metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic97, C97⋊2C4, C2.D97, C194.C2, SmallGroup(388,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C97 — Dic97 |
Generators and relations for Dic97
G = < a,b | a194=1, b2=a97, bab-1=a-1 >
Smallest permutation representation of Dic97
►Regular action on 388 points
Generators in S388
(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)
(1 233 98 330)(2 232 99 329)(3 231 100 328)(4 230 101 327)(5 229 102 326)(6 228 103 325)(7 227 104 324)(8 226 105 323)(9 225 106 322)(10 224 107 321)(11 223 108 320)(12 222 109 319)(13 221 110 318)(14 220 111 317)(15 219 112 316)(16 218 113 315)(17 217 114 314)(18 216 115 313)(19 215 116 312)(20 214 117 311)(21 213 118 310)(22 212 119 309)(23 211 120 308)(24 210 121 307)(25 209 122 306)(26 208 123 305)(27 207 124 304)(28 206 125 303)(29 205 126 302)(30 204 127 301)(31 203 128 300)(32 202 129 299)(33 201 130 298)(34 200 131 297)(35 199 132 296)(36 198 133 295)(37 197 134 294)(38 196 135 293)(39 195 136 292)(40 388 137 291)(41 387 138 290)(42 386 139 289)(43 385 140 288)(44 384 141 287)(45 383 142 286)(46 382 143 285)(47 381 144 284)(48 380 145 283)(49 379 146 282)(50 378 147 281)(51 377 148 280)(52 376 149 279)(53 375 150 278)(54 374 151 277)(55 373 152 276)(56 372 153 275)(57 371 154 274)(58 370 155 273)(59 369 156 272)(60 368 157 271)(61 367 158 270)(62 366 159 269)(63 365 160 268)(64 364 161 267)(65 363 162 266)(66 362 163 265)(67 361 164 264)(68 360 165 263)(69 359 166 262)(70 358 167 261)(71 357 168 260)(72 356 169 259)(73 355 170 258)(74 354 171 257)(75 353 172 256)(76 352 173 255)(77 351 174 254)(78 350 175 253)(79 349 176 252)(80 348 177 251)(81 347 178 250)(82 346 179 249)(83 345 180 248)(84 344 181 247)(85 343 182 246)(86 342 183 245)(87 341 184 244)(88 340 185 243)(89 339 186 242)(90 338 187 241)(91 337 188 240)(92 336 189 239)(93 335 190 238)(94 334 191 237)(95 333 192 236)(96 332 193 235)(97 331 194 234)
G:=sub<Sym(388)| (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), (1,233,98,330)(2,232,99,329)(3,231,100,328)(4,230,101,327)(5,229,102,326)(6,228,103,325)(7,227,104,324)(8,226,105,323)(9,225,106,322)(10,224,107,321)(11,223,108,320)(12,222,109,319)(13,221,110,318)(14,220,111,317)(15,219,112,316)(16,218,113,315)(17,217,114,314)(18,216,115,313)(19,215,116,312)(20,214,117,311)(21,213,118,310)(22,212,119,309)(23,211,120,308)(24,210,121,307)(25,209,122,306)(26,208,123,305)(27,207,124,304)(28,206,125,303)(29,205,126,302)(30,204,127,301)(31,203,128,300)(32,202,129,299)(33,201,130,298)(34,200,131,297)(35,199,132,296)(36,198,133,295)(37,197,134,294)(38,196,135,293)(39,195,136,292)(40,388,137,291)(41,387,138,290)(42,386,139,289)(43,385,140,288)(44,384,141,287)(45,383,142,286)(46,382,143,285)(47,381,144,284)(48,380,145,283)(49,379,146,282)(50,378,147,281)(51,377,148,280)(52,376,149,279)(53,375,150,278)(54,374,151,277)(55,373,152,276)(56,372,153,275)(57,371,154,274)(58,370,155,273)(59,369,156,272)(60,368,157,271)(61,367,158,270)(62,366,159,269)(63,365,160,268)(64,364,161,267)(65,363,162,266)(66,362,163,265)(67,361,164,264)(68,360,165,263)(69,359,166,262)(70,358,167,261)(71,357,168,260)(72,356,169,259)(73,355,170,258)(74,354,171,257)(75,353,172,256)(76,352,173,255)(77,351,174,254)(78,350,175,253)(79,349,176,252)(80,348,177,251)(81,347,178,250)(82,346,179,249)(83,345,180,248)(84,344,181,247)(85,343,182,246)(86,342,183,245)(87,341,184,244)(88,340,185,243)(89,339,186,242)(90,338,187,241)(91,337,188,240)(92,336,189,239)(93,335,190,238)(94,334,191,237)(95,333,192,236)(96,332,193,235)(97,331,194,234)>;
