metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C47⋊C8, C94.C4, C4.2D47, C2.Dic47, C188.2C2, SmallGroup(376,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C47 — C47⋊C8 |
Generators and relations for C47⋊C8
G = < a,b | a47=b8=1, bab-1=a-1 >
Smallest permutation representation of C47⋊C8
►Regular action on 376 points
Generators in S376
(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)(330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376)
(1 330 176 236 58 283 102 189)(2 376 177 282 59 329 103 235)(3 375 178 281 60 328 104 234)(4 374 179 280 61 327 105 233)(5 373 180 279 62 326 106 232)(6 372 181 278 63 325 107 231)(7 371 182 277 64 324 108 230)(8 370 183 276 65 323 109 229)(9 369 184 275 66 322 110 228)(10 368 185 274 67 321 111 227)(11 367 186 273 68 320 112 226)(12 366 187 272 69 319 113 225)(13 365 188 271 70 318 114 224)(14 364 142 270 71 317 115 223)(15 363 143 269 72 316 116 222)(16 362 144 268 73 315 117 221)(17 361 145 267 74 314 118 220)(18 360 146 266 75 313 119 219)(19 359 147 265 76 312 120 218)(20 358 148 264 77 311 121 217)(21 357 149 263 78 310 122 216)(22 356 150 262 79 309 123 215)(23 355 151 261 80 308 124 214)(24 354 152 260 81 307 125 213)(25 353 153 259 82 306 126 212)(26 352 154 258 83 305 127 211)(27 351 155 257 84 304 128 210)(28 350 156 256 85 303 129 209)(29 349 157 255 86 302 130 208)(30 348 158 254 87 301 131 207)(31 347 159 253 88 300 132 206)(32 346 160 252 89 299 133 205)(33 345 161 251 90 298 134 204)(34 344 162 250 91 297 135 203)(35 343 163 249 92 296 136 202)(36 342 164 248 93 295 137 201)(37 341 165 247 94 294 138 200)(38 340 166 246 48 293 139 199)(39 339 167 245 49 292 140 198)(40 338 168 244 50 291 141 197)(41 337 169 243 51 290 95 196)(42 336 170 242 52 289 96 195)(43 335 171 241 53 288 97 194)(44 334 172 240 54 287 98 193)(45 333 173 239 55 286 99 192)(46 332 174 238 56 285 100 191)(47 331 175 237 57 284 101 190)
G:=sub<Sym(376)| (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)(330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376), (1,330,176,236,58,283,102,189)(2,376,177,282,59,329,103,235)(3,375,178,281,60,328,104,234)(4,374,179,280,61,327,105,233)(5,373,180,279,62,326,106,232)(6,372,181,278,63,325,107,231)(7,371,182,277,64,324,108,230)(8,370,183,276,65,323,109,229)(9,369,184,275,66,322,110,228)(10,368,185,274,67,321,111,227)(11,367,186,273,68,320,112,226)(12,366,187,272,69,319,113,225)(13,365,188,271,70,318,114,224)(14,364,142,270,71,317,115,223)(15,363,143,269,72,316,116,222)(16,362,144,268,73,315,117,221)(17,361,145,267,74,314,118,220)(18,360,146,266,75,313,119,219)(19,359,147,265,76,312,120,218)(20,358,148,264,77,311,121,217)(21,357,149,263,78,310,122,216)(22,356,150,262,79,309,123,215)(23,355,151,261,80,308,124,214)(24,354,152,260,81,307,125,213)(25,353,153,259,82,306,126,212)(26,352,154,258,83,305,127,211)(27,351,155,257,84,304,128,210)(28,350,156,256,85,303,129,209)(29,349,157,255,86,302,130,208)(30,348,158,254,87,301,131,207)(31,347,159,253,88,300,132,206)(32,346,160,252,89,299,133,205)(33,345,161,251,90,298,134,204)(34,344,162,250,91,297,135,203)(35,343,163,249,92,296,136,202)(36,342,164,248,93,295,137,201)(37,341,165,247,94,294,138,200)(38,340,166,246,48,293,139,199)(39,339,167,245,49,292,140,198)(40,338,168,244,50,291,141,197)(41,337,169,243,51,290,95,196)(42,336,170,242,52,289,96,195)(43,335,171,241,53,288,97,194)(44,334,172,240,54,287,98,193)(45,333,173,239,55,286,99,192)(46,332,174,238,56,285,100,191)(47,331,175,237,57,284,101,190)>;
