metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C23⋊C16, C46.C8, C92.2C4, C8.2D23, C184.2C2, C4.2Dic23, C2.(C23⋊C8), SmallGroup(368,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C23⋊C16 |
Generators and relations for C23⋊C16
G = < a,b | a23=b16=1, bab-1=a-1 >
Smallest permutation representation of C23⋊C16
►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)
(1 346 177 254 90 300 129 208 44 323 151 231 59 277 101 185)(2 368 178 276 91 322 130 230 45 345 152 253 60 299 102 207)(3 367 179 275 92 321 131 229 46 344 153 252 61 298 103 206)(4 366 180 274 70 320 132 228 24 343 154 251 62 297 104 205)(5 365 181 273 71 319 133 227 25 342 155 250 63 296 105 204)(6 364 182 272 72 318 134 226 26 341 156 249 64 295 106 203)(7 363 183 271 73 317 135 225 27 340 157 248 65 294 107 202)(8 362 184 270 74 316 136 224 28 339 158 247 66 293 108 201)(9 361 162 269 75 315 137 223 29 338 159 246 67 292 109 200)(10 360 163 268 76 314 138 222 30 337 160 245 68 291 110 199)(11 359 164 267 77 313 116 221 31 336 161 244 69 290 111 198)(12 358 165 266 78 312 117 220 32 335 139 243 47 289 112 197)(13 357 166 265 79 311 118 219 33 334 140 242 48 288 113 196)(14 356 167 264 80 310 119 218 34 333 141 241 49 287 114 195)(15 355 168 263 81 309 120 217 35 332 142 240 50 286 115 194)(16 354 169 262 82 308 121 216 36 331 143 239 51 285 93 193)(17 353 170 261 83 307 122 215 37 330 144 238 52 284 94 192)(18 352 171 260 84 306 123 214 38 329 145 237 53 283 95 191)(19 351 172 259 85 305 124 213 39 328 146 236 54 282 96 190)(20 350 173 258 86 304 125 212 40 327 147 235 55 281 97 189)(21 349 174 257 87 303 126 211 41 326 148 234 56 280 98 188)(22 348 175 256 88 302 127 210 42 325 149 233 57 279 99 187)(23 347 176 255 89 301 128 209 43 324 150 232 58 278 100 186)
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), (1,346,177,254,90,300,129,208,44,323,151,231,59,277,101,185)(2,368,178,276,91,322,130,230,45,345,152,253,60,299,102,207)(3,367,179,275,92,321,131,229,46,344,153,252,61,298,103,206)(4,366,180,274,70,320,132,228,24,343,154,251,62,297,104,205)(5,365,181,273,71,319,133,227,25,342,155,250,63,296,105,204)(6,364,182,272,72,318,134,226,26,341,156,249,64,295,106,203)(7,363,183,271,73,317,135,225,27,340,157,248,65,294,107,202)(8,362,184,270,74,316,136,224,28,339,158,247,66,293,108,201)(9,361,162,269,75,315,137,223,29,338,159,246,67,292,109,200)(10,360,163,268,76,314,138,222,30,337,160,245,68,291,110,199)(11,359,164,267,77,313,116,221,31,336,161,244,69,290,111,198)(12,358,165,266,78,312,117,220,32,335,139,243,47,289,112,197)(13,357,166,265,79,311,118,219,33,334,140,242,48,288,113,196)(14,356,167,264,80,310,119,218,34,333,141,241,49,287,114,195)(15,355,168,263,81,309,120,217,35,332,142,240,50,286,115,194)(16,354,169,262,82,308,121,216,36,331,143,239,51,285,93,193)(17,353,170,261,83,307,122,215,37,330,144,238,52,284,94,192)(18,352,171,260,84,306,123,214,38,329,145,237,53,283,95,191)(19,351,172,259,85,305,124,213,39,328,146,236,54,282,96,190)(20,350,173,258,86,304,125,212,40,327,147,235,55,281,97,189)(21,349,174,257,87,303,126,211,41,326,148,234,56,280,98,188)(22,348,175,256,88,302,127,210,42,325,149,233,57,279,99,187)(23,347,176,255,89,301,128,209,43,324,150,232,58,278,100,186)>;