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), (1,233,98,330)(2,232,99,329)(3,231,100,328)(4,230,101,327)(5,229,102,326)(6,228,103,325)(7,227,104,324)(8,226,105,323)(9,225,106,322)(10,224,107,321)(11,223,108,320)(12,222,109,319)(13,221,110,318)(14,220,111,317)(15,219,112,316)(16,218,113,315)(17,217,114,314)(18,216,115,313)(19,215,116,312)(20,214,117,311)(21,213,118,310)(22,212,119,309)(23,211,120,308)(24,210,121,307)(25,209,122,306)(26,208,123,305)(27,207,124,304)(28,206,125,303)(29,205,126,302)(30,204,127,301)(31,203,128,300)(32,202,129,299)(33,201,130,298)(34,200,131,297)(35,199,132,296)(36,198,133,295)(37,197,134,294)(38,196,135,293)(39,195,136,292)(40,388,137,291)(41,387,138,290)(42,386,139,289)(43,385,140,288)(44,384,141,287)(45,383,142,286)(46,382,143,285)(47,381,144,284)(48,380,145,283)(49,379,146,282)(50,378,147,281)(51,377,148,280)(52,376,149,279)(53,375,150,278)(54,374,151,277)(55,373,152,276)(56,372,153,275)(57,371,154,274)(58,370,155,273)(59,369,156,272)(60,368,157,271)(61,367,158,270)(62,366,159,269)(63,365,160,268)(64,364,161,267)(65,363,162,266)(66,362,163,265)(67,361,164,264)(68,360,165,263)(69,359,166,262)(70,358,167,261)(71,357,168,260)(72,356,169,259)(73,355,170,258)(74,354,171,257)(75,353,172,256)(76,352,173,255)(77,351,174,254)(78,350,175,253)(79,349,176,252)(80,348,177,251)(81,347,178,250)(82,346,179,249)(83,345,180,248)(84,344,181,247)(85,343,182,246)(86,342,183,245)(87,341,184,244)(88,340,185,243)(89,339,186,242)(90,338,187,241)(91,337,188,240)(92,336,189,239)(93,335,190,238)(94,334,191,237)(95,333,192,236)(96,332,193,235)(97,331,194,234) );
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)], [(1,233,98,330),(2,232,99,329),(3,231,100,328),(4,230,101,327),(5,229,102,326),(6,228,103,325),(7,227,104,324),(8,226,105,323),(9,225,106,322),(10,224,107,321),(11,223,108,320),(12,222,109,319),(13,221,110,318),(14,220,111,317),(15,219,112,316),(16,218,113,315),(17,217,114,314),(18,216,115,313),(19,215,116,312),(20,214,117,311),(21,213,118,310),(22,212,119,309),(23,211,120,308),(24,210,121,307),(25,209,122,306),(26,208,123,305),(27,207,124,304),(28,206,125,303),(29,205,126,302),(30,204,127,301),(31,203,128,300),(32,202,129,299),(33,201,130,298),(34,200,131,297),(35,199,132,296),(36,198,133,295),(37,197,134,294),(38,196,135,293),(39,195,136,292),(40,388,137,291),(41,387,138,290),(42,386,139,289),(43,385,140,288),(44,384,141,287),(45,383,142,286),(46,382,143,285),(47,381,144,284),(48,380,145,283),(49,379,146,282),(50,378,147,281),(51,377,148,280),(52,376,149,279),(53,375,150,278),(54,374,151,277),(55,373,152,276),(56,372,153,275),(57,371,154,274),(58,370,155,273),(59,369,156,272),(60,368,157,271),(61,367,158,270),(62,366,159,269),(63,365,160,268),(64,364,161,267),(65,363,162,266),(66,362,163,265),(67,361,164,264),(68,360,165,263),(69,359,166,262),(70,358,167,261),(71,357,168,260),(72,356,169,259),(73,355,170,258),(74,354,171,257),(75,353,172,256),(76,352,173,255),(77,351,174,254),(78,350,175,253),(79,349,176,252),(80,348,177,251),(81,347,178,250),(82,346,179,249),(83,345,180,248),(84,344,181,247),(85,343,182,246),(86,342,183,245),(87,341,184,244),(88,340,185,243),(89,339,186,242),(90,338,187,241),(91,337,188,240),(92,336,189,239),(93,335,190,238),(94,334,191,237),(95,333,192,236),(96,332,193,235),(97,331,194,234)]])
100 conjugacy classes
| class | 1 | 2 | 4A | 4B | 97A | ··· | 97AV | 194A | ··· | 194AV |
| order | 1 | 2 | 4 | 4 | 97 | ··· | 97 | 194 | ··· | 194 |
| size | 1 | 1 | 97 | 97 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D97 | Dic97 |
| kernel | Dic97 | C194 | C97 | C2 | C1 |
| # reps | 1 | 1 | 2 | 48 | 48 |
Matrix representation of Dic97 ►in GL3(𝔽389) generated by
| 388 | 0 | 0 |
| 0 | 329 | 388 |
| 0 | 1 | 0 |
| 274 | 0 | 0 |
| 0 | 217 | 387 |
| 0 | 204 | 172 |
G:=sub<GL(3,GF(389))| [388,0,0,0,329,1,0,388,0],[274,0,0,0,217,204,0,387,172] >;
Dic97 in GAP, Magma, Sage, TeX
{\rm Dic}_{97} % in TeX
G:=Group("Dic97"); // GroupNames label
G:=SmallGroup(388,1);
// by ID
G=gap.SmallGroup(388,1);
# by ID
G:=PCGroup([3,-2,-2,-97,6,3458]);
// Polycyclic
G:=Group<a,b|a^194=1,b^2=a^97,b*a*b^-1=a^-1>;
// generators/relations
Export