metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic89, C89⋊2C4, C2.D89, C178.C2, SmallGroup(356,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C89 — Dic89 |
Generators and relations for Dic89
G = < a,b | a178=1, b2=a89, bab-1=a-1 >
Smallest permutation representation of Dic89
►Regular action on 356 points
Generators in S356
(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)
(1 301 90 212)(2 300 91 211)(3 299 92 210)(4 298 93 209)(5 297 94 208)(6 296 95 207)(7 295 96 206)(8 294 97 205)(9 293 98 204)(10 292 99 203)(11 291 100 202)(12 290 101 201)(13 289 102 200)(14 288 103 199)(15 287 104 198)(16 286 105 197)(17 285 106 196)(18 284 107 195)(19 283 108 194)(20 282 109 193)(21 281 110 192)(22 280 111 191)(23 279 112 190)(24 278 113 189)(25 277 114 188)(26 276 115 187)(27 275 116 186)(28 274 117 185)(29 273 118 184)(30 272 119 183)(31 271 120 182)(32 270 121 181)(33 269 122 180)(34 268 123 179)(35 267 124 356)(36 266 125 355)(37 265 126 354)(38 264 127 353)(39 263 128 352)(40 262 129 351)(41 261 130 350)(42 260 131 349)(43 259 132 348)(44 258 133 347)(45 257 134 346)(46 256 135 345)(47 255 136 344)(48 254 137 343)(49 253 138 342)(50 252 139 341)(51 251 140 340)(52 250 141 339)(53 249 142 338)(54 248 143 337)(55 247 144 336)(56 246 145 335)(57 245 146 334)(58 244 147 333)(59 243 148 332)(60 242 149 331)(61 241 150 330)(62 240 151 329)(63 239 152 328)(64 238 153 327)(65 237 154 326)(66 236 155 325)(67 235 156 324)(68 234 157 323)(69 233 158 322)(70 232 159 321)(71 231 160 320)(72 230 161 319)(73 229 162 318)(74 228 163 317)(75 227 164 316)(76 226 165 315)(77 225 166 314)(78 224 167 313)(79 223 168 312)(80 222 169 311)(81 221 170 310)(82 220 171 309)(83 219 172 308)(84 218 173 307)(85 217 174 306)(86 216 175 305)(87 215 176 304)(88 214 177 303)(89 213 178 302)
G:=sub<Sym(356)| (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), (1,301,90,212)(2,300,91,211)(3,299,92,210)(4,298,93,209)(5,297,94,208)(6,296,95,207)(7,295,96,206)(8,294,97,205)(9,293,98,204)(10,292,99,203)(11,291,100,202)(12,290,101,201)(13,289,102,200)(14,288,103,199)(15,287,104,198)(16,286,105,197)(17,285,106,196)(18,284,107,195)(19,283,108,194)(20,282,109,193)(21,281,110,192)(22,280,111,191)(23,279,112,190)(24,278,113,189)(25,277,114,188)(26,276,115,187)(27,275,116,186)(28,274,117,185)(29,273,118,184)(30,272,119,183)(31,271,120,182)(32,270,121,181)(33,269,122,180)(34,268,123,179)(35,267,124,356)(36,266,125,355)(37,265,126,354)(38,264,127,353)(39,263,128,352)(40,262,129,351)(41,261,130,350)(42,260,131,349)(43,259,132,348)(44,258,133,347)(45,257,134,346)(46,256,135,345)(47,255,136,344)(48,254,137,343)(49,253,138,342)(50,252,139,341)(51,251,140,340)(52,250,141,339)(53,249,142,338)(54,248,143,337)(55,247,144,336)(56,246,145,335)(57,245,146,334)(58,244,147,333)(59,243,148,332)(60,242,149,331)(61,241,150,330)(62,240,151,329)(63,239,152,328)(64,238,153,327)(65,237,154,326)(66,236,155,325)(67,235,156,324)(68,234,157,323)(69,233,158,322)(70,232,159,321)(71,231,160,320)(72,230,161,319)(73,229,162,318)(74,228,163,317)(75,227,164,316)(76,226,165,315)(77,225,166,314)(78,224,167,313)(79,223,168,312)(80,222,169,311)(81,221,170,310)(82,220,171,309)(83,219,172,308)(84,218,173,307)(85,217,174,306)(86,216,175,305)(87,215,176,304)(88,214,177,303)(89,213,178,302)>;
