metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: Dic92, C8.D23, C23⋊1Q16, C46.3D4, C2.5D92, C184.1C2, C4.10D46, C92.10C22, Dic46.1C2, SmallGroup(368,7)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic92
G = < a,b | a184=1, b2=a92, bab-1=a-1 >
Smallest permutation representation of Dic92
►Regular action on 368 points
Generators in S368
(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)
(1 355 93 263)(2 354 94 262)(3 353 95 261)(4 352 96 260)(5 351 97 259)(6 350 98 258)(7 349 99 257)(8 348 100 256)(9 347 101 255)(10 346 102 254)(11 345 103 253)(12 344 104 252)(13 343 105 251)(14 342 106 250)(15 341 107 249)(16 340 108 248)(17 339 109 247)(18 338 110 246)(19 337 111 245)(20 336 112 244)(21 335 113 243)(22 334 114 242)(23 333 115 241)(24 332 116 240)(25 331 117 239)(26 330 118 238)(27 329 119 237)(28 328 120 236)(29 327 121 235)(30 326 122 234)(31 325 123 233)(32 324 124 232)(33 323 125 231)(34 322 126 230)(35 321 127 229)(36 320 128 228)(37 319 129 227)(38 318 130 226)(39 317 131 225)(40 316 132 224)(41 315 133 223)(42 314 134 222)(43 313 135 221)(44 312 136 220)(45 311 137 219)(46 310 138 218)(47 309 139 217)(48 308 140 216)(49 307 141 215)(50 306 142 214)(51 305 143 213)(52 304 144 212)(53 303 145 211)(54 302 146 210)(55 301 147 209)(56 300 148 208)(57 299 149 207)(58 298 150 206)(59 297 151 205)(60 296 152 204)(61 295 153 203)(62 294 154 202)(63 293 155 201)(64 292 156 200)(65 291 157 199)(66 290 158 198)(67 289 159 197)(68 288 160 196)(69 287 161 195)(70 286 162 194)(71 285 163 193)(72 284 164 192)(73 283 165 191)(74 282 166 190)(75 281 167 189)(76 280 168 188)(77 279 169 187)(78 278 170 186)(79 277 171 185)(80 276 172 368)(81 275 173 367)(82 274 174 366)(83 273 175 365)(84 272 176 364)(85 271 177 363)(86 270 178 362)(87 269 179 361)(88 268 180 360)(89 267 181 359)(90 266 182 358)(91 265 183 357)(92 264 184 356)
G:=sub<Sym(368)| (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), (1,355,93,263)(2,354,94,262)(3,353,95,261)(4,352,96,260)(5,351,97,259)(6,350,98,258)(7,349,99,257)(8,348,100,256)(9,347,101,255)(10,346,102,254)(11,345,103,253)(12,344,104,252)(13,343,105,251)(14,342,106,250)(15,341,107,249)(16,340,108,248)(17,339,109,247)(18,338,110,246)(19,337,111,245)(20,336,112,244)(21,335,113,243)(22,334,114,242)(23,333,115,241)(24,332,116,240)(25,331,117,239)(26,330,118,238)(27,329,119,237)(28,328,120,236)(29,327,121,235)(30,326,122,234)(31,325,123,233)(32,324,124,232)(33,323,125,231)(34,322,126,230)(35,321,127,229)(36,320,128,228)(37,319,129,227)(38,318,130,226)(39,317,131,225)(40,316,132,224)(41,315,133,223)(42,314,134,222)(43,313,135,221)(44,312,136,220)(45,311,137,219)(46,310,138,218)(47,309,139,217)(48,308,140,216)(49,307,141,215)(50,306,142,214)(51,305,143,213)(52,304,144,212)(53,303,145,211)(54,302,146,210)(55,301,147,209)(56,300,148,208)(57,299,149,207)(58,298,150,206)(59,297,151,205)(60,296,152,204)(61,295,153,203)(62,294,154,202)(63,293,155,201)(64,292,156,200)(65,291,157,199)(66,290,158,198)(67,289,159,197)(68,288,160,196)(69,287,161,195)(70,286,162,194)(71,285,163,193)(72,284,164,192)(73,283,165,191)(74,282,166,190)(75,281,167,189)(76,280,168,188)(77,279,169,187)(78,278,170,186)(79,277,171,185)(80,276,172,368)(81,275,173,367)(82,274,174,366)(83,273,175,365)(84,272,176,364)(85,271,177,363)(86,270,178,362)(87,269,179,361)(88,268,180,360)(89,267,181,359)(90,266,182,358)(91,265,183,357)(92,264,184,356)>;
