direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C3×Dic31, C93⋊2C4, C31⋊3C12, C62.3C6, C6.2D31, C186.2C2, C2.(C3×D31), SmallGroup(372,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C31 — C3×Dic31 |
Generators and relations for C3×Dic31
G = < a,b,c | a3=b62=1, c2=b31, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×Dic31
►Regular action on 372 points
Generators in S372
(1 136 96)(2 137 97)(3 138 98)(4 139 99)(5 140 100)(6 141 101)(7 142 102)(8 143 103)(9 144 104)(10 145 105)(11 146 106)(12 147 107)(13 148 108)(14 149 109)(15 150 110)(16 151 111)(17 152 112)(18 153 113)(19 154 114)(20 155 115)(21 156 116)(22 157 117)(23 158 118)(24 159 119)(25 160 120)(26 161 121)(27 162 122)(28 163 123)(29 164 124)(30 165 63)(31 166 64)(32 167 65)(33 168 66)(34 169 67)(35 170 68)(36 171 69)(37 172 70)(38 173 71)(39 174 72)(40 175 73)(41 176 74)(42 177 75)(43 178 76)(44 179 77)(45 180 78)(46 181 79)(47 182 80)(48 183 81)(49 184 82)(50 185 83)(51 186 84)(52 125 85)(53 126 86)(54 127 87)(55 128 88)(56 129 89)(57 130 90)(58 131 91)(59 132 92)(60 133 93)(61 134 94)(62 135 95)(187 311 249)(188 312 250)(189 313 251)(190 314 252)(191 315 253)(192 316 254)(193 317 255)(194 318 256)(195 319 257)(196 320 258)(197 321 259)(198 322 260)(199 323 261)(200 324 262)(201 325 263)(202 326 264)(203 327 265)(204 328 266)(205 329 267)(206 330 268)(207 331 269)(208 332 270)(209 333 271)(210 334 272)(211 335 273)(212 336 274)(213 337 275)(214 338 276)(215 339 277)(216 340 278)(217 341 279)(218 342 280)(219 343 281)(220 344 282)(221 345 283)(222 346 284)(223 347 285)(224 348 286)(225 349 287)(226 350 288)(227 351 289)(228 352 290)(229 353 291)(230 354 292)(231 355 293)(232 356 294)(233 357 295)(234 358 296)(235 359 297)(236 360 298)(237 361 299)(238 362 300)(239 363 301)(240 364 302)(241 365 303)(242 366 304)(243 367 305)(244 368 306)(245 369 307)(246 370 308)(247 371 309)(248 372 310)
(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)
(1 187 32 218)(2 248 33 217)(3 247 34 216)(4 246 35 215)(5 245 36 214)(6 244 37 213)(7 243 38 212)(8 242 39 211)(9 241 40 210)(10 240 41 209)(11 239 42 208)(12 238 43 207)(13 237 44 206)(14 236 45 205)(15 235 46 204)(16 234 47 203)(17 233 48 202)(18 232 49 201)(19 231 50 200)(20 230 51 199)(21 229 52 198)(22 228 53 197)(23 227 54 196)(24 226 55 195)(25 225 56 194)(26 224 57 193)(27 223 58 192)(28 222 59 191)(29 221 60 190)(30 220 61 189)(31 219 62 188)(63 282 94 251)(64 281 95 250)(65 280 96 249)(66 279 97 310)(67 278 98 309)(68 277 99 308)(69 276 100 307)(70 275 101 306)(71 274 102 305)(72 273 103 304)(73 272 104 303)(74 271 105 302)(75 270 106 301)(76 269 107 300)(77 268 108 299)(78 267 109 298)(79 266 110 297)(80 265 111 296)(81 264 112 295)(82 263 113 294)(83 262 114 293)(84 261 115 292)(85 260 116 291)(86 259 117 290)(87 258 118 289)(88 