metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C49⋊C8, C98.C4, C28.4D7, C4.2D49, C2.Dic49, C196.2C2, C14.1Dic7, C7.(C7⋊C8), SmallGroup(392,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C49 — C49⋊C8 |
Generators and relations for C49⋊C8
G = < a,b | a49=b8=1, bab-1=a-1 >
Smallest permutation representation of C49⋊C8
►Regular action on 392 points
Generators in S392
(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)(344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392)
(1 350 159 281 84 301 126 207)(2 349 160 280 85 300 127 206)(3 348 161 279 86 299 128 205)(4 347 162 278 87 298 129 204)(5 346 163 277 88 297 130 203)(6 345 164 276 89 296 131 202)(7 344 165 275 90 295 132 201)(8 392 166 274 91 343 133 200)(9 391 167 273 92 342 134 199)(10 390 168 272 93 341 135 198)(11 389 169 271 94 340 136 197)(12 388 170 270 95 339 137 245)(13 387 171 269 96 338 138 244)(14 386 172 268 97 337 139 243)(15 385 173 267 98 336 140 242)(16 384 174 266 50 335 141 241)(17 383 175 265 51 334 142 240)(18 382 176 264 52 333 143 239)(19 381 177 263 53 332 144 238)(20 380 178 262 54 331 145 237)(21 379 179 261 55 330 146 236)(22 378 180 260 56 329 147 235)(23 377 181 259 57 328 99 234)(24 376 182 258 58 327 100 233)(25 375 183 257 59 326 101 232)(26 374 184 256 60 325 102 231)(27 373 185 255 61 324 103 230)(28 372 186 254 62 323 104 229)(29 371 187 253 63 322 105 228)(30 370 188 252 64 321 106 227)(31 369 189 251 65 320 107 226)(32 368 190 250 66 319 108 225)(33 367 191 249 67 318 109 224)(34 366 192 248 68 317 110 223)(35 365 193 247 69 316 111 222)(36 364 194 246 70 315 112 221)(37 363 195 294 71 314 113 220)(38 362 196 293 72 313 114 219)(39 361 148 292 73 312 115 218)(40 360 149 291 74 311 116 217)(41 359 150 290 75 310 117 216)(42 358 151 289 76 309 118 215)(43 357 152 288 77 308 119 214)(44 356 153 287 78 307 120 213)(45 355 154 286 79 306 121 212)(46 354 155 285 80 305 122 211)(47 353 156 284 81 304 123 210)(48 352 157 283 82 303 124 209)(49 351 158 282 83 302 125 208)
G:=sub<Sym(392)| (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)(344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,350,159,281,84,301,126,207)(2,349,160,280,85,300,127,206)(3,348,161,279,86,299,128,205)(4,347,162,278,87,298,129,204)(5,346,163,277,88,297,130,203)(6,345,164,276,89,296,131,202)(7,344,165,275,90,295,132,201)(8,392,166,274,91,343,133,200)(9,391,167,273,92,342,134,199)(10,390,168,272,93,341,135,198)(11,389,169,271,94,340,136,197)(12,388,170,270,95,339,137,245)(13,387,171,269,96,338,138,244)(14,386,172,268,97,337,139,243)(15,385,173,267,98,336,140,242)(16,384,174,266,50,335,141,241)(17,383,175,265,51,334,142,240)(18,382,176,264,52,333,143,239)(19,381,177,263,53,332,144,238)(20,380,178,262,54,331,145,237)(21,379,179,261,55,330,146,236)(22,378,180,260,56,329,147,235)(23,377,181,259,57,328,99,234)(24,376,182,258,58,327,100,233)(25,375,183,257,59,326,101,232)(26,374,184,256,60,325,102,231)(27,373,185,255,61,324,103,230)(28,372,186,254,62,323,104,229)(29,371,187,253,63,322,105,228)(30,370,188,252,64,321,106,227)(31,369,189,251,65,320,107,226)(32,368,190,250,66,319,108,225)(33,367,191,249,67,318,109,224)(34,366,192,248,68,317,110,223)(35,365,193,247,69,316,111,222)(36,364,194,246,70,315,112,221)(37,363,195,294,71,314,113,220)(38,362,196,293,72,313,114,219)(39,361,148,292,73,312,115,218)(40,360,149,291,74,311,116,217)(41,359,150,290,75,310,117,216)(42,358,151,289,76,309,118,215)(43,357,152,288,77,308,119,214)(44,356,153,287,78,307,120,213)(45,355,154,286,79,306,121,212)(46,354,155,285,80,305,122,211)(47,353,156,284,81,304,123,210)(48,352,157,283,82,303,124,209)(49,351,158,282,83,302,125,208)>;
