393),(44,394,227,66,166,322,303),(45,304,323,167,67,228,395),(46,396,229,68,168,324,305),(47,306,325,169,69,230,397),(48,398,231,70,170,326,307),(49,308,327,171,71,232,399),(50,400,233,72,172,328,309),(51,310,329,173,73,234,401),(52,402,235,74,174,330,311),(53,312,331,175,75,236,403),(54,404,237,76,176,332,313),(55,314,333,177,77,238,405),(56,406,239,78,178,334,315),(57,316,335,179,79,240,407),(58,408,241,80,180,336,317),(59,318,337,181,81,242,409),(60,410,243,82,182,338,319),(61,320,339,183,83,244,411),(62,412,245,84,184,340,257),(63,258,341,185,85,246,413),(64,414,247,86,186,342,259)], [(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),(385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448)])

160 conjugacy classes

class 1  2 4A4B7A7B7C8A8B8C8D14A14B14C16A···16H28A···28F32A···32P56A···56L64A···64AF112A···112X224A···224AV
order1244777888814141416···1628···2832···3256···5664···64112···112224···224
size111122211112221···12···21···12···27···72···22···2

160 irreducible representations

dim1111111222222
type+++-
imageC1C2C4C8C16C32C64D7Dic7C7⋊C8C7⋊C16C7⋊C32C7⋊C64
kernelC7⋊C64C224C112C56C28C14C7C32C16C8C4C2C1
# reps112481632336122448

Matrix representation of C7⋊C64 in GL3(𝔽449) generated by

100
001
0448354
,
18800
0381135
030968
G:=sub<GL(3,GF(449))| [1,0,0,0,0,448,0,1,354],[188,0,0,0,381,309,0,135,68] >;

C7⋊C64 in GAP, Magma, Sage, TeX

C_7\rtimes C_{64}
% in TeX

G:=Group("C7:C64");
// GroupNames label

G:=SmallGroup(448,1);
// by ID

G=gap.SmallGroup(448,1);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,14,36,58,80,102,18822]);
// Polycyclic

G:=Group<a,b|a^7=b^64=1,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of C7⋊C64 in TeX

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