metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic87, C87⋊3C4, C58.S3, C2.D87, C6.D29, C3⋊Dic29, C29⋊2Dic3, C174.1C2, SmallGroup(348,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C87 — Dic87 |
Generators and relations for Dic87
G = < a,b | a174=1, b2=a87, bab-1=a-1 >
Smallest permutation representation of Dic87
►Regular action on 348 points
Generators in S348
(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)
(1 293 88 206)(2 292 89 205)(3 291 90 204)(4 290 91 203)(5 289 92 202)(6 288 93 201)(7 287 94 200)(8 286 95 199)(9 285 96 198)(10 284 97 197)(11 283 98 196)(12 282 99 195)(13 281 100 194)(14 280 101 193)(15 279 102 192)(16 278 103 191)(17 277 104 190)(18 276 105 189)(19 275 106 188)(20 274 107 187)(21 273 108 186)(22 272 109 185)(23 271 110 184)(24 270 111 183)(25 269 112 182)(26 268 113 181)(27 267 114 180)(28 266 115 179)(29 265 116 178)(30 264 117 177)(31 263 118 176)(32 262 119 175)(33 261 120 348)(34 260 121 347)(35 259 122 346)(36 258 123 345)(37 257 124 344)(38 256 125 343)(39 255 126 342)(40 254 127 341)(41 253 128 340)(42 252 129 339)(43 251 130 338)(44 250 131 337)(45 249 132 336)(46 248 133 335)(47 247 134 334)(48 246 135 333)(49 245 136 332)(50 244 137 331)(51 243 138 330)(52 242 139 329)(53 241 140 328)(54 240 141 327)(55 239 142 326)(56 238 143 325)(57 237 144 324)(58 236 145 323)(59 235 146 322)(60 234 147 321)(61 233 148 320)(62 232 149 319)(63 231 150 318)(64 230 151 317)(65 229 152 316)(66 228 153 315)(67 227 154 314)(68 226 155 313)(69 225 156 312)(70 224 157 311)(71 223 158 310)(72 222 159 309)(73 221 160 308)(74 220 161 307)(75 219 162 306)(76 218 163 305)(77 217 164 304)(78 216 165 303)(79 215 166 302)(80 214 167 301)(81 213 168 300)(82 212 169 299)(83 211 170 298)(84 210 171 297)(85 209 172 296)(86 208 173 295)(87 207 174 294)
G:=sub<Sym(348)| (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), (1,293,88,206)(2,292,89,205)(3,291,90,204)(4,290,91,203)(5,289,92,202)(6,288,93,201)(7,287,94,200)(8,286,95,199)(9,285,96,198)(10,284,97,197)(11,283,98,196)(12,282,99,195)(13,281,100,194)(14,280,101,193)(15,279,102,192)(16,278,103,191)(17,277,104,190)(18,276,105,189)(19,275,106,188)(20,274,107,187)(21,273,108,186)(22,272,109,185)(23,271,110,184)(24,270,111,183)(25,269,112,182)(26,268,113,181)(27,267,114,180)(28,266,115,179)(29,265,116,178)(30,264,117,177)(31,263,118,176)(32,262,119,175)(33,261,120,348)(34,260,121,347)(35,259,122,346)(36,258,123,345)(37,257,124,344)(38,256,125,343)(39,255,126,342)(40,254,127,341)(41,253,128,340)(42,252,129,339)(43,251,130,338)(44,250,131,337)(45,249,132,336)(46,248,133,335)(47,247,134,334)(48,246,135,333)(49,245,136,332)(50,244,137,331)(51,243,138,330)(52,242,139,329)(53,241,140,328)(54,240,141,327)(55,239,142,326)(56,238,143,325)(57,237,144,324)(58,236,145,323)(59,235,146,322)(60,234,147,321)(61,233,148,320)(62,232,149,319)(63,231,150,318)(64,230,151,317)(65,229,152,316)(66,228,153,315)(67,227,154,314)(68,226,155,313)(69,225,156,312)(70,224,157,311)(71,223,158,310)(72,222,159,309)(73,221,160,308)(74,220,161,307)(75,219,162,306)(76,218,163,305)(77,217,164,304)(78,216,165,303)(79,215,166,302)(80,214,167,301)(81,213,168,300)(82,212,169,299)(83,211,170,298)(84,210,171,297)(85,209,172,296)(86,208,173,295)(87,207,174,294)>;