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)(344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392), (1,350,159,281,84,301,126,207)(2,349,160,280,85,300,127,206)(3,348,161,279,86,299,128,205)(4,347,162,278,87,298,129,204)(5,346,163,277,88,297,130,203)(6,345,164,276,89,296,131,202)(7,344,165,275,90,295,132,201)(8,392,166,274,91,343,133,200)(9,391,167,273,92,342,134,199)(10,390,168,272,93,341,135,198)(11,389,169,271,94,340,136,197)(12,388,170,270,95,339,137,245)(13,387,171,269,96,338,138,244)(14,386,172,268,97,337,139,243)(15,385,173,267,98,336,140,242)(16,384,174,266,50,335,141,241)(17,383,175,265,51,334,142,240)(18,382,176,264,52,333,143,239)(19,381,177,263,53,332,144,238)(20,380,178,262,54,331,145,237)(21,379,179,261,55,330,146,236)(22,378,180,260,56,329,147,235)(23,377,181,259,57,328,99,234)(24,376,182,258,58,327,100,233)(25,375,183,257,59,326,101,232)(26,374,184,256,60,325,102,231)(27,373,185,255,61,324,103,230)(28,372,186,254,62,323,104,229)(29,371,187,253,63,322,105,228)(30,370,188,252,64,321,106,227)(31,369,189,251,65,320,107,226)(32,368,190,250,66,319,108,225)(33,367,191,249,67,318,109,224)(34,366,192,248,68,317,110,223)(35,365,193,247,69,316,111,222)(36,364,194,246,70,315,112,221)(37,363,195,294,71,314,113,220)(38,362,196,293,72,313,114,219)(39,361,148,292,73,312,115,218)(40,360,149,291,74,311,116,217)(41,359,150,290,75,310,117,216)(42,358,151,289,76,309,118,215)(43,357,152,288,77,308,119,214)(44,356,153,287,78,307,120,213)(45,355,154,286,79,306,121,212)(46,354,155,285,80,305,122,211)(47,353,156,284,81,304,123,210)(48,352,157,283,82,303,124,209)(49,351,158,282,83,302,125,208) );
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),(344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392)], [(1,350,159,281,84,301,126,207),(2,349,160,280,85,300,127,206),(3,348,161,279,86,299,128,205),(4,347,162,278,87,298,129,204),(5,346,163,277,88,297,130,203),(6,345,164,276,89,296,131,202),(7,344,165,275,90,295,132,201),(8,392,166,274,91,343,133,200),(9,391,167,273,92,342,134,199),(10,390,168,272,93,341,135,198),(11,389,169,271,94,340,136,197),(12,388,170,270,95,339,137,245),(13,387,171,269,96,338,138,244),(14,386,172,268,97,337,139,243),(15,385,173,267,98,336,140,242),(16,384,174,266,50,335,141,241),(17,383,175,265,51,334,142,240),(18,382,176,264,52,333,143,239),(19,381,177,263,53,332,144,238),(20,380,178,262,54,331,145,237),(21,379,179,261,55,330,146,236),(22,378,180,260,56,329,147,235),(23,377,181,259,57,328,99,234),(24,376,182,258,58,327,100,233),(25,375,183,257,59,326,101,232),(26,374,184,256,60,325,102,231),(27,373,185,255,61,324,103,230),(28,372,186,254,62,323,104,229),(29,371,187,253,63,322,105,228),(30,370,188,252,64,321,106,227),(31,369,189,251,65,320,107,226),(32,368,190,250,66,319,108,225),(33,367,191,249,67,318,109,224),(34,366,192,248,68,317,110,223),(35,365,193,247,69,316,111,222),(36,364,194,246,70,315,112,221),(37,363,195,294,71,314,113,220),(38,362,196,293,72,313,114,219),(39,361,148,292,73,312,115,218),(40,360,149,291,74,311,116,217),(41,359,150,290,75,310,117,216),(42,358,151,289,76,309,118,215),(43,357,152,288,77,308,119,214),(44,356,153,287,78,307,120,213),(45,355,154,286,79,306,121,212),(46,354,155,285,80,305,122,211),(47,353,156,284,81,304,123,210),(48,352,157,283,82,303,124,209),(49,351,158,282,83,302,125,208)]])
104 conjugacy classes
| class | 1 | 2 | 4A | 4B | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 14A | 14B | 14C | 28A | ··· | 28F | 49A | ··· | 49U | 98A | ··· | 98U | 196A | ··· | 196AP |
| order | 1 | 2 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 14 | 14 | 14 | 28 | ··· | 28 | 49 | ··· | 49 | 98 | ··· | 98 | 196 | ··· | 196 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 49 | 49 | 49 | 49 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | ||||
| image | C1 | C2 | C4 | C8 | D7 | Dic7 | C7⋊C8 | D49 | Dic49 | C49⋊C8 |
| kernel | C49⋊C8 | C196 | C98 | C49 | C28 | C14 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 3 | 3 | 6 | 21 | 21 | 42 |
Matrix representation of C49⋊C8 ►in GL3(𝔽3137) generated by
| 1 | 0 | 0 |
| 0 | 242 | 2797 |
| 0 | 1227 | 2152 |
| 1941 | 0 | 0 |
| 0 | 1724 | 293 |
| 0 | 1494 | 1413 |
G:=sub<GL(3,GF(3137))| [1,0,0,0,242,1227,0,2797,2152],[1941,0,0,0,1724,1494,0,293,1413] >;
C49⋊C8 in GAP, Magma, Sage, TeX
C_{49}\rtimes C_8 % in TeX
G:=Group("C49:C8"); // GroupNames label
G:=SmallGroup(392,1);
// by ID
G=gap.SmallGroup(392,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-7,-7,10,26,2083,858,8404]);
// Polycyclic
G:=Group<a,b|a^49=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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