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), (1,301,90,212)(2,300,91,211)(3,299,92,210)(4,298,93,209)(5,297,94,208)(6,296,95,207)(7,295,96,206)(8,294,97,205)(9,293,98,204)(10,292,99,203)(11,291,100,202)(12,290,101,201)(13,289,102,200)(14,288,103,199)(15,287,104,198)(16,286,105,197)(17,285,106,196)(18,284,107,195)(19,283,108,194)(20,282,109,193)(21,281,110,192)(22,280,111,191)(23,279,112,190)(24,278,113,189)(25,277,114,188)(26,276,115,187)(27,275,116,186)(28,274,117,185)(29,273,118,184)(30,272,119,183)(31,271,120,182)(32,270,121,181)(33,269,122,180)(34,268,123,179)(35,267,124,356)(36,266,125,355)(37,265,126,354)(38,264,127,353)(39,263,128,352)(40,262,129,351)(41,261,130,350)(42,260,131,349)(43,259,132,348)(44,258,133,347)(45,257,134,346)(46,256,135,345)(47,255,136,344)(48,254,137,343)(49,253,138,342)(50,252,139,341)(51,251,140,340)(52,250,141,339)(53,249,142,338)(54,248,143,337)(55,247,144,336)(56,246,145,335)(57,245,146,334)(58,244,147,333)(59,243,148,332)(60,242,149,331)(61,241,150,330)(62,240,151,329)(63,239,152,328)(64,238,153,327)(65,237,154,326)(66,236,155,325)(67,235,156,324)(68,234,157,323)(69,233,158,322)(70,232,159,321)(71,231,160,320)(72,230,161,319)(73,229,162,318)(74,228,163,317)(75,227,164,316)(76,226,165,315)(77,225,166,314)(78,224,167,313)(79,223,168,312)(80,222,169,311)(81,221,170,310)(82,220,171,309)(83,219,172,308)(84,218,173,307)(85,217,174,306)(86,216,175,305)(87,215,176,304)(88,214,177,303)(89,213,178,302) );
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)], [(1,301,90,212),(2,300,91,211),(3,299,92,210),(4,298,93,209),(5,297,94,208),(6,296,95,207),(7,295,96,206),(8,294,97,205),(9,293,98,204),(10,292,99,203),(11,291,100,202),(12,290,101,201),(13,289,102,200),(14,288,103,199),(15,287,104,198),(16,286,105,197),(17,285,106,196),(18,284,107,195),(19,283,108,194),(20,282,109,193),(21,281,110,192),(22,280,111,191),(23,279,112,190),(24,278,113,189),(25,277,114,188),(26,276,115,187),(27,275,116,186),(28,274,117,185),(29,273,118,184),(30,272,119,183),(31,271,120,182),(32,270,121,181),(33,269,122,180),(34,268,123,179),(35,267,124,356),(36,266,125,355),(37,265,126,354),(38,264,127,353),(39,263,128,352),(40,262,129,351),(41,261,130,350),(42,260,131,349),(43,259,132,348),(44,258,133,347),(45,257,134,346),(46,256,135,345),(47,255,136,344),(48,254,137,343),(49,253,138,342),(50,252,139,341),(51,251,140,340),(52,250,141,339),(53,249,142,338),(54,248,143,337),(55,247,144,336),(56,246,145,335),(57,245,146,334),(58,244,147,333),(59,243,148,332),(60,242,149,331),(61,241,150,330),(62,240,151,329),(63,239,152,328),(64,238,153,327),(65,237,154,326),(66,236,155,325),(67,235,156,324),(68,234,157,323),(69,233,158,322),(70,232,159,321),(71,231,160,320),(72,230,161,319),(73,229,162,318),(74,228,163,317),(75,227,164,316),(76,226,165,315),(77,225,166,314),(78,224,167,313),(79,223,168,312),(80,222,169,311),(81,221,170,310),(82,220,171,309),(83,219,172,308),(84,218,173,307),(85,217,174,306),(86,216,175,305),(87,215,176,304),(88,214,177,303),(89,213,178,302)]])
92 conjugacy classes
| class | 1 | 2 | 4A | 4B | 89A | ··· | 89AR | 178A | ··· | 178AR |
| order | 1 | 2 | 4 | 4 | 89 | ··· | 89 | 178 | ··· | 178 |
| size | 1 | 1 | 89 | 89 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | - | |
| image | C1 | C2 | C4 | D89 | Dic89 |
| kernel | Dic89 | C178 | C89 | C2 | C1 |
| # reps | 1 | 1 | 2 | 44 | 44 |
Matrix representation of Dic89 ►in GL2(𝔽1069) generated by
| 11 | 1 |
| 1068 | 0 |
| 487 | 430 |
| 418 | 582 |
G:=sub<GL(2,GF(1069))| [11,1068,1,0],[487,418,430,582] >;
Dic89 in GAP, Magma, Sage, TeX
{\rm Dic}_{89} % in TeX
G:=Group("Dic89"); // GroupNames label
G:=SmallGroup(356,1);
// by ID
G=gap.SmallGroup(356,1);
# by ID
G:=PCGroup([3,-2,-2,-89,6,3170]);
// Polycyclic
G:=Group<a,b|a^178=1,b^2=a^89,b*a*b^-1=a^-1>;
// generators/relations
Export