metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic86, C43⋊Q8, C4.D43, C2.3D86, C172.1C2, Dic43.C2, C86.1C22, SmallGroup(344,3)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic86
G = < a,b | a172=1, b2=a86, bab-1=a-1 >
Smallest permutation representation of Dic86
►Regular action on 344 points
Generators in S344
(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)
(1 275 87 189)(2 274 88 188)(3 273 89 187)(4 272 90 186)(5 271 91 185)(6 270 92 184)(7 269 93 183)(8 268 94 182)(9 267 95 181)(10 266 96 180)(11 265 97 179)(12 264 98 178)(13 263 99 177)(14 262 100 176)(15 261 101 175)(16 260 102 174)(17 259 103 173)(18 258 104 344)(19 257 105 343)(20 256 106 342)(21 255 107 341)(22 254 108 340)(23 253 109 339)(24 252 110 338)(25 251 111 337)(26 250 112 336)(27 249 113 335)(28 248 114 334)(29 247 115 333)(30 246 116 332)(31 245 117 331)(32 244 118 330)(33 243 119 329)(34 242 120 328)(35 241 121 327)(36 240 122 326)(37 239 123 325)(38 238 124 324)(39 237 125 323)(40 236 126 322)(41 235 127 321)(42 234 128 320)(43 233 129 319)(44 232 130 318)(45 231 131 317)(46 230 132 316)(47 229 133 315)(48 228 134 314)(49 227 135 313)(50 226 136 312)(51 225 137 311)(52 224 138 310)(53 223 139 309)(54 222 140 308)(55 221 141 307)(56 220 142 306)(57 219 143 305)(58 218 144 304)(59 217 145 303)(60 216 146 302)(61 215 147 301)(62 214 148 300)(63 213 149 299)(64 212 150 298)(65 211 151 297)(66 210 152 296)(67 209 153 295)(68 208 154 294)(69 207 155 293)(70 206 156 292)(71 205 157 291)(72 204 158 290)(73 203 159 289)(74 202 160 288)(75 201 161 287)(76 200 162 286)(77 199 163 285)(78 198 164 284)(79 197 165 283)(80 196 166 282)(81 195 167 281)(82 194 168 280)(83 193 169 279)(84 192 170 278)(85 191 171 277)(86 190 172 276)
G:=sub<Sym(344)| (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), (1,275,87,189)(2,274,88,188)(3,273,89,187)(4,272,90,186)(5,271,91,185)(6,270,92,184)(7,269,93,183)(8,268,94,182)(9,267,95,181)(10,266,96,180)(11,265,97,179)(12,264,98,178)(13,263,99,177)(14,262,100,176)(15,261,101,175)(16,260,102,174)(17,259,103,173)(18,258,104,344)(19,257,105,343)(20,256,106,342)(21,255,107,341)(22,254,108,340)(23,253,109,339)(24,252,110,338)(25,251,111,337)(26,250,112,336)(27,249,113,335)(28,248,114,334)(29,247,115,333)(30,246,116,332)(31,245,117,331)(32,244,118,330)(33,243,119,329)(34,242,120,328)(35,241,121,327)(36,240,122,326)(37,239,123,325)(38,238,124,324)(39,237,125,323)(40,236,126,322)(41,235,127,321)(42,234,128,320)(43,233,129,319)(44,232,130,318)(45,231,131,317)(46,230,132,316)(47,229,133,315)(48,228,134,314)(49,227,135,313)(50,226,136,312)(51,225,137,311)(52,224,138,310)(53,223,139,309)(54,222,140,308)(55,221,141,307)(56,220,142,306)(57,219,143,305)(58,218,144,304)(59,217,145,303)(60,216,146,302)(61,215,147,301)(62,214,148,300)(63,213,149,299)(64,212,150,298)(65,211,151,297)(66,210,152,296)(67,209,153,295)(68,208,154,294)(69,207,155,293)(70,206,156,292)(71,205,157,291)(72,204,158,290)(73,203,159,289)(74,202,160,288)(75,201,161,287)(76,200,162,286)(77,199,163,285)(78,198,164,284)(79,197,165,283)(80,196,166,282)(81,195,167,281)(82,194,168,280)(83,193,169,279)(84,192,170,278)(85,191,171,277)(86,190,172,276)>;