257 119 288)(89 256 120 287)(90 255 121 286)(91 254 122 285)(92 253 123 284)(93 252 124 283)(125 322 156 353)(126 321 157 352)(127 320 158 351)(128 319 159 350)(129 318 160 349)(130 317 161 348)(131 316 162 347)(132 315 163 346)(133 314 164 345)(134 313 165 344)(135 312 166 343)(136 311 167 342)(137 372 168 341)(138 371 169 340)(139 370 170 339)(140 369 171 338)(141 368 172 337)(142 367 173 336)(143 366 174 335)(144 365 175 334)(145 364 176 333)(146 363 177 332)(147 362 178 331)(148 361 179 330)(149 360 180 329)(150 359 181 328)(151 358 182 327)(152 357 183 326)(153 356 184 325)(154 355 185 324)(155 354 186 323)
G:=sub<Sym(372)| (1,136,96)(2,137,97)(3,138,98)(4,139,99)(5,140,100)(6,141,101)(7,142,102)(8,143,103)(9,144,104)(10,145,105)(11,146,106)(12,147,107)(13,148,108)(14,149,109)(15,150,110)(16,151,111)(17,152,112)(18,153,113)(19,154,114)(20,155,115)(21,156,116)(22,157,117)(23,158,118)(24,159,119)(25,160,120)(26,161,121)(27,162,122)(28,163,123)(29,164,124)(30,165,63)(31,166,64)(32,167,65)(33,168,66)(34,169,67)(35,170,68)(36,171,69)(37,172,70)(38,173,71)(39,174,72)(40,175,73)(41,176,74)(42,177,75)(43,178,76)(44,179,77)(45,180,78)(46,181,79)(47,182,80)(48,183,81)(49,184,82)(50,185,83)(51,186,84)(52,125,85)(53,126,86)(54,127,87)(55,128,88)(56,129,89)(57,130,90)(58,131,91)(59,132,92)(60,133,93)(61,134,94)(62,135,95)(187,311,249)(188,312,250)(189,313,251)(190,314,252)(191,315,253)(192,316,254)(193,317,255)(194,318,256)(195,319,257)(196,320,258)(197,321,259)(198,322,260)(199,323,261)(200,324,262)(201,325,263)(202,326,264)(203,327,265)(204,328,266)(205,329,267)(206,330,268)(207,331,269)(208,332,270)(209,333,271)(210,334,272)(211,335,273)(212,336,274)(213,337,275)(214,338,276)(215,339,277)(216,340,278)(217,341,279)(218,342,280)(219,343,281)(220,344,282)(221,345,283)(222,346,284)(223,347,285)(224,348,286)(225,349,287)(226,350,288)(227,351,289)(228,352,290)(229,353,291)(230,354,292)(231,355,293)(232,356,294)(233,357,295)(234,358,296)(235,359,297)(236,360,298)(237,361,299)(238,362,300)(239,363,301)(240,364,302)(241,365,303)(242,366,304)(243,367,305)(244,368,306)(245,369,307)(246,370,308)(247,371,309)(248,372,310), (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), (1,187,32,218)(2,248,33,217)(3,247,34,216)(4,246,35,215)(5,245,36,214)(6,244,37,213)(7,243,38,212)(8,242,39,211)(9,241,40,210)(10,240,41,209)(11,239,42,208)(12,238,43,207)(13,237,44,206)(14,236,45,205)(15,235,46,204)(16,234,47,203)(17,233,48,202)(18,232,49,201)(19,231,50,200)(20,230,51,199)(21,229,52,198)(22,228,53,197)(23,227,54,196)(24,226,55,195)(25,225,56,194)(26,224,57,193)(27,223,58,192)(28,222,59,191)(29,221,60,190)(30,220,61,189)(31,219,62,188)(63,282,94,251)(64,281,95,250)(65,280,96,249)(66,279,97,310)(67,278,98,309)(68,277,99,308)(69,276,100,307)(70,275,101,306)(71,274,102,305)(72,273,103