direct product, abelian, monomial, 2-elementary
Aliases: C4×C92, SmallGroup(368,19)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C92 |
| C1 — C4×C92 |
| C1 — C4×C92 |
Generators and relations for C4×C92
G = < a,b | a4=b92=1, ab=ba >
Smallest permutation representation of C4×C92
►Regular action on 368 points
Generators in S368
(1 187 96 307)(2 188 97 308)(3 189 98 309)(4 190 99 310)(5 191 100 311)(6 192 101 312)(7 193 102 313)(8 194 103 314)(9 195 104 315)(10 196 105 316)(11 197 106 317)(12 198 107 318)(13 199 108 319)(14 200 109 320)(15 201 110 321)(16 202 111 322)(17 203 112 323)(18 204 113 324)(19 205 114 325)(20 206 115 326)(21 207 116 327)(22 208 117 328)(23 209 118 329)(24 210 119 330)(25 211 120 331)(26 212 121 332)(27 213 122 333)(28 214 123 334)(29 215 124 335)(30 216 125 336)(31 217 126 337)(32 218 127 338)(33 219 128 339)(34 220 129 340)(35 221 130 341)(36 222 131 342)(37 223 132 343)(38 224 133 344)(39 225 134 345)(40 226 135 346)(41 227 136 347)(42 228 137 348)(43 229 138 349)(44 230 139 350)(45 231 140 351)(46 232 141 352)(47 233 142 353)(48 234 143 354)(49 235 144 355)(50 236 145 356)(51 237 146 357)(52 238 147 358)(53 239 148 359)(54 240 149 360)(55 241 150 361)(56 242 151 362)(57 243 152 363)(58 244 153 364)(59 245 154 365)(60 246 155 366)(61 247 156 367)(62 248 157 368)(63 249 158 277)(64 250 159 278)(65 251 160 279)(66 252 161 280)(67 253 162 281)(68 254 163 282)(69 255 164 283)(70 256 165 284)(71 257 166 285)(72 258 167 286)(73 259 168 287)(74 260 169 288)(75 261 170 289)(76 262 171 290)(77 263 172 291)(78 264 173 292)(79 265 174 293)(80 266 175 294)(81 267 176 295)(82 268 177 296)(83 269 178 297)(84 270 179 298)(85 271 180 299)(86 272 181 300)(87 273 182 301)(88 274 183 302)(89 275 184 303)(90 276 93 304)(91 185 94 305)(92 186 95 306)
(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)
G:=sub<Sym(368)| (1,187,96,307)(2,188,97,308)(3,189,98,309)(4,190,99,310)(5,191,100,311)(6,192,101,312)(7,193,102,313)(8,194,103,314)(9,195,104,315)(10,196,105,316)(11,197,106,317)(12,198,107,318)(13,199,108,319)(14,200,109,320)(15,201,110,321)(16,202,111,322)(17,203,112,323)(18,204,113,324)(19,205,114,325)(20,206,115,326)(21,207,116,327)(22,208,117,328)(23,209,118,329)(24,210,119,330)(25,211,120,331)(26,212,121,332)(27,213,122,333)(28,214,123,334)(29,215,124,335)(30,216,125,336)(31,217,126,337)(32,218,127,338)(33,219,128,339)(34,220,129,340)(35,221,130,341)(36,222,131,342)(37,223,132,343)(38,224,133,344)(39,225,134,345)(40,226,135,346)(41,227,136,347)(42,228,137,348)(43,229,138,349)(44,230,139,350)(45,231,140,351)(46,232,141,352)(47,233,142,353)(48,234,143,354)(49,235,144,355)(50,236,145,356)(51,237,146,357)(52,238,147,358)(53,239,148,359)(54,240,149,360)(55,241,150,361)(56,242,151,362)(57,243,152,363)(58,244,153,364)(59,245,154,365)(60,246,155,366)(61,247,156,367)(62,248,157,368)(63,249,158,277)(64,250,159,278)(65,251,160,279)(66,252,161,280)(67,253,162,281)(68,254,163,282)(69,255,164,283)(70,256,165,284)(71,257,166,285)(72,258,167,286)(73,259,168,287)(74,260,169,288)(75,261,170,289)(76,262,171,290)(77,263,172,291)(78,264,173,292)(79,265,174,293)(80,266,175,294)(81,267,176,295)(82,268,177,296)(83,269,178,297)(84,270,179,298)(85,271,180,299)(86,272,181,300)(87,273,182,301)(88,274,183,302)(89,275,184,303)(90,276,93,304)(91,185,94,305)(92,186,95,306), (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)>;