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), (1,275,87,189)(2,274,88,188)(3,273,89,187)(4,272,90,186)(5,271,91,185)(6,270,92,184)(7,269,93,183)(8,268,94,182)(9,267,95,181)(10,266,96,180)(11,265,97,179)(12,264,98,178)(13,263,99,177)(14,262,100,176)(15,261,101,175)(16,260,102,174)(17,259,103,173)(18,258,104,344)(19,257,105,343)(20,256,106,342)(21,255,107,341)(22,254,108,340)(23,253,109,339)(24,252,110,338)(25,251,111,337)(26,250,112,336)(27,249,113,335)(28,248,114,334)(29,247,115,333)(30,246,116,332)(31,245,117,331)(32,244,118,330)(33,243,119,329)(34,242,120,328)(35,241,121,327)(36,240,122,326)(37,239,123,325)(38,238,124,324)(39,237,125,323)(40,236,126,322)(41,235,127,321)(42,234,128,320)(43,233,129,319)(44,232,130,318)(45,231,131,317)(46,230,132,316)(47,229,133,315)(48,228,134,314)(49,227,135,313)(50,226,136,312)(51,225,137,311)(52,224,138,310)(53,223,139,309)(54,222,140,308)(55,221,141,307)(56,220,142,306)(57,219,143,305)(58,218,144,304)(59,217,145,303)(60,216,146,302)(61,215,147,301)(62,214,148,300)(63,213,149,299)(64,212,150,298)(65,211,151,297)(66,210,152,296)(67,209,153,295)(68,208,154,294)(69,207,155,293)(70,206,156,292)(71,205,157,291)(72,204,158,290)(73,203,159,289)(74,202,160,288)(75,201,161,287)(76,200,162,286)(77,199,163,285)(78,198,164,284)(79,197,165,283)(80,196,166,282)(81,195,167,281)(82,194,168,280)(83,193,169,279)(84,192,170,278)(85,191,171,277)(86,190,172,276) );
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)], [(1,275,87,189),(2,274,88,188),(3,273,89,187),(4,272,90,186),(5,271,91,185),(6,270,92,184),(7,269,93,183),(8,268,94,182),(9,267,95,181),(10,266,96,180),(11,265,97,179),(12,264,98,178),(13,263,99,177),(14,262,100,176),(15,261,101,175),(16,260,102,174),(17,259,103,173),(18,258,104,344),(19,257,105,343),(20,256,106,342),(21,255,107,341),(22,254,108,340),(23,253,109,339),(24,252,110,338),(25,251,111,337),(26,250,112,336),(27,249,113,335),(28,248,114,334),(29,247,115,333),(30,246,116,332),(31,245,117,331),(32,244,118,330),(33,243,119,329),(34,242,120,328),(35,241,121,327),(36,240,122,326),(37,239,123,325),(38,238,124,324),(39,237,125,323),(40,236,126,322),(41,235,127,321),(42,234,128,320),(43,233,129,319),(44,232,130,318),(45,231,131,317),(46,230,132,316),(47,229,133,315),(48,228,134,314),(49,227,135,313),(50,226,136,312),(51,225,137,311),(52,224,138,310),(53,223,139,309),(54,222,140,308),(55,221,141,307),(56,220,142,306),(57,219,143,305),(58,218,144,304),(59,217,145,303),(60,216,146,302),(61,215,147,301),(62,214,148,300),(63,213,149,299),(64,212,150,298),(65,211,151,297),(66,210,152,296),(67,209,153,295),(68,208,154,294),(69,207,155,293),(70,206,156,292),(71,205,157,291),(72,204,158,290),(73,203,159,289),(74,202,160,288),(75,201,161,287),(76,200,162,286),(77,199,163,285),(78,198,164,284),(79,197,165,283),(80,196,166,282),(81,195,167,281),(82,194,168,280),(83,193,169,279),(84,192,170,278),(85,191,171,277),(86,190,172,276)]])
89 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 43A | ··· | 43U | 86A | ··· | 86U | 172A | ··· | 172AP |
| order | 1 | 2 | 4 | 4 | 4 | 43 | ··· | 43 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 2 | 86 | 86 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
89 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | + | - |
| image | C1 | C2 | C2 | Q8 | D43 | D86 | Dic86 |
| kernel | Dic86 | Dic43 | C172 | C43 | C4 | C2 | C1 |
| # reps | 1 | 2 | 1 | 1 | 21 | 21 | 42 |
Matrix representation of Dic86 ►in GL2(𝔽173) generated by
| 19 | 116 |
| 57 | 84 |
| 32 | 53 |
| 59 | 141 |
G:=sub<GL(2,GF(173))| [19,57,116,84],[32,59,53,141] >;
Dic86 in GAP, Magma, Sage, TeX
{\rm Dic}_{86} % in TeX
G:=Group("Dic86"); // GroupNames label
G:=SmallGroup(344,3);
// by ID
G=gap.SmallGroup(344,3);
# by ID
G:=PCGroup([4,-2,-2,-2,-43,16,49,21,5379]);
// Polycyclic
G:=Group<a,b|a^172=1,b^2=a^86,b*a*b^-1=a^-1>;
// generators/relations
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