,304)(73,272,104,303)(74,271,105,302)(75,270,106,301)(76,269,107,300)(77,268,108,299)(78,267,109,298)(79,266,110,297)(80,265,111,296)(81,264,112,295)(82,263,113,294)(83,262,114,293)(84,261,115,292)(85,260,116,291)(86,259,117,290)(87,258,118,289)(88,257,119,288)(89,256,120,287)(90,255,121,286)(91,254,122,285)(92,253,123,284)(93,252,124,283)(125,322,156,353)(126,321,157,352)(127,320,158,351)(128,319,159,350)(129,318,160,349)(130,317,161,348)(131,316,162,347)(132,315,163,346)(133,314,164,345)(134,313,165,344)(135,312,166,343)(136,311,167,342)(137,372,168,341)(138,371,169,340)(139,370,170,339)(140,369,171,338)(141,368,172,337)(142,367,173,336)(143,366,174,335)(144,365,175,334)(145,364,176,333)(146,363,177,332)(147,362,178,331)(148,361,179,330)(149,360,180,329)(150,359,181,328)(151,358,182,327)(152,357,183,326)(153,356,184,325)(154,355,185,324)(155,354,186,323)>;
G:=Group( (1,136,96)(2,137,97)(3,138,98)(4,139,99)(5,140,100)(6,141,101)(7,142,102)(8,143,103)(9,144,104)(10,145,105)(11,146,106)(12,147,107)(13,148,108)(14,149,109)(15,150,110)(16,151,111)(17,152,112)(18,153,113)(19,154,114)(20,155,115)(21,156,116)(22,157,117)(23,158,118)(24,159,119)(25,160,120)(26,161,121)(27,162,122)(28,163,123)(29,164,124)(30,165,63)(31,166,64)(32,167,65)(33,168,66)(34,169,67)(35,170,68)(36,171,69)(37,172,70)(38,173,71)(39,174,72)(40,175,73)(41,176,74)(42,177,75)(43,178,76)(44,179,77)(45,180,78)(46,181,79)(47,182,80)(48,183,81)(49,184,82)(50,185,83)(51,186,84)(52,125,85)(53,126,86)(54,127,87)(55,128,88)(56,129,89)(57,130,90)(58,131,91)(59,132,92)(60,133,93)(61,134,94)(62,135,95)(187,311,249)(188,312,250)(189,313,251)(190,314,252)(191,315,253)(192,316,254)(193,317,255)(194,318,256)(195,319,257)(196,320,258)(197,321,259)(198,322,260)(199,323,261)(200,324,262)(201,325,263)(202,326,264)(203,327,265)(204,328,266)(205,329,267)(206,330,268)(207,331,269)(208,332,270)(209,333,271)(210,334,272)(211,335,273)(212,336,274)(213,337,275)(214,338,276)(215,339,277)(216,340,278)(217,341,279)(218,342,280)(219,343,281)(220,344,282)(221,345,283)(222,346,284)(223,347,285)(224,348,286)(225,349,287)(226,350,288)(227,351,289)(228,352,290)(229,353,291)(230,354,292)(231,355,293)(232,356,294)(233,357,295)(234,358,296)(235,359,297)(236,360,298)(237,361,299)(238,362,300)(239,363,301)(240,364,302)(241,365,303)(242,366,304)(243,367,305)(244,368,306)(245,369,307)(246,370,308)(247,371,309)(248,372,310), (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), (1,187,32,218)(2,248,33,217)(3,247,34,216)(4,246,35,215)(5,245,36,214)(6,244,37,213)(7,243,38,212)(8,242,39,211)(9,241,40,210)(10,240,41,209)(11,239,42,208)(12,238,43,207)(13,237,44,206)(14,236,45,205)(15,235,46,204)(16,234,47,203)(17,233,48,202)(18,232,49,201)(19,231,50,200)(20,230,51,199)(21,229,52,198)(22,228,53,197)(23,227,54,196)(24,226,55,195)(25,225,56,