G:=Group( (1,187,96,307)(2,188,97,308)(3,189,98,309)(4,190,99,310)(5,191,100,311)(6,192,101,312)(7,193,102,313)(8,194,103,314)(9,195,104,315)(10,196,105,316)(11,197,106,317)(12,198,107,318)(13,199,108,319)(14,200,109,320)(15,201,110,321)(16,202,111,322)(17,203,112,323)(18,204,113,324)(19,205,114,325)(20,206,115,326)(21,207,116,327)(22,208,117,328)(23,209,118,329)(24,210,119,330)(25,211,120,331)(26,212,121,332)(27,213,122,333)(28,214,123,334)(29,215,124,335)(30,216,125,336)(31,217,126,337)(32,218,127,338)(33,219,128,339)(34,220,129,340)(35,221,130,341)(36,222,131,342)(37,223,132,343)(38,224,133,344)(39,225,134,345)(40,226,135,346)(41,227,136,347)(42,228,137,348)(43,229,138,349)(44,230,139,350)(45,231,140,351)(46,232,141,352)(47,233,142,353)(48,234,143,354)(49,235,144,355)(50,236,145,356)(51,237,146,357)(52,238,147,358)(53,239,148,359)(54,240,149,360)(55,241,150,361)(56,242,151,362)(57,243,152,363)(58,244,153,364)(59,245,154,365)(60,246,155,366)(61,247,156,367)(62,248,157,368)(63,249,158,277)(64,250,159,278)(65,251,160,279)(66,252,161,280)(67,253,162,281)(68,254,163,282)(69,255,164,283)(70,256,165,284)(71,257,166,285)(72,258,167,286)(73,259,168,287)(74,260,169,288)(75,261,170,289)(76,262,171,290)(77,263,172,291)(78,264,173,292)(79,265,174,293)(80,266,175,294)(81,267,176,295)(82,268,177,296)(83,269,178,297)(84,270,179,298)(85,271,180,299)(86,272,181,300)(87,273,182,301)(88,274,183,302)(89,275,184,303)(90,276,93,304)(91,185,94,305)(92,186,95,306), (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) );
G=PermutationGroup([[(1,187,96,307),(2,188,97,308),(3,189,98,309),(4,190,99,310),(5,191,100,311),(6,192,101,312),(7,193,102,313),(8,194,103,314),(9,195,104,315),(10,196,105,316),(11,197,106,317),(12,198,107,318),(13,199,108,319),(14,200,109,320),(15,201,110,321),(16,202,111,322),(17,203,112,323),(18,204,113,324),(19,205,114,325),(20,206,115,326),(21,207,116,327),(22,208,117,328),(23,209,118,329),(24,210,119,330),(25,211,120,331),(26,212,121,332),(27,213,122,333),(28,214,123,334),(29,215,124,335),(30,216,125,336),(31,217,126,337),(32,218,127,338),(33,219,128,339),(34,220,129,340),(35,221,130,341),(36,222,131,342),(37,223,132,343),(38,224,133,344),(39,225,134,345),(40,226,135,346),(41,227,136,347),(42,228,137,348),(43,229,138,349),(44,230,139,350),(45,231,140,351),(46,232,141,352),(47,233,142,353),(48,234,143,354),(49,235,144,355),(50,236,145,356),(51,237,146,357),(52,238,147,358),(53,239,148,359),(54,240,149,360),(55,241,150,361),(56,242,151,362),(57,243,152,363),(58,244,153,364),(59,245,154,365),(60,246,155,366),(61,247,156,367),(62,248,157,368),(63,249,158,277),(64,250,159,278),(65,251,160,279),(66,252,161,280),(67,253,162,281),(68,254,163,282),(69,255,164,283),(70,256,165,284),(71,257,166,285),(72,258,167,286),(73,259,168,287),(74,260,169,288),(75,261,170,289),(76,262,171,290),(77,263,172,291),(78,264,173,292),(79,265,174,293),(80,266,175,294),(81,267,176,295),(82,268,177,296),(83,269,178,297),(84,270,179,298),(85,271,180,299),(86,272,181,300),(87,273,182,301),(88,274,183,302),(89,275,184,303),(90,276,93,304),(91,185,94,305),(92,186,95,306)], [(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)]])
368 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 23A | ··· | 23V | 46A | ··· | 46BN | 92A | ··· | 92JD |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
368 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||
| image | C1 | C2 | C4 | C23 | C46 | C92 |
| kernel | C4×C92 | C2×C92 | C92 | C42 | C2×C4 | C4 |
| # reps | 1 | 3 | 12 | 22 | 66 | 264 |
Matrix representation of C4×C92 ►in GL2(𝔽277) generated by
| 60 | 0 |
| 0 | 60 |
| 149 | 0 |
| 0 | 113 |
G:=sub<GL(2,GF(277))| [60,0,0,60],[149,0,0,113] >;
C4×C92 in GAP, Magma, Sage, TeX
C_4\times C_{92} % in TeX
G:=Group("C4xC92"); // GroupNames label
G:=SmallGroup(368,19);
// by ID
G=gap.SmallGroup(368,19);
# by ID
G:=PCGroup([5,-2,-2,-23,-2,-2,460,926]);
// Polycyclic
G:=Group<a,b|a^4=b^92=1,a*b=b*a>;
// generators/relations
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