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), (1,346,177,254,90,300,129,208,44,323,151,231,59,277,101,185)(2,368,178,276,91,322,130,230,45,345,152,253,60,299,102,207)(3,367,179,275,92,321,131,229,46,344,153,252,61,298,103,206)(4,366,180,274,70,320,132,228,24,343,154,251,62,297,104,205)(5,365,181,273,71,319,133,227,25,342,155,250,63,296,105,204)(6,364,182,272,72,318,134,226,26,341,156,249,64,295,106,203)(7,363,183,271,73,317,135,225,27,340,157,248,65,294,107,202)(8,362,184,270,74,316,136,224,28,339,158,247,66,293,108,201)(9,361,162,269,75,315,137,223,29,338,159,246,67,292,109,200)(10,360,163,268,76,314,138,222,30,337,160,245,68,291,110,199)(11,359,164,267,77,313,116,221,31,336,161,244,69,290,111,198)(12,358,165,266,78,312,117,220,32,335,139,243,47,289,112,197)(13,357,166,265,79,311,118,219,33,334,140,242,48,288,113,196)(14,356,167,264,80,310,119,218,34,333,141,241,49,287,114,195)(15,355,168,263,81,309,120,217,35,332,142,240,50,286,115,194)(16,354,169,262,82,308,121,216,36,331,143,239,51,285,93,193)(17,353,170,261,83,307,122,215,37,330,144,238,52,284,94,192)(18,352,171,260,84,306,123,214,38,329,145,237,53,283,95,191)(19,351,172,259,85,305,124,213,39,328,146,236,54,282,96,190)(20,350,173,258,86,304,125,212,40,327,147,235,55,281,97,189)(21,349,174,257,87,303,126,211,41,326,148,234,56,280,98,188)(22,348,175,256,88,302,127,210,42,325,149,233,57,279,99,187)(23,347,176,255,89,301,128,209,43,324,150,232,58,278,100,186) );
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)], [(1,346,177,254,90,300,129,208,44,323,151,231,59,277,101,185),(2,368,178,276,91,322,130,230,45,345,152,253,60,299,102,207),(3,367,179,275,92,321,131,229,46,344,153,252,61,298,103,206),(4,366,180,274,70,320,132,228,24,343,154,251,62,297,104,205),(5,365,181,273,71,319,133,227,25,342,155,250,63,296,105,204),(6,364,182,272,72,318,134,226,26,341,156,249,64,295,106,203),(7,363,183,271,73,317,135,225,27,340,157,248,65,294,107,202),(8,362,184,270,74,316,136,224,28,339,158,247,66,293,108,201),(9,361,162,269,75,315,137,223,29,338,159,246,67,292,109,200),(10,360,163,268,76,314,138,222,30,337,160,245,68,291,110,199),(11,359,164,267,77,313,116,221,31,336,161,244,69,290,111,198),(12,358,165,266,78,312,117,220,32,335,139,243,47,289,112,197),(13,357,166,265,79,311,118,219,33,334,140,242,48,288,113,196),(14,356,167,264,80,310,119,218,34,333,141,241,49,287,114,195),(15,355,168,263,81,309,120,217,35,332,142,240,50,286,115,194),(16,354,169,262,82,308,121,216,36,331,143,239,51,285,93,193),(17,353,170,261,83,307,122,215,37,330,144,238,52,284,94,192),(18,352,171,260,84,306,123,214,38,329,145,237,53,283,95,191),(19,351,172,259,85,305,124,213,39,328,146,236,54,282,96,190),(20,350,173,258,86,304,125,212,40,327,147,235,55,281,97,189),(21,349,174,257,87,303,126,211,41,326,148,234,56,280,98,188),(22,348,175,256,88,302,127,210,42,325,149,233,57,279,99,187),(23,347,176,255,89,301,128,209,43,324,150,232,58,278,100,186)]])
104 conjugacy classes
| class | 1 | 2 | 4A | 4B | 8A | 8B | 8C | 8D | 16A | ··· | 16H | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V | 184A | ··· | 184AR |
| order | 1 | 2 | 4 | 4 | 8 | 8 | 8 | 8 | 16 | ··· | 16 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 23 | ··· | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | |||||
| image | C1 | C2 | C4 | C8 | C16 | D23 | Dic23 | C23⋊C8 | C23⋊C16 |
| kernel | C23⋊C16 | C184 | C92 | C46 | C23 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 11 | 11 | 22 | 44 |
Matrix representation of C23⋊C16 ►in GL2(𝔽3313) generated by
| 0 | 1 |
| 3312 | 1146 |
| 541 | 1639 |
| 2094 | 2772 |
G:=sub<GL(2,GF(3313))| [0,3312,1,1146],[541,2094,1639,2772] >;
C23⋊C16 in GAP, Magma, Sage, TeX
C_{23}\rtimes C_{16} % in TeX
G:=Group("C23:C16"); // GroupNames label
G:=SmallGroup(368,1);
// by ID
G=gap.SmallGroup(368,1);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-23,10,26,42,8804]);
// Polycyclic
G:=Group<a,b|a^23=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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