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G = Dic109order 436 = 22·109

Dicyclic group

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

Aliases: Dic109, C1092C4, C218.C2, C2.D109, SmallGroup(436,1)

Series: Derived Chief Lower central Upper central

C1C109 — Dic109
C1C109C218 — Dic109
C109 — Dic109
C1C2

Generators and relations for Dic109
 G = < a,b | a218=1, b2=a109, bab-1=a-1 >

109C4

Smallest permutation representation of Dic109
Regular action on 436 points
Generators in S436
(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)
(1 225 110 334)(2 224 111 333)(3 223 112 332)(4 222 113 331)(5 221 114 330)(6 220 115 329)(7 219 116 328)(8 436 117 327)(9 435 118 326)(10 434 119 325)(11 433 120 324)(12 432 121 323)(13 431 122 322)(14 430 123 321)(15 429 124 320)(16 428 125 319)(17 427 126 318)(18 426 127 317)(19 425 128 316)(20 424 129 315)(21 423 130 314)(22 422 131 313)(23 421 132 312)(24 420 133 311)(25 419 134 310)(26 418 135 309)(27 417 136 308)(28 416 137 307)(29 415 138 306)(30 414 139 305)(31 413 140 304)(32 412 141 303)(33 411 142 302)(34 410 143 301)(35 409 144 300)(36 408 145 299)(37 407 146 298)(38 406 147 297)(39 405 148 296)(40 404 149 295)(41 403 150 294)(42 402 151 293)(43 401 152 292)(44 400 153 291)(45 399 154 290)(46 398 155 289)(47 397 156 288)(48 396 157 287)(49 395 158 286)(50 394 159 285)(51 393 160 284)(52 392 161 283)(53 391 162 282)(54 390 163 281)(55 389 164 280)(56 388 165 279)(57 387 166 278)(58 386 167 277)(59 385 168 276)(60 384 169 275)(61 383 170 274)(62 382 171 273)(63 381 172 272)(64 380 173 271)(65 379 174 270)(66 378 175 269)(67 377 176 268)(68 376 177 267)(69 375 178 266)(70 374 179 265)(71 373 180 264)(72 372 181 263)(73 371 182 262)(74 370 183 261)(75 369 184 260)(76 368 185 259)(77 367 186 258)(78 366 187 257)(79 365 188 256)(80 364 189 255)(81 363 190 254)(82 362 191 253)(83 361 192 252)(84 360 193 251)(85 359 194 250)(86 358 195 249)(87 357 196 248)(88 356 197 247)(89 355 198 246)(90 354 199 245)(91 353 200 244)(92 352 201 243)(93 351 202 242)(94 350 203 241)(95 349 204 240)(96 348 205 239)(97 347 206 238)(98 346 207 237)(99 345 208 236)(100 344 209 235)(101 343 210 234)(102 342 211 233)(103 341 212 232)(104 340 213 231)(105 339 214 230)(106 338 215 229)(107 337 216 228)(108 336 217 227)(109 335 218 226)

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

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

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

112 conjugacy classes

class 1  2 4A4B109A···109BB218A···218BB
order1244109···109218···218
size111091092···22···2

112 irreducible representations

dim11122
type+++-
imageC1C2C4D109Dic109
kernelDic109C218C109C2C1
# reps1125454

Matrix representation of Dic109 in GL3(𝔽2617) generated by

261600
017312616
010
,
66700
017681445
02580849
G:=sub<GL(3,GF(2617))| [2616,0,0,0,1731,1,0,2616,0],[667,0,0,0,1768,2580,0,1445,849] >;

Dic109 in GAP, Magma, Sage, TeX

{\rm Dic}_{109}
% in TeX

G:=Group("Dic109");
// GroupNames label

G:=SmallGroup(436,1);
// by ID

G=gap.SmallGroup(436,1);
# by ID

G:=PCGroup([3,-2,-2,-109,6,3890]);
// Polycyclic

G:=Group<a,b|a^218=1,b^2=a^109,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic109 in TeX

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