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), (1,355,93,263)(2,354,94,262)(3,353,95,261)(4,352,96,260)(5,351,97,259)(6,350,98,258)(7,349,99,257)(8,348,100,256)(9,347,101,255)(10,346,102,254)(11,345,103,253)(12,344,104,252)(13,343,105,251)(14,342,106,250)(15,341,107,249)(16,340,108,248)(17,339,109,247)(18,338,110,246)(19,337,111,245)(20,336,112,244)(21,335,113,243)(22,334,114,242)(23,333,115,241)(24,332,116,240)(25,331,117,239)(26,330,118,238)(27,329,119,237)(28,328,120,236)(29,327,121,235)(30,326,122,234)(31,325,123,233)(32,324,124,232)(33,323,125,231)(34,322,126,230)(35,321,127,229)(36,320,128,228)(37,319,129,227)(38,318,130,226)(39,317,131,225)(40,316,132,224)(41,315,133,223)(42,314,134,222)(43,313,135,221)(44,312,136,220)(45,311,137,219)(46,310,138,218)(47,309,139,217)(48,308,140,216)(49,307,141,215)(50,306,142,214)(51,305,143,213)(52,304,144,212)(53,303,145,211)(54,302,146,210)(55,301,147,209)(56,300,148,208)(57,299,149,207)(58,298,150,206)(59,297,151,205)(60,296,152,204)(61,295,153,203)(62,294,154,202)(63,293,155,201)(64,292,156,200)(65,291,157,199)(66,290,158,198)(67,289,159,197)(68,288,160,196)(69,287,161,195)(70,286,162,194)(71,285,163,193)(72,284,164,192)(73,283,165,191)(74,282,166,190)(75,281,167,189)(76,280,168,188)(77,279,169,187)(78,278,170,186)(79,277,171,185)(80,276,172,368)(81,275,173,367)(82,274,174,366)(83,273,175,365)(84,272,176,364)(85,271,177,363)(86,270,178,362)(87,269,179,361)(88,268,180,360)(89,267,181,359)(90,266,182,358)(91,265,183,357)(92,264,184,356) );
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)], [(1,355,93,263),(2,354,94,262),(3,353,95,261),(4,352,96,260),(5,351,97,259),(6,350,98,258),(7,349,99,257),(8,348,100,256),(9,347,101,255),(10,346,102,254),(11,345,103,253),(12,344,104,252),(13,343,105,251),(14,342,106,250),(15,341,107,249),(16,340,108,248),(17,339,109,247),(18,338,110,246),(19,337,111,245),(20,336,112,244),(21,335,113,243),(22,334,114,242),(23,333,115,241),(24,332,116,240),(25,331,117,239),(26,330,118,238),(27,329,119,237),(28,328,120,236),(29,327,121,235),(30,326,122,234),(31,325,123,233),(32,324,124,232),(33,323,125,231),(34,322,126,230),(35,321,127,229),(36,320,128,228),(37,319,129,227),(38,318,130,226),(39,317,131,225),(40,316,132,224),(41,315,133,223),(42,314,134,222),(43,313,135,221),(44,312,136,220),(45,311,137,219),(46,310,138,218),(47,309,139,217),(48,308,140,216),(49,307,141,215),(50,306,142,214),(51,305,143,213),(52,304,144,212),(53,303,145,211),(54,302,146,210),(55,301,147,209),(56,300,148,208),(57,299,149,207),(58,298,150,206),(59,297,151,205),(60,296,152,204),(61,295,153,203),(62,294,154,202),(63,293,155,201),(64,292,156,200),(65,291,157,199),(66,290,158,198),(67,289,159,197),(68,288,160,196),(69,287,161,195),(70,286,162,194),(71,285,163,193),(72,284,164,192),(73,283,165,191),(74,282,166,190),(75,281,167,189),(76,280,168,188),(77,279,169,187),(78,278,170,186),(79,277,171,185),(80,276,172,368),(81,275,173,367),(82,274,174,366),(83,273,175,365),(84,272,176,364),(85,271,177,363),(86,270,178,362),(87,269,179,361),(88,268,180,360),(89,267,181,359),(90,266,182,358),(91,265,183,357),(92,264,184,356)]])
95 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 8A | 8B | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V | 184A | ··· | 184AR |
| order | 1 | 2 | 4 | 4 | 4 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 2 | 92 | 92 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
95 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | + | - |
| image | C1 | C2 | C2 | D4 | Q16 | D23 | D46 | D92 | Dic92 |
| kernel | Dic92 | C184 | Dic46 | C46 | C23 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 11 | 11 | 22 | 44 |
Matrix representation of Dic92 ►in GL4(𝔽1289) generated by
| 161 | 1 | 0 | 0 |
| 769 | 285 | 0 | 0 |
| 0 | 0 | 1083 | 1120 |
| 0 | 0 | 1122 | 301 |
| 506 | 1230 | 0 | 0 |
| 1128 | 783 | 0 | 0 |
| 0 | 0 | 189 | 81 |
| 0 | 0 | 1214 | 1100 |
G:=sub<GL(4,GF(1289))| [161,769,0,0,1,285,0,0,0,0,1083,1122,0,0,1120,301],[506,1128,0,0,1230,783,0,0,0,0,189,1214,0,0,81,1100] >;
Dic92 in GAP, Magma, Sage, TeX
{\rm Dic}_{92} % in TeX
G:=Group("Dic92"); // GroupNames label
G:=SmallGroup(368,7);
// by ID
G=gap.SmallGroup(368,7);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,40,61,66,182,42,8804]);
// Polycyclic
G:=Group<a,b|a^184=1,b^2=a^92,b*a*b^-1=a^-1>;
// generators/relations
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