Copied to
clipboard

G = Dic103order 412 = 22·103

Dicyclic group

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

Aliases: Dic103, C103⋊C4, C206.C2, C2.D103, SmallGroup(412,1)

Series: Derived Chief Lower central Upper central

C1C103 — Dic103
C1C103C206 — Dic103
C103 — Dic103
C1C2

Generators and relations for Dic103
 G = < a,b | a206=1, b2=a103, bab-1=a-1 >

103C4

Smallest permutation representation of Dic103
Regular action on 412 points
Generators in S412
(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)
(1 374 104 271)(2 373 105 270)(3 372 106 269)(4 371 107 268)(5 370 108 267)(6 369 109 266)(7 368 110 265)(8 367 111 264)(9 366 112 263)(10 365 113 262)(11 364 114 261)(12 363 115 260)(13 362 116 259)(14 361 117 258)(15 360 118 257)(16 359 119 256)(17 358 120 255)(18 357 121 254)(19 356 122 253)(20 355 123 252)(21 354 124 251)(22 353 125 250)(23 352 126 249)(24 351 127 248)(25 350 128 247)(26 349 129 246)(27 348 130 245)(28 347 131 244)(29 346 132 243)(30 345 133 242)(31 344 134 241)(32 343 135 240)(33 342 136 239)(34 341 137 238)(35 340 138 237)(36 339 139 236)(37 338 140 235)(38 337 141 234)(39 336 142 233)(40 335 143 232)(41 334 144 231)(42 333 145 230)(43 332 146 229)(44 331 147 228)(45 330 148 227)(46 329 149 226)(47 328 150 225)(48 327 151 224)(49 326 152 223)(50 325 153 222)(51 324 154 221)(52 323 155 220)(53 322 156 219)(54 321 157 218)(55 320 158 217)(56 319 159 216)(57 318 160 215)(58 317 161 214)(59 316 162 213)(60 315 163 212)(61 314 164 211)(62 313 165 210)(63 312 166 209)(64 311 167 208)(65 310 168 207)(66 309 169 412)(67 308 170 411)(68 307 171 410)(69 306 172 409)(70 305 173 408)(71 304 174 407)(72 303 175 406)(73 302 176 405)(74 301 177 404)(75 300 178 403)(76 299 179 402)(77 298 180 401)(78 297 181 400)(79 296 182 399)(80 295 183 398)(81 294 184 397)(82 293 185 396)(83 292 186 395)(84 291 187 394)(85 290 188 393)(86 289 189 392)(87 288 190 391)(88 287 191 390)(89 286 192 389)(90 285 193 388)(91 284 194 387)(92 283 195 386)(93 282 196 385)(94 281 197 384)(95 280 198 383)(96 279 199 382)(97 278 200 381)(98 277 201 380)(99 276 202 379)(100 275 203 378)(101 274 204 377)(102 273 205 376)(103 272 206 375)

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

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

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

106 conjugacy classes

class 1  2 4A4B103A···103AY206A···206AY
order1244103···103206···206
size111031032···22···2

106 irreducible representations

dim11122
type+++-
imageC1C2C4D103Dic103
kernelDic103C206C103C2C1
# reps1125151

Matrix representation of Dic103 in GL3(𝔽1237) generated by

123600
05311236
010
,
54600
0680204
080557
G:=sub<GL(3,GF(1237))| [1236,0,0,0,531,1,0,1236,0],[546,0,0,0,680,80,0,204,557] >;

Dic103 in GAP, Magma, Sage, TeX

{\rm Dic}_{103}
% in TeX

G:=Group("Dic103");
// GroupNames label

G:=SmallGroup(412,1);
// by ID

G=gap.SmallGroup(412,1);
# by ID

G:=PCGroup([3,-2,-2,-103,6,3674]);
// Polycyclic

G:=Group<a,b|a^206=1,b^2=a^103,b*a*b^-1=a^-1>;
// generators/relations

Export

Subgroup lattice of Dic103 in TeX

׿
×
𝔽