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G = C7×C63order 441 = 32·72

Abelian group of type [7,63]

direct product, abelian, monomial, 7-elementary

Aliases: C7×C63, SmallGroup(441,8)

Series: Derived Chief Lower central Upper central

C1 — C7×C63
C1C3C21C7×C21 — C7×C63
C1 — C7×C63
C1 — C7×C63

Generators and relations for C7×C63
 G = < a,b | a7=b63=1, ab=ba >


Smallest permutation representation of C7×C63
Regular action on 441 points
Generators in S441
(1 424 262 333 167 250 106)(2 425 263 334 168 251 107)(3 426 264 335 169 252 108)(4 427 265 336 170 190 109)(5 428 266 337 171 191 110)(6 429 267 338 172 192 111)(7 430 268 339 173 193 112)(8 431 269 340 174 194 113)(9 432 270 341 175 195 114)(10 433 271 342 176 196 115)(11 434 272 343 177 197 116)(12 435 273 344 178 198 117)(13 436 274 345 179 199 118)(14 437 275 346 180 200 119)(15 438 276 347 181 201 120)(16 439 277 348 182 202 121)(17 440 278 349 183 203 122)(18 441 279 350 184 204 123)(19 379 280 351 185 205 124)(20 380 281 352 186 206 125)(21 381 282 353 187 207 126)(22 382 283 354 188 208 64)(23 383 284 355 189 209 65)(24 384 285 356 127 210 66)(25 385 286 357 128 211 67)(26 386 287 358 129 212 68)(27 387 288 359 130 213 69)(28 388 289 360 131 214 70)(29 389 290 361 132 215 71)(30 390 291 362 133 216 72)(31 391 292 363 134 217 73)(32 392 293 364 135 218 74)(33 393 294 365 136 219 75)(34 394 295 366 137 220 76)(35 395 296 367 138 221 77)(36 396 297 368 139 222 78)(37 397 298 369 140 223 79)(38 398 299 370 141 224 80)(39 399 300 371 142 225 81)(40 400 301 372 143 226 82)(41 401 302 373 144 227 83)(42 402 303 374 145 228 84)(43 403 304 375 146 229 85)(44 404 305 376 147 230 86)(45 405 306 377 148 231 87)(46 406 307 378 149 232 88)(47 407 308 316 150 233 89)(48 408 309 317 151 234 90)(49 409 310 318 152 235 91)(50 410 311 319 153 236 92)(51 411 312 320 154 237 93)(52 412 313 321 155 238 94)(53 413 314 322 156 239 95)(54 414 315 323 157 240 96)(55 415 253 324 158 241 97)(56 416 254 325 159 242 98)(57 417 255 326 160 243 99)(58 418 256 327 161 244 100)(59 419 257 328 162 245 101)(60 420 258 329 163 246 102)(61 421 259 330 164 247 103)(62 422 260 331 165 248 104)(63 423 261 332 166 249 105)
(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)

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

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

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

441 conjugacy classes

class 1 3A3B7A···7AV9A···9F21A···21CR63A···63KB
order1337···79···921···2163···63
size1111···11···11···11···1

441 irreducible representations

dim111111
type+
imageC1C3C7C9C21C63
kernelC7×C63C7×C21C63C72C21C7
# reps1248696288

Matrix representation of C7×C63 in GL2(𝔽127) generated by

640
08
,
740
088
G:=sub<GL(2,GF(127))| [64,0,0,8],[74,0,0,88] >;

C7×C63 in GAP, Magma, Sage, TeX

C_7\times C_{63}
% in TeX

G:=Group("C7xC63");
// GroupNames label

G:=SmallGroup(441,8);
// by ID

G=gap.SmallGroup(441,8);
# by ID

G:=PCGroup([4,-3,-7,-7,-3,588]);
// Polycyclic

G:=Group<a,b|a^7=b^63=1,a*b=b*a>;
// generators/relations

Export

Subgroup lattice of C7×C63 in TeX

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