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)(330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376), (1,330,176,236,58,283,102,189)(2,376,177,282,59,329,103,235)(3,375,178,281,60,328,104,234)(4,374,179,280,61,327,105,233)(5,373,180,279,62,326,106,232)(6,372,181,278,63,325,107,231)(7,371,182,277,64,324,108,230)(8,370,183,276,65,323,109,229)(9,369,184,275,66,322,110,228)(10,368,185,274,67,321,111,227)(11,367,186,273,68,320,112,226)(12,366,187,272,69,319,113,225)(13,365,188,271,70,318,114,224)(14,364,142,270,71,317,115,223)(15,363,143,269,72,316,116,222)(16,362,144,268,73,315,117,221)(17,361,145,267,74,314,118,220)(18,360,146,266,75,313,119,219)(19,359,147,265,76,312,120,218)(20,358,148,264,77,311,121,217)(21,357,149,263,78,310,122,216)(22,356,150,262,79,309,123,215)(23,355,151,261,80,308,124,214)(24,354,152,260,81,307,125,213)(25,353,153,259,82,306,126,212)(26,352,154,258,83,305,127,211)(27,351,155,257,84,304,128,210)(28,350,156,256,85,303,129,209)(29,349,157,255,86,302,130,208)(30,348,158,254,87,301,131,207)(31,347,159,253,88,300,132,206)(32,346,160,252,89,299,133,205)(33,345,161,251,90,298,134,204)(34,344,162,250,91,297,135,203)(35,343,163,249,92,296,136,202)(36,342,164,248,93,295,137,201)(37,341,165,247,94,294,138,200)(38,340,166,246,48,293,139,199)(39,339,167,245,49,292,140,198)(40,338,168,244,50,291,141,197)(41,337,169,243,51,290,95,196)(42,336,170,242,52,289,96,195)(43,335,171,241,53,288,97,194)(44,334,172,240,54,287,98,193)(45,333,173,239,55,286,99,192)(46,332,174,238,56,285,100,191)(47,331,175,237,57,284,101,190) );
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),(330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376)], [(1,330,176,236,58,283,102,189),(2,376,177,282,59,329,103,235),(3,375,178,281,60,328,104,234),(4,374,179,280,61,327,105,233),(5,373,180,279,62,326,106,232),(6,372,181,278,63,325,107,231),(7,371,182,277,64,324,108,230),(8,370,183,276,65,323,109,229),(9,369,184,275,66,322,110,228),(10,368,185,274,67,321,111,227),(11,367,186,273,68,320,112,226),(12,366,187,272,69,319,113,225),(13,365,188,271,70,318,114,224),(14,364,142,270,71,317,115,223),(15,363,143,269,72,316,116,222),(16,362,144,268,73,315,117,221),(17,361,145,267,74,314,118,220),(18,360,146,266,75,313,119,219),(19,359,147,265,76,312,120,218),(20,358,148,264,77,311,121,217),(21,357,149,263,78,310,122,216),(22,356,150,262,79,309,123,215),(23,355,151,261,80,308,124,214),(24,354,152,260,81,307,125,213),(25,353,153,259,82,306,126,212),(26,352,154,258,83,305,127,211),(27,351,155,257,84,304,128,210),(28,350,156,256,85,303,129,209),(29,349,157,255,86,302,130,208),(30,348,158,254,87,301,131,207),(31,347,159,253,88,300,132,206),(32,346,160,252,89,299,133,205),(33,345,161,251,90,298,134,204),(34,344,162,250,91,297,135,203),(35,343,163,249,92,296,136,202),(36,342,164,248,93,295,137,201),(37,341,165,247,94,294,138,200),(38,340,166,246,48,293,139,199),(39,339,167,245,49,292,140,198),(40,338,168,244,50,291,141,197),(41,337,169,243,51,290,95,196),(42,336,170,242,52,289,96,195),(43,335,171,241,53,288,97,194),(44,334,172,240,54,287,98,193),(45,333,173,239,55,286,99,192),(46,332,174,238,56,285,100,191),(47,331,175,237,57,284,101,190)]])
100 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 47A | ··· | 47W | 94A | ··· | 94W | 188A | ··· | 188AT |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 47 | ··· | 47 | 94 | ··· | 94 | 188 | ··· | 188 |
| size | 1 | 1 | 1 | 1 | 47 | 47 | 47 | 47 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
100 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D47 | Dic47 | C47⋊C8 |
| kernel | C47⋊C8 | C188 | C94 | C47 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 23 | 23 | 46 |
Matrix representation of C47⋊C8 ►in GL3(𝔽1129) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1128 | 873 |
| 1098 | 0 | 0 |
| 0 | 618 | 523 |
| 0 | 375 | 511 |
G:=sub<GL(3,GF(1129))| [1,0,0,0,0,1128,0,1,873],[1098,0,0,0,618,375,0,523,511] >;
C47⋊C8 in GAP, Magma, Sage, TeX
C_{47}\rtimes C_8 % in TeX
G:=Group("C47:C8"); // GroupNames label
G:=SmallGroup(376,1);
// by ID
G=gap.SmallGroup(376,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-47,8,21,5891]);
// Polycyclic
G:=Group<a,b|a^47=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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