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G = Dic117order 468 = 22·32·13

Dicyclic group

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

Aliases: Dic117, C26.D9, C9⋊Dic13, C1173C4, C2.D117, C18.D13, C78.1S3, C6.1D39, C132Dic9, C3.Dic39, C234.1C2, C39.2Dic3, SmallGroup(468,5)

Series: Derived Chief Lower central Upper central

C1C117 — Dic117
C1C3C39C117C234 — Dic117
C117 — Dic117
C1C2

Generators and relations for Dic117
 G = < a,b | a234=1, b2=a117, bab-1=a-1 >

117C4
39Dic3
9Dic13
13Dic9
3Dic39

Smallest permutation representation of Dic117
Regular action on 468 points
Generators in S468
(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)
(1 402 118 285)(2 401 119 284)(3 400 120 283)(4 399 121 282)(5 398 122 281)(6 397 123 280)(7 396 124 279)(8 395 125 278)(9 394 126 277)(10 393 127 276)(11 392 128 275)(12 391 129 274)(13 390 130 273)(14 389 131 272)(15 388 132 271)(16 387 133 270)(17 386 134 269)(18 385 135 268)(19 384 136 267)(20 383 137 266)(21 382 138 265)(22 381 139 264)(23 380 140 263)(24 379 141 262)(25 378 142 261)(26 377 143 260)(27 376 144 259)(28 375 145 258)(29 374 146 257)(30 373 147 256)(31 372 148 255)(32 371 149 254)(33 370 150 253)(34 369 151 252)(35 368 152 251)(36 367 153 250)(37 366 154 249)(38 365 155 248)(39 364 156 247)(40 363 157 246)(41 362 158 245)(42 361 159 244)(43 360 160 243)(44 359 161 242)(45 358 162 241)(46 357 163 240)(47 356 164 239)(48 355 165 238)(49 354 166 237)(50 353 167 236)(51 352 168 235)(52 351 169 468)(53 350 170 467)(54 349 171 466)(55 348 172 465)(56 347 173 464)(57 346 174 463)(58 345 175 462)(59 344 176 461)(60 343 177 460)(61 342 178 459)(62 341 179 458)(63 340 180 457)(64 339 181 456)(65 338 182 455)(66 337 183 454)(67 336 184 453)(68 335 185 452)(69 334 186 451)(70 333 187 450)(71 332 188 449)(72 331 189 448)(73 330 190 447)(74 329 191 446)(75 328 192 445)(76 327 193 444)(77 326 194 443)(78 325 195 442)(79 324 196 441)(80 323 197 440)(81 322 198 439)(82 321 199 438)(83 320 200 437)(84 319 201 436)(85 318 202 435)(86 317 203 434)(87 316 204 433)(88 315 205 432)(89 314 206 431)(90 313 207 430)(91 312 208 429)(92 311 209 428)(93 310 210 427)(94 309 211 426)(95 308 212 425)(96 307 213 424)(97 306 214 423)(98 305 215 422)(99 304 216 421)(100 303 217 420)(101 302 218 419)(102 301 219 418)(103 300 220 417)(104 299 221 416)(105 298 222 415)(106 297 223 414)(107 296 224 413)(108 295 225 412)(109 294 226 411)(110 293 227 410)(111 292 228 409)(112 291 229 408)(113 290 230 407)(114 289 231 406)(115 288 232 405)(116 287 233 404)(117 286 234 403)

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

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

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

120 conjugacy classes

class 1  2  3 4A4B 6 9A9B9C13A···13F18A18B18C26A···26F39A···39L78A···78L117A···117AJ234A···234AJ
order12344699913···1318181826···2639···3978···78117···117234···234
size11211711722222···22222···22···22···22···22···2

120 irreducible representations

dim1112222222222
type+++-++--+-+-
imageC1C2C4S3Dic3D9D13Dic9Dic13D39Dic39D117Dic117
kernelDic117C234C117C78C39C26C18C13C9C6C3C2C1
# reps11211363612123636

Matrix representation of Dic117 in GL2(𝔽937) generated by

739194
743545
,
328216
825609
G:=sub<GL(2,GF(937))| [739,743,194,545],[328,825,216,609] >;

Dic117 in GAP, Magma, Sage, TeX

{\rm Dic}_{117}
% in TeX

G:=Group("Dic117");
// GroupNames label

G:=SmallGroup(468,5);
// by ID

G=gap.SmallGroup(468,5);
# by ID

G:=PCGroup([5,-2,-2,-3,-13,-3,10,2462,1182,2883,7804]);
// Polycyclic

G:=Group<a,b|a^234=1,b^2=a^117,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic117 in TeX

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