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G = Dic104order 416 = 25·13

Dicyclic group

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: Dic104, C16.D13, C131Q32, C4.3D52, C26.3D8, C208.1C2, C8.15D26, C52.26D4, C2.5D104, Dic52.1C2, C104.16C22, SmallGroup(416,8)

Series: Derived Chief Lower central Upper central

C1C104 — Dic104
C1C13C26C52C104Dic52 — Dic104
C13C26C52C104 — Dic104
C1C2C4C8C16

Generators and relations for Dic104
 G = < a,b | a208=1, b2=a104, bab-1=a-1 >

52C4
52C4
26Q8
26Q8
4Dic13
4Dic13
13Q16
13Q16
2Dic26
2Dic26
13Q32

Smallest permutation representation of Dic104
Regular action on 416 points
Generators in S416
(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)
(1 306 105 410)(2 305 106 409)(3 304 107 408)(4 303 108 407)(5 302 109 406)(6 301 110 405)(7 300 111 404)(8 299 112 403)(9 298 113 402)(10 297 114 401)(11 296 115 400)(12 295 116 399)(13 294 117 398)(14 293 118 397)(15 292 119 396)(16 291 120 395)(17 290 121 394)(18 289 122 393)(19 288 123 392)(20 287 124 391)(21 286 125 390)(22 285 126 389)(23 284 127 388)(24 283 128 387)(25 282 129 386)(26 281 130 385)(27 280 131 384)(28 279 132 383)(29 278 133 382)(30 277 134 381)(31 276 135 380)(32 275 136 379)(33 274 137 378)(34 273 138 377)(35 272 139 376)(36 271 140 375)(37 270 141 374)(38 269 142 373)(39 268 143 372)(40 267 144 371)(41 266 145 370)(42 265 146 369)(43 264 147 368)(44 263 148 367)(45 262 149 366)(46 261 150 365)(47 260 151 364)(48 259 152 363)(49 258 153 362)(50 257 154 361)(51 256 155 360)(52 255 156 359)(53 254 157 358)(54 253 158 357)(55 252 159 356)(56 251 160 355)(57 250 161 354)(58 249 162 353)(59 248 163 352)(60 247 164 351)(61 246 165 350)(62 245 166 349)(63 244 167 348)(64 243 168 347)(65 242 169 346)(66 241 170 345)(67 240 171 344)(68 239 172 343)(69 238 173 342)(70 237 174 341)(71 236 175 340)(72 235 176 339)(73 234 177 338)(74 233 178 337)(75 232 179 336)(76 231 180 335)(77 230 181 334)(78 229 182 333)(79 228 183 332)(80 227 184 331)(81 226 185 330)(82 225 186 329)(83 224 187 328)(84 223 188 327)(85 222 189 326)(86 221 190 325)(87 220 191 324)(88 219 192 323)(89 218 193 322)(90 217 194 321)(91 216 195 320)(92 215 196 319)(93 214 197 318)(94 213 198 317)(95 212 199 316)(96 211 200 315)(97 210 201 314)(98 209 202 313)(99 416 203 312)(100 415 204 311)(101 414 205 310)(102 413 206 309)(103 412 207 308)(104 411 208 307)

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

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

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

107 conjugacy classes

class 1  2 4A4B4C8A8B13A···13F16A16B16C16D26A···26F52A···52L104A···104X208A···208AV
order124448813···131616161626···2652···52104···104208···208
size112104104222···222222···22···22···22···2

107 irreducible representations

dim11122222222
type++++++-+++-
imageC1C2C2D4D8D13Q32D26D52D104Dic104
kernelDic104C208Dic52C52C26C16C13C8C4C2C1
# reps11212646122448

Matrix representation of Dic104 in GL2(𝔽1249) generated by

801304
9451012
,
1052478
1126197
G:=sub<GL(2,GF(1249))| [801,945,304,1012],[1052,1126,478,197] >;

Dic104 in GAP, Magma, Sage, TeX

{\rm Dic}_{104}
% in TeX

G:=Group("Dic104");
// GroupNames label

G:=SmallGroup(416,8);
// by ID

G=gap.SmallGroup(416,8);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-13,96,73,79,218,122,579,69,13829]);
// Polycyclic

G:=Group<a,b|a^208=1,b^2=a^104,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic104 in TeX

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