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), (1,293,88,206)(2,292,89,205)(3,291,90,204)(4,290,91,203)(5,289,92,202)(6,288,93,201)(7,287,94,200)(8,286,95,199)(9,285,96,198)(10,284,97,197)(11,283,98,196)(12,282,99,195)(13,281,100,194)(14,280,101,193)(15,279,102,192)(16,278,103,191)(17,277,104,190)(18,276,105,189)(19,275,106,188)(20,274,107,187)(21,273,108,186)(22,272,109,185)(23,271,110,184)(24,270,111,183)(25,269,112,182)(26,268,113,181)(27,267,114,180)(28,266,115,179)(29,265,116,178)(30,264,117,177)(31,263,118,176)(32,262,119,175)(33,261,120,348)(34,260,121,347)(35,259,122,346)(36,258,123,345)(37,257,124,344)(38,256,125,343)(39,255,126,342)(40,254,127,341)(41,253,128,340)(42,252,129,339)(43,251,130,338)(44,250,131,337)(45,249,132,336)(46,248,133,335)(47,247,134,334)(48,246,135,333)(49,245,136,332)(50,244,137,331)(51,243,138,330)(52,242,139,329)(53,241,140,328)(54,240,141,327)(55,239,142,326)(56,238,143,325)(57,237,144,324)(58,236,145,323)(59,235,146,322)(60,234,147,321)(61,233,148,320)(62,232,149,319)(63,231,150,318)(64,230,151,317)(65,229,152,316)(66,228,153,315)(67,227,154,314)(68,226,155,313)(69,225,156,312)(70,224,157,311)(71,223,158,310)(72,222,159,309)(73,221,160,308)(74,220,161,307)(75,219,162,306)(76,218,163,305)(77,217,164,304)(78,216,165,303)(79,215,166,302)(80,214,167,301)(81,213,168,300)(82,212,169,299)(83,211,170,298)(84,210,171,297)(85,209,172,296)(86,208,173,295)(87,207,174,294) );
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)], [(1,293,88,206),(2,292,89,205),(3,291,90,204),(4,290,91,203),(5,289,92,202),(6,288,93,201),(7,287,94,200),(8,286,95,199),(9,285,96,198),(10,284,97,197),(11,283,98,196),(12,282,99,195),(13,281,100,194),(14,280,101,193),(15,279,102,192),(16,278,103,191),(17,277,104,190),(18,276,105,189),(19,275,106,188),(20,274,107,187),(21,273,108,186),(22,272,109,185),(23,271,110,184),(24,270,111,183),(25,269,112,182),(26,268,113,181),(27,267,114,180),(28,266,115,179),(29,265,116,178),(30,264,117,177),(31,263,118,176),(32,262,119,175),(33,261,120,348),(34,260,121,347),(35,259,122,346),(36,258,123,345),(37,257,124,344),(38,256,125,343),(39,255,126,342),(40,254,127,341),(41,253,128,340),(42,252,129,339),(43,251,130,338),(44,250,131,337),(45,249,132,336),(46,248,133,335),(47,247,134,334),(48,246,135,333),(49,245,136,332),(50,244,137,331),(51,243,138,330),(52,242,139,329),(53,241,140,328),(54,240,141,327),(55,239,142,326),(56,238,143,325),(57,237,144,324),(58,236,145,323),(59,235,146,322),(60,234,147,321),(61,233,148,320),(62,232,149,319),(63,231,150,318),(64,230,151,317),(65,229,152,316),(66,228,153,315),(67,227,154,314),(68,226,155,313),(69,225,156,312),(70,224,157,311),(71,223,158,310),(72,222,159,309),(73,221,160,308),(74,220,161,307),(75,219,162,306),(76,218,163,305),(77,217,164,304),(78,216,165,303),(79,215,166,302),(80,214,167,301),(81,213,168,300),(82,212,169,299),(83,211,170,298),(84,210,171,297),(85,209,172,296),(86,208,173,295),(87,207,174,294)]])
90 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 29A | ··· | 29N | 58A | ··· | 58N | 87A | ··· | 87AB | 174A | ··· | 174AB |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 29 | ··· | 29 | 58 | ··· | 58 | 87 | ··· | 87 | 174 | ··· | 174 |
| size | 1 | 1 | 2 | 87 | 87 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D29 | Dic29 | D87 | Dic87 |
| kernel | Dic87 | C174 | C87 | C58 | C29 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 14 | 14 | 28 | 28 |
Matrix representation of Dic87 ►in GL4(𝔽349) generated by
| 259 | 1 | 0 | 0 |
| 348 | 0 | 0 | 0 |
| 0 | 0 | 77 | 118 |
| 0 | 0 | 317 | 164 |
| 285 | 222 | 0 | 0 |
| 46 | 64 | 0 | 0 |
| 0 | 0 | 75 | 43 |
| 0 | 0 | 64 | 274 |
G:=sub<GL(4,GF(349))| [259,348,0,0,1,0,0,0,0,0,77,317,0,0,118,164],[285,46,0,0,222,64,0,0,0,0,75,64,0,0,43,274] >;
Dic87 in GAP, Magma, Sage, TeX
{\rm Dic}_{87} % in TeX
G:=Group("Dic87"); // GroupNames label
G:=SmallGroup(348,3);
// by ID
G=gap.SmallGroup(348,3);
# by ID
G:=PCGroup([4,-2,-2,-3,-29,8,98,5379]);
// Polycyclic
G:=Group<a,b|a^174=1,b^2=a^87,b*a*b^-1=a^-1>;
// generators/relations
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