metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C92⋊1C4, C4⋊Dic23, C2.1D92, C46.4D4, C46.2Q8, C2.2Dic46, C22.5D46, C23⋊2(C4⋊C4), (C2×C92).3C2, C46.8(C2×C4), (C2×C4).3D23, (C2×C46).5C22, C2.4(C2×Dic23), (C2×Dic23).2C2, SmallGroup(368,12)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C92⋊C4
G = < a,b | a92=b4=1, bab-1=a-1 >
Smallest permutation representation of C92⋊C4
►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)
(1 294 232 98)(2 293 233 97)(3 292 234 96)(4 291 235 95)(5 290 236 94)(6 289 237 93)(7 288 238 184)(8 287 239 183)(9 286 240 182)(10 285 241 181)(11 284 242 180)(12 283 243 179)(13 282 244 178)(14 281 245 177)(15 280 246 176)(16 279 247 175)(17 278 248 174)(18 277 249 173)(19 368 250 172)(20 367 251 171)(21 366 252 170)(22 365 253 169)(23 364 254 168)(24 363 255 167)(25 362 256 166)(26 361 257 165)(27 360 258 164)(28 359 259 163)(29 358 260 162)(30 357 261 161)(31 356 262 160)(32 355 263 159)(33 354 264 158)(34 353 265 157)(35 352 266 156)(36 351 267 155)(37 350 268 154)(38 349 269 153)(39 348 270 152)(40 347 271 151)(41 346 272 150)(42 345 273 149)(43 344 274 148)(44 343 275 147)(45 342 276 146)(46 341 185 145)(47 340 186 144)(48 339 187 143)(49 338 188 142)(50 337 189 141)(51 336 190 140)(52 335 191 139)(53 334 192 138)(54 333 193 137)(55 332 194 136)(56 331 195 135)(57 330 196 134)(58 329 197 133)(59 328 198 132)(60 327 199 131)(61 326 200 130)(62 325 201 129)(63 324 202 128)(64 323 203 127)(65 322 204 126)(66 321 205 125)(67 320 206 124)(68 319 207 123)(69 318 208 122)(70 317 209 121)(71 316 210 120)(72 315 211 119)(73 314 212 118)(74 313 213 117)(75 312 214 116)(76 311 215 115)(77 310 216 114)(78 309 217 113)(79 308 218 112)(80 307 219 111)(81 306 220 110)(82 305 221 109)(83 304 222 108)(84 303 223 107)(85 302 224 106)(86 301 225 105)(87 300 226 104)(88 299 227 103)(89 298 228 102)(90 297 229 101)(91 296 230 100)(92 295 231 99)
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), (1,294,232,98)(2,293,233,97)(3,292,234,96)(4,291,235,95)(5,290,236,94)(6,289,237,93)(7,288,238,184)(8,287,239,183)(9,286,240,182)(10,285,241,181)(11,284,242,180)(12,283,243,179)(13,282,244,178)(14,281,245,177)(15,280,246,176)(16,279,247,175)(17,278,248,174)(18,277,249,173)(19,368,250,172)(20,367,251,171)(21,366,252,170)(22,365,253,169)(23,364,254,168)(24,363,255,167)(25,362,256,166)(26,361,257,165)(27,360,258,164)(28,359,259,163)(29,358,260,162)(30,357,261,161)(31,356,262,160)(32,355,263,159)(33,354,264,158)(34,353,265,157)(35,352,266,156)(36,351,267,155)(37,350,268,154)(38,349,269,153)(39,348,270,152)(40,347,271,151)(41,346,272,150)(42,345,273,149)(43,344,274,148)(44,343,275,147)(45,342,276,146)(46,341,185,145)(47,340,186,144)(48,339,187,143)(49,338,188,142)(50,337,189,141)(51,336,190,140)(52,335,191,139)(53,334,192,138)(54,333,193,137)(55,332,194,136)(56,331,195,135)(57,330,196,134)(58,329,197,133)(59,328,198,132)(60,327,199,131)(61,326,200,130)(62,325,201,129)(63,324,202,128)(64,323,203,127)(65,322,204,126)(66,321,205,125)(67,320,206,124)(68,319,207,123)(69,318,208,122)(70,317,209,121)(71,316,210,120)(72,315,211,119)(73,314,212,118)(74,313,213,117)(75,312,214,116)(76,311,215,115)(77,310,216,114)(78,309,217,113)(79,308,218,112)(80,307,219,111)(81,306,220,110)(82,305,221,109)(83,304,222,108)(84,303,223,107)(85,302,224,106)(86,301,225,105)(87,300,226,104)(88,299,227,103)(89,298,228,102)(90,297,229,101)(91,296,230,100)(92,295,231,99)>;
