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G = Dic108order 432 = 24·33

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic108, C8.D27, C271Q16, C72.1S3, C24.1D9, C54.1D4, C6.1D36, C4.8D54, C3.Dic36, C9.Dic12, C216.1C2, C2.3D108, C18.1D12, C36.51D6, C12.51D18, C108.8C22, Dic54.1C2, SmallGroup(432,4)

Series: Derived Chief Lower central Upper central

C1C108 — Dic108
C1C3C9C27C54C108Dic54 — Dic108
C27C54C108 — Dic108
C1C2C4C8

Generators and relations for Dic108
 G = < a,b | a216=1, b2=a108, bab-1=a-1 >

54C4
54C4
27Q8
27Q8
18Dic3
18Dic3
27Q16
9Dic6
9Dic6
6Dic9
6Dic9
9Dic12
3Dic18
3Dic18
2Dic27
2Dic27
3Dic36

Smallest permutation representation of Dic108
Regular action on 432 points
Generators in S432
(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)
(1 369 109 261)(2 368 110 260)(3 367 111 259)(4 366 112 258)(5 365 113 257)(6 364 114 256)(7 363 115 255)(8 362 116 254)(9 361 117 253)(10 360 118 252)(11 359 119 251)(12 358 120 250)(13 357 121 249)(14 356 122 248)(15 355 123 247)(16 354 124 246)(17 353 125 245)(18 352 126 244)(19 351 127 243)(20 350 128 242)(21 349 129 241)(22 348 130 240)(23 347 131 239)(24 346 132 238)(25 345 133 237)(26 344 134 236)(27 343 135 235)(28 342 136 234)(29 341 137 233)(30 340 138 232)(31 339 139 231)(32 338 140 230)(33 337 141 229)(34 336 142 228)(35 335 143 227)(36 334 144 226)(37 333 145 225)(38 332 146 224)(39 331 147 223)(40 330 148 222)(41 329 149 221)(42 328 150 220)(43 327 151 219)(44 326 152 218)(45 325 153 217)(46 324 154 432)(47 323 155 431)(48 322 156 430)(49 321 157 429)(50 320 158 428)(51 319 159 427)(52 318 160 426)(53 317 161 425)(54 316 162 424)(55 315 163 423)(56 314 164 422)(57 313 165 421)(58 312 166 420)(59 311 167 419)(60 310 168 418)(61 309 169 417)(62 308 170 416)(63 307 171 415)(64 306 172 414)(65 305 173 413)(66 304 174 412)(67 303 175 411)(68 302 176 410)(69 301 177 409)(70 300 178 408)(71 299 179 407)(72 298 180 406)(73 297 181 405)(74 296 182 404)(75 295 183 403)(76 294 184 402)(77 293 185 401)(78 292 186 400)(79 291 187 399)(80 290 188 398)(81 289 189 397)(82 288 190 396)(83 287 191 395)(84 286 192 394)(85 285 193 393)(86 284 194 392)(87 283 195 391)(88 282 196 390)(89 281 197 389)(90 280 198 388)(91 279 199 387)(92 278 200 386)(93 277 201 385)(94 276 202 384)(95 275 203 383)(96 274 204 382)(97 273 205 381)(98 272 206 380)(99 271 207 379)(100 270 208 378)(101 269 209 377)(102 268 210 376)(103 267 211 375)(104 266 212 374)(105 265 213 373)(106 264 214 372)(107 263 215 371)(108 262 216 370)

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

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

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

111 conjugacy classes

class 1  2  3 4A4B4C 6 8A8B9A9B9C12A12B18A18B18C24A24B24C24D27A···27I36A···36F54A···54I72A···72L108A···108R216A···216AJ
order12344468899912121818182424242427···2736···3654···5472···72108···108216···216
size11221081082222222222222222···22···22···22···22···22···2

111 irreducible representations

dim11122222222222222
type++++++-+++-+++-+-
imageC1C2C2S3D4D6Q16D9D12D18Dic12D27D36D54Dic36D108Dic108
kernelDic108C216Dic54C72C54C36C27C24C18C12C9C8C6C4C3C2C1
# reps11211123234969121836

Matrix representation of Dic108 in GL2(𝔽433) generated by

346102
331244
,
168288
120265
G:=sub<GL(2,GF(433))| [346,331,102,244],[168,120,288,265] >;

Dic108 in GAP, Magma, Sage, TeX

{\rm Dic}_{108}
% in TeX

G:=Group("Dic108");
// GroupNames label

G:=SmallGroup(432,4);
// by ID

G=gap.SmallGroup(432,4);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b|a^216=1,b^2=a^108,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic108 in TeX

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