metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic93, C93⋊1C4, C62.S3, C6.D31, C2.D93, C3⋊Dic31, C31⋊Dic3, C186.1C2, SmallGroup(372,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C93 — Dic93 |
Generators and relations for Dic93
G = < a,b | a186=1, b2=a93, bab-1=a-1 >
Smallest permutation representation of Dic93
►Regular action on 372 points
Generators in S372
(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)
(1 230 94 323)(2 229 95 322)(3 228 96 321)(4 227 97 320)(5 226 98 319)(6 225 99 318)(7 224 100 317)(8 223 101 316)(9 222 102 315)(10 221 103 314)(11 220 104 313)(12 219 105 312)(13 218 106 311)(14 217 107 310)(15 216 108 309)(16 215 109 308)(17 214 110 307)(18 213 111 306)(19 212 112 305)(20 211 113 304)(21 210 114 303)(22 209 115 302)(23 208 116 301)(24 207 117 300)(25 206 118 299)(26 205 119 298)(27 204 120 297)(28 203 121 296)(29 202 122 295)(30 201 123 294)(31 200 124 293)(32 199 125 292)(33 198 126 291)(34 197 127 290)(35 196 128 289)(36 195 129 288)(37 194 130 287)(38 193 131 286)(39 192 132 285)(40 191 133 284)(41 190 134 283)(42 189 135 282)(43 188 136 281)(44 187 137 280)(45 372 138 279)(46 371 139 278)(47 370 140 277)(48 369 141 276)(49 368 142 275)(50 367 143 274)(51 366 144 273)(52 365 145 272)(53 364 146 271)(54 363 147 270)(55 362 148 269)(56 361 149 268)(57 360 150 267)(58 359 151 266)(59 358 152 265)(60 357 153 264)(61 356 154 263)(62 355 155 262)(63 354 156 261)(64 353 157 260)(65 352 158 259)(66 351 159 258)(67 350 160 257)(68 349 161 256)(69 348 162 255)(70 347 163 254)(71 346 164 253)(72 345 165 252)(73 344 166 251)(74 343 167 250)(75 342 168 249)(76 341 169 248)(77 340 170 247)(78 339 171 246)(79 338 172 245)(80 337 173 244)(81 336 174 243)(82 335 175 242)(83 334 176 241)(84 333 177 240)(85 332 178 239)(86 331 179 238)(87 330 180 237)(88 329 181 236)(89 328 182 235)(90 327 183 234)(91 326 184 233)(92 325 185 232)(93 324 186 231)
G:=sub<Sym(372)| (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), (1,230,94,323)(2,229,95,322)(3,228,96,321)(4,227,97,320)(5,226,98,319)(6,225,99,318)(7,224,100,317)(8,223,101,316)(9,222,102,315)(10,221,103,314)(11,220,104,313)(12,219,105,312)(13,218,106,311)(14,217,107,310)(15,216,108,309)(16,215,109,308)(17,214,110,307)(18,213,111,306)(19,212,112,305)(20,211,113,304)(21,210,114,303)(22,209,115,302)(23,208,116,301)(24,207,117,300)(25,206,118,299)(26,205,119,298)(27,204,120,297)(28,203,121,296)(29,202,122,295)(30,201,123,294)(31,200,124,293)(32,199,125,292)(33,198,126,291)(34,197,127,290)(35,196,128,289)(36,195,129,288)(37,194,130,287)(38,193,131,286)(39,192,132,285)(40,191,133,284)(41,190,134,283)(42,189,135,282)(43,188,136,281)(44,187,137,280)(45,372,138,279)(46,371,139,278)(47,370,140,277)(48,369,141,276)(49,368,142,275)(50,367,143,274)(51,366,144,273)(52,365,145,272)(53,364,146,271)(54,363,147,270)(55,362,148,269)(56,361,149,268)(57,360,150,267)(58,359,151,266)(59,358,152,265)(60,357,153,264)(61,356,154,263)(62,355,155,262)(63,354,156,261)(64,353,157,260)(65,352,158,259)(66,351,159,258)(67,350,160,257)(68,349,161,256)(69,348,162,255)(70,347,163,254)(71,346,164,253)(72,345,165,252)(73,344,166,251)(74,343,167,250)(75,342,168,249)(76,341,169,248)(77,340,170,247)(78,339,171,246)(79,338,172,245)(80,337,173,244)(81,336,174,243)(82,335,175,242)(83,334,176,241)(84,333,177,240)(85,332,178,239)(86,331,179,238)(87,330,180,237)(88,329,181,236)(89,328,182,235)(90,327,183,234)(91,326,184,233)(92,325,185,232)(93,324,186,231)>;
