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G = Dic118order 472 = 23·59

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic118, C59⋊Q8, C4.D59, C236.1C2, C2.3D118, Dic59.C2, C118.1C22, SmallGroup(472,3)

Series: Derived Chief Lower central Upper central

C1C118 — Dic118
C1C59C118Dic59 — Dic118
C59C118 — Dic118
C1C2C4

Generators and relations for Dic118
 G = < a,b | a236=1, b2=a118, bab-1=a-1 >

59C4
59C4
59Q8

Smallest permutation representation of Dic118
Regular action on 472 points
Generators in S472
(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)(237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472)
(1 368 119 250)(2 367 120 249)(3 366 121 248)(4 365 122 247)(5 364 123 246)(6 363 124 245)(7 362 125 244)(8 361 126 243)(9 360 127 242)(10 359 128 241)(11 358 129 240)(12 357 130 239)(13 356 131 238)(14 355 132 237)(15 354 133 472)(16 353 134 471)(17 352 135 470)(18 351 136 469)(19 350 137 468)(20 349 138 467)(21 348 139 466)(22 347 140 465)(23 346 141 464)(24 345 142 463)(25 344 143 462)(26 343 144 461)(27 342 145 460)(28 341 146 459)(29 340 147 458)(30 339 148 457)(31 338 149 456)(32 337 150 455)(33 336 151 454)(34 335 152 453)(35 334 153 452)(36 333 154 451)(37 332 155 450)(38 331 156 449)(39 330 157 448)(40 329 158 447)(41 328 159 446)(42 327 160 445)(43 326 161 444)(44 325 162 443)(45 324 163 442)(46 323 164 441)(47 322 165 440)(48 321 166 439)(49 320 167 438)(50 319 168 437)(51 318 169 436)(52 317 170 435)(53 316 171 434)(54 315 172 433)(55 314 173 432)(56 313 174 431)(57 312 175 430)(58 311 176 429)(59 310 177 428)(60 309 178 427)(61 308 179 426)(62 307 180 425)(63 306 181 424)(64 305 182 423)(65 304 183 422)(66 303 184 421)(67 302 185 420)(68 301 186 419)(69 300 187 418)(70 299 188 417)(71 298 189 416)(72 297 190 415)(73 296 191 414)(74 295 192 413)(75 294 193 412)(76 293 194 411)(77 292 195 410)(78 291 196 409)(79 290 197 408)(80 289 198 407)(81 288 199 406)(82 287 200 405)(83 286 201 404)(84 285 202 403)(85 284 203 402)(86 283 204 401)(87 282 205 400)(88 281 206 399)(89 280 207 398)(90 279 208 397)(91 278 209 396)(92 277 210 395)(93 276 211 394)(94 275 212 393)(95 274 213 392)(96 273 214 391)(97 272 215 390)(98 271 216 389)(99 270 217 388)(100 269 218 387)(101 268 219 386)(102 267 220 385)(103 266 221 384)(104 265 222 383)(105 264 223 382)(106 263 224 381)(107 262 225 380)(108 261 226 379)(109 260 227 378)(110 259 228 377)(111 258 229 376)(112 257 230 375)(113 256 231 374)(114 255 232 373)(115 254 233 372)(116 253 234 371)(117 252 235 370)(118 251 236 369)