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), (1,294,232,98)(2,293,233,97)(3,292,234,96)(4,291,235,95)(5,290,236,94)(6,289,237,93)(7,288,238,184)(8,287,239,183)(9,286,240,182)(10,285,241,181)(11,284,242,180)(12,283,243,179)(13,282,244,178)(14,281,245,177)(15,280,246,176)(16,279,247,175)(17,278,248,174)(18,277,249,173)(19,368,250,172)(20,367,251,171)(21,366,252,170)(22,365,253,169)(23,364,254,168)(24,363,255,167)(25,362,256,166)(26,361,257,165)(27,360,258,164)(28,359,259,163)(29,358,260,162)(30,357,261,161)(31,356,262,160)(32,355,263,159)(33,354,264,158)(34,353,265,157)(35,352,266,156)(36,351,267,155)(37,350,268,154)(38,349,269,153)(39,348,270,152)(40,347,271,151)(41,346,272,150)(42,345,273,149)(43,344,274,148)(44,343,275,147)(45,342,276,146)(46,341,185,145)(47,340,186,144)(48,339,187,143)(49,338,188,142)(50,337,189,141)(51,336,190,140)(52,335,191,139)(53,334,192,138)(54,333,193,137)(55,332,194,136)(56,331,195,135)(57,330,196,134)(58,329,197,133)(59,328,198,132)(60,327,199,131)(61,326,200,130)(62,325,201,129)(63,324,202,128)(64,323,203,127)(65,322,204,126)(66,321,205,125)(67,320,206,124)(68,319,207,123)(69,318,208,122)(70,317,209,121)(71,316,210,120)(72,315,211,119)(73,314,212,118)(74,313,213,117)(75,312,214,116)(76,311,215,115)(77,310,216,114)(78,309,217,113)(79,308,218,112)(80,307,219,111)(81,306,220,110)(82,305,221,109)(83,304,222,108)(84,303,223,107)(85,302,224,106)(86,301,225,105)(87,300,226,104)(88,299,227,103)(89,298,228,102)(90,297,229,101)(91,296,230,100)(92,295,231,99) );
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)], [(1,294,232,98),(2,293,233,97),(3,292,234,96),(4,291,235,95),(5,290,236,94),(6,289,237,93),(7,288,238,184),(8,287,239,183),(9,286,240,182),(10,285,241,181),(11,284,242,180),(12,283,243,179),(13,282,244,178),(14,281,245,177),(15,280,246,176),(16,279,247,175),(17,278,248,174),(18,277,249,173),(19,368,250,172),(20,367,251,171),(21,366,252,170),(22,365,253,169),(23,364,254,168),(24,363,255,167),(25,362,256,166),(26,361,257,165),(27,360,258,164),(28,359,259,163),(29,358,260,162),(30,357,261,161),(31,356,262,160),(32,355,263,159),(33,354,264,158),(34,353,265,157),(35,352,266,156),(36,351,267,155),(37,350,268,154),(38,349,269,153),(39,348,270,152),(40,347,271,151),(41,346,272,150),(42,345,273,149),(43,344,274,148),(44,343,275,147),(45,342,276,146),(46,341,185,145),(47,340,186,144),(48,339,187,143),(49,338,188,142),(50,337,189,141),(51,336,190,140),(52,335,191,139),(53,334,192,138),(54,333,193,137),(55,332,194,136),(56,331,195,135),(57,330,196,134),(58,329,197,133),(59,328,198,132),(60,327,199,131),(61,326,200,130),(62,325,201,129),(63,324,202,128),(64,323,203,127),(65,322,204,126),(66,321,205,125),(67,320,206,124),(68,319,207,123),(69,318,208,122),(70,317,209,121),(71,316,210,120),(72,315,211,119),(73,314,212,118),(74,313,213,117),(75,312,214,116),(76,311,215,115),(77,310,216,114),(78,309,217,113),(79,308,218,112),(80,307,219,111),(81,306,220,110),(82,305,221,109),(83,304,222,108),(84,303,223,107),(85,302,224,106),(86,301,225,105),(87,300,226,104),(88,299,227,103),(89,298,228,102),(90,297,229,101),(91,296,230,100),(92,295,231,99)]])
98 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 23A | ··· | 23K | 46A | ··· | 46AG | 92A | ··· | 92AR |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 46 | 46 | 46 | 46 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
98 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 | D23 | Dic23 | D46 | Dic46 | D92 |
| kernel | C92⋊C4 | C2×Dic23 | C2×C92 | C92 | C46 | C46 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 11 | 22 | 11 | 22 | 22 |
Matrix representation of C92⋊C4 ►in GL3(𝔽277) generated by
| 1 | 0 | 0 |
| 0 | 112 | 21 |
| 0 | 256 | 95 |
| 60 | 0 | 0 |
| 0 | 46 | 94 |
| 0 | 4 | 231 |
G:=sub<GL(3,GF(277))| [1,0,0,0,112,256,0,21,95],[60,0,0,0,46,4,0,94,231] >;
C92⋊C4 in GAP, Magma, Sage, TeX
C_{92}\rtimes C_4 % in TeX
G:=Group("C92:C4"); // GroupNames label
G:=SmallGroup(368,12);
// by ID
G=gap.SmallGroup(368,12);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,20,101,46,8804]);
// Polycyclic
G:=Group<a,b|a^92=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
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