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), (1,230,94,323)(2,229,95,322)(3,228,96,321)(4,227,97,320)(5,226,98,319)(6,225,99,318)(7,224,100,317)(8,223,101,316)(9,222,102,315)(10,221,103,314)(11,220,104,313)(12,219,105,312)(13,218,106,311)(14,217,107,310)(15,216,108,309)(16,215,109,308)(17,214,110,307)(18,213,111,306)(19,212,112,305)(20,211,113,304)(21,210,114,303)(22,209,115,302)(23,208,116,301)(24,207,117,300)(25,206,118,299)(26,205,119,298)(27,204,120,297)(28,203,121,296)(29,202,122,295)(30,201,123,294)(31,200,124,293)(32,199,125,292)(33,198,126,291)(34,197,127,290)(35,196,128,289)(36,195,129,288)(37,194,130,287)(38,193,131,286)(39,192,132,285)(40,191,133,284)(41,190,134,283)(42,189,135,282)(43,188,136,281)(44,187,137,280)(45,372,138,279)(46,371,139,278)(47,370,140,277)(48,369,141,276)(49,368,142,275)(50,367,143,274)(51,366,144,273)(52,365,145,272)(53,364,146,271)(54,363,147,270)(55,362,148,269)(56,361,149,268)(57,360,150,267)(58,359,151,266)(59,358,152,265)(60,357,153,264)(61,356,154,263)(62,355,155,262)(63,354,156,261)(64,353,157,260)(65,352,158,259)(66,351,159,258)(67,350,160,257)(68,349,161,256)(69,348,162,255)(70,347,163,254)(71,346,164,253)(72,345,165,252)(73,344,166,251)(74,343,167,250)(75,342,168,249)(76,341,169,248)(77,340,170,247)(78,339,171,246)(79,338,172,245)(80,337,173,244)(81,336,174,243)(82,335,175,242)(83,334,176,241)(84,333,177,240)(85,332,178,239)(86,331,179,238)(87,330,180,237)(88,329,181,236)(89,328,182,235)(90,327,183,234)(91,326,184,233)(92,325,185,232)(93,324,186,231) );
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)], [(1,230,94,323),(2,229,95,322),(3,228,96,321),(4,227,97,320),(5,226,98,319),(6,225,99,318),(7,224,100,317),(8,223,101,316),(9,222,102,315),(10,221,103,314),(11,220,104,313),(12,219,105,312),(13,218,106,311),(14,217,107,310),(15,216,108,309),(16,215,109,308),(17,214,110,307),(18,213,111,306),(19,212,112,305),(20,211,113,304),(21,210,114,303),(22,209,115,302),(23,208,116,301),(24,207,117,300),(25,206,118,299),(26,205,119,298),(27,204,120,297),(28,203,121,296),(29,202,122,295),(30,201,123,294),(31,200,124,293),(32,199,125,292),(33,198,126,291),(34,197,127,290),(35,196,128,289),(36,195,129,288),(37,194,130,287),(38,193,131,286),(39,192,132,285),(40,191,133,284),(41,190,134,283),(42,189,135,282),(43,188,136,281),(44,187,137,280),(45,372,138,279),(46,371,139,278),(47,370,140,277),(48,369,141,276),(49,368,142,275),(50,367,143,274),(51,366,144,273),(52,365,145,272),(53,364,146,271),(54,363,147,270),(55,362,148,269),(56,361,149,268),(57,360,150,267),(58,359,151,266),(59,358,152,265),(60,357,153,264),(61,356,154,263),(62,355,155,262),(63,354,156,261),(64,353,157,260),(65,352,158,259),(66,351,159,258),(67,350,160,257),(68,349,161,256),(69,348,162,255),(70,347,163,254),(71,346,164,253),(72,345,165,252),(73,344,166,251),(74,343,167,250),(75,342,168,249),(76,341,169,248),(77,340,170,247),(78,339,171,246),(79,338,172,245),(80,337,173,244),(81,336,174,243),(82,335,175,242),(83,334,176,241),(84,333,177,240),(85,332,178,239),(86,331,179,238),(87,330,180,237),(88,329,181,236),(89,328,182,235),(90,327,183,234),(91,326,184,233),(92,325,185,232),(93,324,186,231)]])
96 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 31A | ··· | 31O | 62A | ··· | 62O | 93A | ··· | 93AD | 186A | ··· | 186AD |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 | 186 | ··· | 186 |
| size | 1 | 1 | 2 | 93 | 93 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |
| image | C1 | C2 | C4 | S3 | Dic3 | D31 | Dic31 | D93 | Dic93 |
| kernel | Dic93 | C186 | C93 | C62 | C31 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 1 | 15 | 15 | 30 | 30 |
Matrix representation of Dic93 ►in GL2(𝔽373) generated by
| 83 | 175 |
| 198 | 31 |
| 81 | 147 |
| 270 | 292 |
G:=sub<GL(2,GF(373))| [83,198,175,31],[81,270,147,292] >;
Dic93 in GAP, Magma, Sage, TeX
{\rm Dic}_{93} % in TeX
G:=Group("Dic93"); // GroupNames label
G:=SmallGroup(372,5);
// by ID
G=gap.SmallGroup(372,5);
# by ID
G:=PCGroup([4,-2,-2,-3,-31,8,98,5763]);
// Polycyclic
G:=Group<a,b|a^186=1,b^2=a^93,b*a*b^-1=a^-1>;
// generators/relations
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