G:=sub<Sym(472)| (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)(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472), (1,368,119,250)(2,367,120,249)(3,366,121,248)(4,365,122,247)(5,364,123,246)(6,363,124,245)(7,362,125,244)(8,361,126,243)(9,360,127,242)(10,359,128,241)(11,358,129,240)(12,357,130,239)(13,356,131,238)(14,355,132,237)(15,354,133,472)(16,353,134,471)(17,352,135,470)(18,351,136,469)(19,350,137,468)(20,349,138,467)(21,348,139,466)(22,347,140,465)(23,346,141,464)(24,345,142,463)(25,344,143,462)(26,343,144,461)(27,342,145,460)(28,341,146,459)(29,340,147,458)(30,339,148,457)(31,338,149,456)(32,337,150,455)(33,336,151,454)(34,335,152,453)(35,334,153,452)(36,333,154,451)(37,332,155,450)(38,331,156,449)(39,330,157,448)(40,329,158,447)(41,328,159,446)(42,327,160,445)(43,326,161,444)(44,325,162,443)(45,324,163,442)(46,323,164,441)(47,322,165,440)(48,321,166,439)(49,320,167,438)(50,319,168,437)(51,318,169,436)(52,317,170,435)(53,316,171,434)(54,315,172,433)(55,314,173,432)(56,313,174,431)(57,312,175,430)(58,311,176,429)(59,310,177,428)(60,309,178,427)(61,308,179,426)(62,307,180,425)(63,306,181,424)(64,305,182,423)(65,304,183,422)(66,303,184,421)(67,302,185,420)(68,301,186,419)(69,300,187,418)(70,299,188,417)(71,298,189,416)(72,297,190,415)(73,296,191,414)(74,295,192,413)(75,294,193,412)(76,293,194,411)(77,292,195,410)(78,291,196,409)(79,290,197,408)(80,289,198,407)(81,288,199,406)(82,287,200,405)(83,286,201,404)(84,285,202,403)(85,284,203,402)(86,283,204,401)(87,282,205,400)(88,281,206,399)(89,280,207,398)(90,279,208,397)(91,278,209,396)(92,277,210,395)(93,276,211,394)(94,275,212,393)(95,274,213,392)(96,273,214,391)(97,272,215,390)(98,271,216,389)(99,270,217,388)(100,269,218,387)(101,268,219,386)(102,267,220,385)(103,266,221,384)(104,265,222,383)(105,264,223,382)(106,263,224,381)(107,262,225,380)(108,261,226,379)(109,260,227,378)(110,259,228,377)(111,258,229,376)(112,257,230,375)(113,256,231,374)(114,255,232,373)(115,254,233,372)(116,253,234,371)(117,252,235,370)(118,251,236,369)>;

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)(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472), (1,368,119,250)(2,367,120,249)(3,366,121,248)(4,365,122,247)(5,364,123,246)(6,363,124,245)(7,362,125,244)(8,361,126,243)(9,360,127,242)(10,359,128,241)(11,358,129,240)(12,357,130,239)(13,356,131,238)(14,355,132,237)(15,354,133,472)(16,353,134,471)(17,352,135,470)(18,351,136,469)(19,350,137,468)(20,349,138,467)(21,348,139,466)(22,347,140,465)(23,346,141,464)(24,345,142,463)(25,344,143,462)(26,343,144,461)(27,342,145,460)(28,341,146,459)(29,340,147,458)(30,339,148,457)(31,338,149,456)(32,337,150,455)(33,336,151,454)(34,335,152,453)(35,334,153,452)(36,333,154,451)(37,332,155,450)(38,331,156,449)(39,330,157,448)(40,329,158,447)(41,328,159,446)(42,327,160,445)(43,326,161,444)(44,325,162,443)(45,324,163,442)(46,323,164,441)(47,322,165,440)(48,321,166,439)(49,320,167,438)(50,319,168,437)(51,318,169,436)(52,317,170,435)(53,316,171,434)(54,315,172,433)(55,314,173,432)(56,313,174,431)(57,312,175,430)(58,311,176,429)(59,310,177,428)(60,309,178,427)(61,308,179,426)(62,307,180,425)(63,306,181,424)(64,305,182,423)(65,304,183,422)(66,303,184,421)(67,302,185,420)(68,301,186,419)(69,300,187,418)(70,299,188,417)(71,298,189,416)(72,297,190,415)(73,296,191,414)(74,295,192,413)(75,294,193,412)(76,293,194,411)(77,292,195,410)(78,291,196,409)(79,290,197,408)(80,289,198,407)(81,288,199,406)(82,287,200,405)(83,286,201,404)(84,285,202,403)(85,284,203,402)(86,283,204,401)(87,282,205,400)(88,281,206,399)(89,280,207,398)(90,279,208,397)(91,278,209,396)(92,277,210,395)(93,276,211,394)(94,275,212,393)(95,274,213,392)(96,273,214,391)(97,272,215,390)(98,271,216,389)(99,270,217,388)(100,269,218,387)(101,268,219,386)(102,267,220,385)(103,266,221,384)(104,265,222,383)(105,264,223,382)(106,263,224,381)(107,262,225,380)(108,261,226,379)(109,260,227,378)(110,259,228,377)(111,258,229,376)(112,257,230,375)(113,256,231,374)(114,255,232,373)(115,254,233,372)(116,253,234,371)(117,252,235,370)(118,251,236,369) );

