metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C43⋊C8, C86.C4, C4.2D43, C2.Dic43, C172.2C2, SmallGroup(344,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C43 — C43⋊C8 |
Generators and relations for C43⋊C8
G = < a,b | a43=b8=1, bab-1=a-1 >
Smallest permutation representation of C43⋊C8
►Regular action on 344 points
Generators in S344
(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)
(1 302 141 216 73 259 110 173)(2 344 142 258 74 301 111 215)(3 343 143 257 75 300 112 214)(4 342 144 256 76 299 113 213)(5 341 145 255 77 298 114 212)(6 340 146 254 78 297 115 211)(7 339 147 253 79 296 116 210)(8 338 148 252 80 295 117 209)(9 337 149 251 81 294 118 208)(10 336 150 250 82 293 119 207)(11 335 151 249 83 292 120 206)(12 334 152 248 84 291 121 205)(13 333 153 247 85 290 122 204)(14 332 154 246 86 289 123 203)(15 331 155 245 44 288 124 202)(16 330 156 244 45 287 125 201)(17 329 157 243 46 286 126 200)(18 328 158 242 47 285 127 199)(19 327 159 241 48 284 128 198)(20 326 160 240 49 283 129 197)(21 325 161 239 50 282 87 196)(22 324 162 238 51 281 88 195)(23 323 163 237 52 280 89 194)(24 322 164 236 53 279 90 193)(25 321 165 235 54 278 91 192)(26 320 166 234 55 277 92 191)(27 319 167 233 56 276 93 190)(28 318 168 232 57 275 94 189)(29 317 169 231 58 274 95 188)(30 316 170 230 59 273 96 187)(31 315 171 229 60 272 97 186)(32 314 172 228 61 271 98 185)(33 313 130 227 62 270 99 184)(34 312 131 226 63 269 100 183)(35 311 132 225 64 268 101 182)(36 310 133 224 65 267 102 181)(37 309 134 223 66 266 103 180)(38 308 135 222 67 265 104 179)(39 307 136 221 68 264 105 178)(40 306 137 220 69 263 106 177)(41 305 138 219 70 262 107 176)(42 304 139 218 71 261 108 175)(43 303 140 217 72 260 109 174)
G:=sub<Sym(344)| (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), (1,302,141,216,73,259,110,173)(2,344,142,258,74,301,111,215)(3,343,143,257,75,300,112,214)(4,342,144,256,76,299,113,213)(5,341,145,255,77,298,114,212)(6,340,146,254,78,297,115,211)(7,339,147,253,79,296,116,210)(8,338,148,252,80,295,117,209)(9,337,149,251,81,294,118,208)(10,336,150,250,82,293,119,207)(11,335,151,249,83,292,120,206)(12,334,152,248,84,291,121,205)(13,333,153,247,85,290,122,204)(14,332,154,246,86,289,123,203)(15,331,155,245,44,288,124,202)(16,330,156,244,45,287,125,201)(17,329,157,243,46,286,126,200)(18,328,158,242,47,285,127,199)(19,327,159,241,48,284,128,198)(20,326,160,240,49,283,129,197)(21,325,161,239,50,282,87,196)(22,324,162,238,51,281,88,195)(23,323,163,237,52,280,89,194)(24,322,164,236,53,279,90,193)(25,321,165,235,54,278,91,192)(26,320,166,234,55,277,92,191)(27,319,167,233,56,276,93,190)(28,318,168,232,57,275,94,189)(29,317,169,231,58,274,95,188)(30,316,170,230,59,273,96,187)(31,315,171,229,60,272,97,186)(32,314,172,228,61,271,98,185)(33,313,130,227,62,270,99,184)(34,312,131,226,63,269,100,183)(35,311,132,225,64,268,101,182)(36,310,133,224,65,267,102,181)(37,309,134,223,66,266,103,180)(38,308,135,222,67,265,104,179)(39,307,136,221,68,264,105,178)(40,306,137,220,69,263,106,177)(41,305,138,219,70,262,107,176)(42,304,139,218,71,261,108,175)(43,303,140,217,72,260,109,174)>;