194)(26,224,57,193)(27,223,58,192)(28,222,59,191)(29,221,60,190)(30,220,61,189)(31,219,62,188)(63,282,94,251)(64,281,95,250)(65,280,96,249)(66,279,97,310)(67,278,98,309)(68,277,99,308)(69,276,100,307)(70,275,101,306)(71,274,102,305)(72,273,103,304)(73,272,104,303)(74,271,105,302)(75,270,106,301)(76,269,107,300)(77,268,108,299)(78,267,109,298)(79,266,110,297)(80,265,111,296)(81,264,112,295)(82,263,113,294)(83,262,114,293)(84,261,115,292)(85,260,116,291)(86,259,117,290)(87,258,118,289)(88,257,119,288)(89,256,120,287)(90,255,121,286)(91,254,122,285)(92,253,123,284)(93,252,124,283)(125,322,156,353)(126,321,157,352)(127,320,158,351)(128,319,159,350)(129,318,160,349)(130,317,161,348)(131,316,162,347)(132,315,163,346)(133,314,164,345)(134,313,165,344)(135,312,166,343)(136,311,167,342)(137,372,168,341)(138,371,169,340)(139,370,170,339)(140,369,171,338)(141,368,172,337)(142,367,173,336)(143,366,174,335)(144,365,175,334)(145,364,176,333)(146,363,177,332)(147,362,178,331)(148,361,179,330)(149,360,180,329)(150,359,181,328)(151,358,182,327)(152,357,183,326)(153,356,184,325)(154,355,185,324)(155,354,186,323) );
G=PermutationGroup([[(1,136,96),(2,137,97),(3,138,98),(4,139,99),(5,140,100),(6,141,101),(7,142,102),(8,143,103),(9,144,104),(10,145,105),(11,146,106),(12,147,107),(13,148,108),(14,149,109),(15,150,110),(16,151,111),(17,152,112),(18,153,113),(19,154,114),(20,155,115),(21,156,116),(22,157,117),(23,158,118),(24,159,119),(25,160,120),(26,161,121),(27,162,122),(28,163,123),(29,164,124),(30,165,63),(31,166,64),(32,167,65),(33,168,66),(34,169,67),(35,170,68),(36,171,69),(37,172,70),(38,173,71),(39,174,72),(40,175,73),(41,176,74),(42,177,75),(43,178,76),(44,179,77),(45,180,78),(46,181,79),(47,182,80),(48,183,81),(49,184,82),(50,185,83),(51,186,84),(52,125,85),(53,126,86),(54,127,87),(55,128,88),(56,129,89),(57,130,90),(58,131,91),(59,132,92),(60,133,93),(61,134,94),(62,135,95),(187,311,249),(188,312,250),(189,313,251),(190,314,252),(191,315,253),(192,316,254),(193,317,255),(194,318,256),(195,319,257),(196,320,258),(197,321,259),(198,322,260),(199,323,261),(200,324,262),(201,325,263),(202,326,264),(203,327,265),(204,328,266),(205,329,267),(206,330,268),(207,331,269),(208,332,270),(209,333,271),(210,334,272),(211,335,273),(212,336,274),(213,337,275),(214,338,276),(215,339,277),(216,340,278),(217,341,279),(218,342,280),(219,343,281),(220,344,282),(221,345,283),(222,346,284),(223,347,285),(224,348,286),(225,349,287),(226,350,288),(227,351,289),(228,352,290),(229,353,291),(230,354,292),(231,355,293),(232,356,294),(233,357,295),(234,358,296),(235,359,297),(236,360,298),(237,361,299),(238,362,300),(239,363,301),(240,364,302),(241,365,303),(242,366,304),(243,367,305),(244,368,306),(245,369,307),(246,370,308),(247,371,309),(248,372,310)], [(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