metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic98, C49⋊Q8, C4.D49, C2.3D98, C28.1D7, C7.Dic14, C196.1C2, C14.6D14, Dic49.C2, C98.1C22, SmallGroup(392,3)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic98
G = < a,b | a196=1, b2=a98, bab-1=a-1 >
Smallest permutation representation of Dic98
►Regular action on 392 points
Generators in S392
(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)
(1 261 99 359)(2 260 100 358)(3 259 101 357)(4 258 102 356)(5 257 103 355)(6 256 104 354)(7 255 105 353)(8 254 106 352)(9 253 107 351)(10 252 108 350)(11 251 109 349)(12 250 110 348)(13 249 111 347)(14 248 112 346)(15 247 113 345)(16 246 114 344)(17 245 115 343)(18 244 116 342)(19 243 117 341)(20 242 118 340)(21 241 119 339)(22 240 120 338)(23 239 121 337)(24 238 122 336)(25 237 123 335)(26 236 124 334)(27 235 125 333)(28 234 126 332)(29 233 127 331)(30 232 128 330)(31 231 129 329)(32 230 130 328)(33 229 131 327)(34 228 132 326)(35 227 133 325)(36 226 134 324)(37 225 135 323)(38 224 136 322)(39 223 137 321)(40 222 138 320)(41 221 139 319)(42 220 140 318)(43 219 141 317)(44 218 142 316)(45 217 143 315)(46 216 144 314)(47 215 145 313)(48 214 146 312)(49 213 147 311)(50 212 148 310)(51 211 149 309)(52 210 150 308)(53 209 151 307)(54 208 152 306)(55 207 153 305)(56 206 154 304)(57 205 155 303)(58 204 156 302)(59 203 157 301)(60 202 158 300)(61 201 159 299)(62 200 160 298)(63 199 161 297)(64 198 162 296)(65 197 163 295)(66 392 164 294)(67 391 165 293)(68 390 166 292)(69 389 167 291)(70 388 168 290)(71 387 169 289)(72 386 170 288)(73 385 171 287)(74 384 172 286)(75 383 173 285)(76 382 174 284)(77 381 175 283)(78 380 176 282)(79 379 177 281)(80 378 178 280)(81 377 179 279)(82 376 180 278)(83 375 181 277)(84 374 182 276)(85 373 183 275)(86 372 184 274)(87 371 185 273)(88 370 186 272)(89 369 187 271)(90 368 188 270)(91 367 189 269)(92 366 190 268)(93 365 191 267)(94 364 192 266)(95 363 193 265)(96 362 194 264)(97 361 195 263)(98 360 196 262)
G:=sub<Sym(392)| (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), (1,261,99,359)(2,260,100,358)(3,259,101,357)(4,258,102,356)(5,257,103,355)(6,256,104,354)(7,255,105,353)(8,254,106,352)(9,253,107,351)(10,252,108,350)(11,251,109,349)(12,250,110,348)(13,249,111,347)(14,248,112,346)(15,247,113,345)(16,246,114,344)(17,245,115,343)(18,244,116,342)(19,243,117,341)(20,242,118,340)(21,241,119,339)(22,240,120,338)(23,239,121,337)(24,238,122,336)(25,237,123,335)(26,236,124,334)(27,235,125,333)(28,234,126,332)(29,233,127,331)(30,232,128,330)(31,231,129,329)(32,230,130,328)(33,229,131,327)(34,228,132,326)(35,227,133,325)(36,226,134,324)(37,225,135,323)(38,224,136,322)(39,223,137,321)(40,222,138,320)(41,221,139,319)(42,220,140,318)(43,219,141,317)(44,218,142,316)(45,217,143,315)(46,216,144,314)(47,215,145,313)(48,214,146,312)(49,213,147,311)(50,212,148,310)(51,211,149,309)(52,210,150,308)(53,209,151,307)(54,208,152,306)(55,207,153,305)(56,206,154,304)(57,205,155,303)(58,204,156,302)(59,203,157,301)(60,202,158,300)(61,201,159,299)(62,200,160,298)(63,199,161,297)(64,198,162,296)(65,197,163,295)(66,392,164,294)(67,391,165,293)(68,390,166,292)(69,389,167,291)(70,388,168,290)(71,387,169,289)(72,386,170,288)(73,385,171,287)(74,384,172,286)(75,383,173,285)(76,382,174,284)(77,381,175,283)(78,380,176,282)(79,379,177,281)(80,378,178,280)(81,377,179,279)(82,376,180,278)(83,375,181,277)(84,374,182,276)(85,373,183,275)(86,372,184,274)(87,371,185,273)(88,370,186,272)(89,369,187,271)(90,368,188,270)(91,367,189,269)(92,366,190,268)(93,365,191,267)(94,364,192,266)(95,363,193,265)(96,362,194,264)(97,361,195,263)(98,360,196,262)>;