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), (1,302,141,216,73,259,110,173)(2,344,142,258,74,301,111,215)(3,343,143,257,75,300,112,214)(4,342,144,256,76,299,113,213)(5,341,145,255,77,298,114,212)(6,340,146,254,78,297,115,211)(7,339,147,253,79,296,116,210)(8,338,148,252,80,295,117,209)(9,337,149,251,81,294,118,208)(10,336,150,250,82,293,119,207)(11,335,151,249,83,292,120,206)(12,334,152,248,84,291,121,205)(13,333,153,247,85,290,122,204)(14,332,154,246,86,289,123,203)(15,331,155,245,44,288,124,202)(16,330,156,244,45,287,125,201)(17,329,157,243,46,286,126,200)(18,328,158,242,47,285,127,199)(19,327,159,241,48,284,128,198)(20,326,160,240,49,283,129,197)(21,325,161,239,50,282,87,196)(22,324,162,238,51,281,88,195)(23,323,163,237,52,280,89,194)(24,322,164,236,53,279,90,193)(25,321,165,235,54,278,91,192)(26,320,166,234,55,277,92,191)(27,319,167,233,56,276,93,190)(28,318,168,232,57,275,94,189)(29,317,169,231,58,274,95,188)(30,316,170,230,59,273,96,187)(31,315,171,229,60,272,97,186)(32,314,172,228,61,271,98,185)(33,313,130,227,62,270,99,184)(34,312,131,226,63,269,100,183)(35,311,132,225,64,268,101,182)(36,310,133,224,65,267,102,181)(37,309,134,223,66,266,103,180)(38,308,135,222,67,265,104,179)(39,307,136,221,68,264,105,178)(40,306,137,220,69,263,106,177)(41,305,138,219,70,262,107,176)(42,304,139,218,71,261,108,175)(43,303,140,217,72,260,109,174) );
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)], [(1,302,141,216,73,259,110,173),(2,344,142,258,74,301,111,215),(3,343,143,257,75,300,112,214),(4,342,144,256,76,299,113,213),(5,341,145,255,77,298,114,212),(6,340,146,254,78,297,115,211),(7,339,147,253,79,296,116,210),(8,338,148,252,80,295,117,209),(9,337,149,251,81,294,118,208),(10,336,150,250,82,293,119,207),(11,335,151,249,83,292,120,206),(12,334,152,248,84,291,121,205),(13,333,153,247,85,290,122,204),(14,332,154,246,86,289,123,203),(15,331,155,245,44,288,124,202),(16,330,156,244,45,287,125,201),(17,329,157,243,46,286,126,200),(18,328,158,242,47,285,127,199),(19,327,159,241,48,284,128,198),(20,326,160,240,49,283,129,197),(21,325,161,239,50,282,87,196),(22,324,162,238,51,281,88,195),(23,323,163,237,52,280,89,194),(24,322,164,236,53,279,90,193),(25,321,165,235,54,278,91,192),(26,320,166,234,55,277,92,191),(27,319,167,233,56,276,93,190),(28,318,168,232,57,275,94,189),(29,317,169,231,58,274,95,188),(30,316,170,230,59,273,96,187),(31,315,171,229,60,272,97,186),(32,314,172,228,61,271,98,185),(33,313,130,227,62,270,99,184),(34,312,131,226,63,269,100,183),(35,311,132,225,64,268,101,182),(36,310,133,224,65,267,102,181),(37,309,134,223,66,266,103,180),(38,308,135,222,67,265,104,179),(39,307,136,221,68,264,105,178),(40,306,137,220,69,263,106,177),(41,305,138,219,70,262,107,176),(42,304,139,218,71,261,108,175),(43,303,140,217,72,260,109,174)]])
92 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 43A | ··· | 43U | 86A | ··· | 86U | 172A | ··· | 172AP |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 43 | ··· | 43 | 86 | ··· | 86 | 172 | ··· | 172 |
| size | 1 | 1 | 1 | 1 | 43 | 43 | 43 | 43 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | - | |||
| image | C1 | C2 | C4 | C8 | D43 | Dic43 | C43⋊C8 |
| kernel | C43⋊C8 | C172 | C86 | C43 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 21 | 21 | 42 |
Matrix representation of C43⋊C8 ►in GL3(𝔽1033) generated by
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1032 | 125 |
| 635 | 0 | 0 |
| 0 | 123 | 633 |
| 0 | 513 | 910 |
G:=sub<GL(3,GF(1033))| [1,0,0,0,0,1032,0,1,125],[635,0,0,0,123,513,0,633,910] >;
C43⋊C8 in GAP, Magma, Sage, TeX
C_{43}\rtimes C_8 % in TeX
G:=Group("C43:C8"); // GroupNames label
G:=SmallGroup(344,1);
// by ID
G=gap.SmallGroup(344,1);
# by ID
G:=PCGroup([4,-2,-2,-2,-43,8,21,5379]);
// Polycyclic
G:=Group<a,b|a^43=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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