)], [(1,187,32,218),(2,248,33,217),(3,247,34,216),(4,246,35,215),(5,245,36,214),(6,244,37,213),(7,243,38,212),(8,242,39,211),(9,241,40,210),(10,240,41,209),(11,239,42,208),(12,238,43,207),(13,237,44,206),(14,236,45,205),(15,235,46,204),(16,234,47,203),(17,233,48,202),(18,232,49,201),(19,231,50,200),(20,230,51,199),(21,229,52,198),(22,228,53,197),(23,227,54,196),(24,226,55,195),(25,225,56,194),(26,224,57,193),(27,223,58,192),(28,222,59,191),(29,221,60,190),(30,220,61,189),(31,219,62,188),(63,282,94,251),(64,281,95,250),(65,280,96,249),(66,279,97,310),(67,278,98,309),(68,277,99,308),(69,276,100,307),(70,275,101,306),(71,274,102,305),(72,273,103,304),(73,272,104,303),(74,271,105,302),(75,270,106,301),(76,269,107,300),(77,268,108,299),(78,267,109,298),(79,266,110,297),(80,265,111,296),(81,264,112,295),(82,263,113,294),(83,262,114,293),(84,261,115,292),(85,260,116,291),(86,259,117,290),(87,258,118,289),(88,257,119,288),(89,256,120,287),(90,255,121,286),(91,254,122,285),(92,253,123,284),(93,252,124,283),(125,322,156,353),(126,321,157,352),(127,320,158,351),(128,319,159,350),(129,318,160,349),(130,317,161,348),(131,316,162,347),(132,315,163,346),(133,314,164,345),(134,313,165,344),(135,312,166,343),(136,311,167,342),(137,372,168,341),(138,371,169,340),(139,370,170,339),(140,369,171,338),(141,368,172,337),(142,367,173,336),(143,366,174,335),(144,365,175,334),(145,364,176,333),(146,363,177,332),(147,362,178,331),(148,361,179,330),(149,360,180,329),(150,359,181,328),(151,358,182,327),(152,357,183,326),(153,356,184,325),(154,355,185,324),(155,354,186,323)]])
102 conjugacy classes
| class | 1 | 2 | 3A | 3B | 4A | 4B | 6A | 6B | 12A | 12B | 12C | 12D | 31A | ··· | 31O | 62A | ··· | 62O | 93A | ··· | 93AD | 186A | ··· | 186AD |
| order | 1 | 2 | 3 | 3 | 4 | 4 | 6 | 6 | 12 | 12 | 12 | 12 | 31 | ··· | 31 | 62 | ··· | 62 | 93 | ··· | 93 | 186 | ··· | 186 |
| size | 1 | 1 | 1 | 1 | 31 | 31 | 1 | 1 | 31 | 31 | 31 | 31 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
102 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | ||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D31 | Dic31 | C3×D31 | C3×Dic31 |
| kernel | C3×Dic31 | C186 | Dic31 | C93 | C62 | C31 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 4 | 15 | 15 | 30 | 30 |
Matrix representation of C3×Dic31 ►in GL2(𝔽373) generated by
| 88 | 0 |
| 0 | 88 |
| 0 | 372 |
| 1 | 241 |
| 42 | 300 |
| 249 | 331 |
G:=sub<GL(2,GF(373))| [88,0,0,88],[0,1,372,241],[42,249,300,331] >;
C3×Dic31 in GAP, Magma, Sage, TeX
C_3\times {\rm Dic}_{31} % in TeX
G:=Group("C3xDic31"); // GroupNames label
G:=SmallGroup(372,4);
// by ID
G=gap.SmallGroup(372,4);
# by ID
G:=PCGroup([4,-2,-3,-2,-31,24,5763]);
// Polycyclic
G:=Group<a,b,c|a^3=b^62=1,c^2=b^31,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
Export