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),(237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,312,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,412,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,441,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472)], [(1,368,119,250),(2,367,120,249),(3,366,121,248),(4,365,122,247),(5,364,123,246),(6,363,124,245),(7,362,125,244),(8,361,126,243),(9,360,127,242),(10,359,128,241),(11,358,129,240),(12,357,130,239),(13,356,131,238),(14,355,132,237),(15,354,133,472),(16,353,134,471),(17,352,135,470),(18,351,136,469),(19,350,137,468),(20,349,138,467),(21,348,139,466),(22,347,140,465),(23,346,141,464),(24,345,142,463),(25,344,143,462),(26,343,144,461),(27,342,145,460),(28,341,146,459),(29,340,147,458),(30,339,148,457),(31,338,149,456),(32,337,150,455),(33,336,151,454),(34,335,152,453),(35,334,153,452),(36,333,154,451),(37,332,155,450),(38,331,156,449),(39,330,157,448),(40,329,158,447),(41,328,159,446),(42,327,160,445),(43,326,161,444),(44,325,162,443),(45,324,163,442),(46,323,164,441),(47,322,165,440),(48,321,166,439),(49,320,167,438),(50,319,168,437),(51,318,169,436),(52,317,170,435),(53,316,171,434),(54,315,172,433),(55,314,173,432),(56,313,174,431),(57,312,175,430),(58,311,176,429),(59,310,177,428),(60,309,178,427),(61,308,179,426),(62,307,180,425),(63,306,181,424),(64,305,182,423),(65,304,183,422),(66,303,184,421),(67,302,185,420),(68,301,186,419),(69,300,187,418),(70,299,188,417),(71,298,189,416),(72,297,190,415),(73,296,191,414),(74,295,192,413),(75,294,193,412),(76,293,194,411),(77,292,195,410),(78,291,196,409),(79,290,197,408),(80,289,198,407),(81,288,199,406),(82,287,200,405),(83,286,201,404),(84,285,202,403),(85,284,203,402),(86,283,204,401),(87,282,205,400),(88,281,206,399),(89,280,207,398),(90,279,208,397),(91,278,209,396),(92,277,210,395),(93,276,211,394),(94,275,212,393),(95,274,213,392),(96,273,214,391),(97,272,215,390),(98,271,216,389),(99,270,217,388),(100,269,218,387),(101,268,219,386),(102,267,220,385),(103,266,221,384),(104,265,222,383),(105,264,223,382),(106,263,224,381),(107,262,225,380),(108,261,226,379),(109,260,227,378),(110,259,228,377),(111,258,229,376),(112,257,230,375),(113,256,231,374),(114,255,232,373),(115,254,233,372),(116,253,234,371),(117,252,235,370),(118,251,236,369)])

121 conjugacy classes

class 1  2 4A4B4C59A···59AC118A···118AC236A···236BF
order1244459···59118···118236···236
size1121181182···22···22···2

121 irreducible representations

dim1112222
type+++-++-
imageC1C2C2Q8D59D118Dic118
kernelDic118Dic59C236C59C4C2C1
# reps1211292958

Matrix representation of Dic118 in GL2(𝔽709) generated by

351501
2084
,
287140
70422
G:=sub<GL(2,GF(709))| [351,208,501,4],[287,70,140,422] >;

Dic118 in GAP, Magma, Sage, TeX

{\rm Dic}_{118}
% in TeX

G:=Group("Dic118");
// GroupNames label

G:=SmallGroup(472,3);
// by ID

G=gap.SmallGroup(472,3);
# by ID

G:=PCGroup([4,-2,-2,-2,-59,16,49,21,7427]);
// Polycyclic

G:=Group<a,b|a^236=1,b^2=a^118,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic118 in TeX

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