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), (1,261,99,359)(2,260,100,358)(3,259,101,357)(4,258,102,356)(5,257,103,355)(6,256,104,354)(7,255,105,353)(8,254,106,352)(9,253,107,351)(10,252,108,350)(11,251,109,349)(12,250,110,348)(13,249,111,347)(14,248,112,346)(15,247,113,345)(16,246,114,344)(17,245,115,343)(18,244,116,342)(19,243,117,341)(20,242,118,340)(21,241,119,339)(22,240,120,338)(23,239,121,337)(24,238,122,336)(25,237,123,335)(26,236,124,334)(27,235,125,333)(28,234,126,332)(29,233,127,331)(30,232,128,330)(31,231,129,329)(32,230,130,328)(33,229,131,327)(34,228,132,326)(35,227,133,325)(36,226,134,324)(37,225,135,323)(38,224,136,322)(39,223,137,321)(40,222,138,320)(41,221,139,319)(42,220,140,318)(43,219,141,317)(44,218,142,316)(45,217,143,315)(46,216,144,314)(47,215,145,313)(48,214,146,312)(49,213,147,311)(50,212,148,310)(51,211,149,309)(52,210,150,308)(53,209,151,307)(54,208,152,306)(55,207,153,305)(56,206,154,304)(57,205,155,303)(58,204,156,302)(59,203,157,301)(60,202,158,300)(61,201,159,299)(62,200,160,298)(63,199,161,297)(64,198,162,296)(65,197,163,295)(66,392,164,294)(67,391,165,293)(68,390,166,292)(69,389,167,291)(70,388,168,290)(71,387,169,289)(72,386,170,288)(73,385,171,287)(74,384,172,286)(75,383,173,285)(76,382,174,284)(77,381,175,283)(78,380,176,282)(79,379,177,281)(80,378,178,280)(81,377,179,279)(82,376,180,278)(83,375,181,277)(84,374,182,276)(85,373,183,275)(86,372,184,274)(87,371,185,273)(88,370,186,272)(89,369,187,271)(90,368,188,270)(91,367,189,269)(92,366,190,268)(93,365,191,267)(94,364,192,266)(95,363,193,265)(96,362,194,264)(97,361,195,263)(98,360,196,262) );
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)], [(1,261,99,359),(2,260,100,358),(3,259,101,357),(4,258,102,356),(5,257,103,355),(6,256,104,354),(7,255,105,353),(8,254,106,352),(9,253,107,351),(10,252,108,350),(11,251,109,349),(12,250,110,348),(13,249,111,347),(14,248,112,346),(15,247,113,345),(16,246,114,344),(17,245,115,343),(18,244,116,342),(19,243,117,341),(20,242,118,340),(21,241,119,339),(22,240,120,338),(23,239,121,337),(24,238,122,336),(25,237,123,335),(26,236,124,334),(27,235,125,333),(28,234,126,332),(29,233,127,331),(30,232,128,330),(31,231,129,329),(32,230,130,328),(33,229,131,327),(34,228,132,326),(35,227,133,325),(36,226,134,324),(37,225,135,323),(38,224,136,322),(39,223,137,321),(40,222,138,320),(41,221,139,319),(42,220,140,318),(43,219,141,317),(44,218,142,316),(45,217,143,315),(46,216,144,314),(47,215,145,313),(48,214,146,312),(49,213,147,311),(50,212,148,310),(51,211,149,309),(52,210,150,308),(53,209,151,307),(54,208,152,306),(55,207,153,305),(56,206,154,304),(57,205,155,303),(58,204,156,302),(59,203,157,301),(60,202,158,300),(61,201,159,299),(62,200,160,298),(63,199,161,297),(64,198,162,296),(65,197,163,295),(66,392,164,294),(67,391,165,293),(68,390,166,292),(69,389,167,291),(70,388,168,290),(71,387,169,289),(72,386,170,288),(73,385,171,287),(74,384,172,286),(75,383,173,285),(76,382,174,284),(77,381,175,283),(78,380,176,282),(79,379,177,281),(80,378,178,280),(81,377,179,279),(82,376,180,278),(83,375,181,277),(84,374,182,276),(85,373,183,275),(86,372,184,274),(87,371,185,273),(88,370,186,272),(89,369,187,271),(90,368,188,270),(91,367,189,269),(92,366,190,268),(93,365,191,267),(94,364,192,266),(95,363,193,265),(96,362,194,264),(97,361,195,263),(98,360,196,262)]])
101 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 14A | 14B | 14C | 28A | ··· | 28F | 49A | ··· | 49U | 98A | ··· | 98U | 196A | ··· | 196AP |
| order | 1 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 14 | 14 | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 2 | 98 | 98 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
101 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D7 | D14 | Dic14 | D49 | D98 | Dic98 |
| kernel | Dic98 | Dic49 | C196 | C49 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 3 | 3 | 6 | 21 | 21 | 42 |
Matrix representation of Dic98 ►in GL2(𝔽197) generated by
| 114 | 14 |
| 183 | 159 |
| 12 | 187 |
| 113 | 185 |
G:=sub<GL(2,GF(197))| [114,183,14,159],[12,113,187,185] >;
Dic98 in GAP, Magma, Sage, TeX
{\rm Dic}_{98} % in TeX
G:=Group("Dic98"); // GroupNames label
G:=SmallGroup(392,3);
// by ID
G=gap.SmallGroup(392,3);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,20,61,26,2083,858,8404]);
// Polycyclic
G:=Group<a,b|a^196=1,b^2=a^98,b*a*b^-1=a^-1>;
// generators/relations
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