metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C21⋊1C16, C42.1C8, C84.2C4, C56.2S3, C24.3D7, C8.2D21, C168.3C2, C28.2Dic3, C12.2Dic7, C4.2Dic21, C3⋊(C7⋊C16), C7⋊(C3⋊C16), C6.(C7⋊C8), C14.(C3⋊C8), C2.(C21⋊C8), SmallGroup(336,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C21 — C21⋊C16 |
Generators and relations for C21⋊C16
G = < a,b | a21=b16=1, bab-1=a-1 >
Smallest permutation representation of C21⋊C16
►Regular action on 336 points
Generators in S336
(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)
(1 333 153 245 66 289 114 207 25 306 134 228 44 264 94 182)(2 332 154 244 67 288 115 206 26 305 135 227 45 263 95 181)(3 331 155 243 68 287 116 205 27 304 136 226 46 262 96 180)(4 330 156 242 69 286 117 204 28 303 137 225 47 261 97 179)(5 329 157 241 70 285 118 203 29 302 138 224 48 260 98 178)(6 328 158 240 71 284 119 202 30 301 139 223 49 259 99 177)(7 327 159 239 72 283 120 201 31 300 140 222 50 258 100 176)(8 326 160 238 73 282 121 200 32 299 141 221 51 257 101 175)(9 325 161 237 74 281 122 199 33 298 142 220 52 256 102 174)(10 324 162 236 75 280 123 198 34 297 143 219 53 255 103 173)(11 323 163 235 76 279 124 197 35 296 144 218 54 254 104 172)(12 322 164 234 77 278 125 196 36 295 145 217 55 253 105 171)(13 321 165 233 78 277 126 195 37 315 146 216 56 273 85 170)(14 320 166 232 79 276 106 194 38 314 147 215 57 272 86 169)(15 319 167 252 80 275 107 193 39 313 127 214 58 271 87 189)(16 318 168 251 81 274 108 192 40 312 128 213 59 270 88 188)(17 317 148 250 82 294 109 191 41 311 129 212 60 269 89 187)(18 316 149 249 83 293 110 190 42 310 130 211 61 268 90 186)(19 336 150 248 84 292 111 210 22 309 131 231 62 267 91 185)(20 335 151 247 64 291 112 209 23 308 132 230 63 266 92 184)(21 334 152 246 65 290 113 208 24 307 133 229 43 265 93 183)
G:=sub<Sym(336)| (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), (1,333,153,245,66,289,114,207,25,306,134,228,44,264,94,182)(2,332,154,244,67,288,115,206,26,305,135,227,45,263,95,181)(3,331,155,243,68,287,116,205,27,304,136,226,46,262,96,180)(4,330,156,242,69,286,117,204,28,303,137,225,47,261,97,179)(5,329,157,241,70,285,118,203,29,302,138,224,48,260,98,178)(6,328,158,240,71,284,119,202,30,301,139,223,49,259,99,177)(7,327,159,239,72,283,120,201,31,300,140,222,50,258,100,176)(8,326,160,238,73,282,121,200,32,299,141,221,51,257,101,175)(9,325,161,237,74,281,122,199,33,298,142,220,52,256,102,174)(10,324,162,236,75,280,123,198,34,297,143,219,53,255,103,173)(11,323,163,235,76,279,124,197,35,296,144,218,54,254,104,172)(12,322,164,234,77,278,125,196,36,295,145,217,55,253,105,171)(13,321,165,233,78,277,126,195,37,315,146,216,56,273,85,170)(14,320,166,232,79,276,106,194,38,314,147,215,57,272,86,169)(15,319,167,252,80,275,107,193,39,313,127,214,58,271,87,189)(16,318,168,251,81,274,108,192,40,312,128,213,59,270,88,188)(17,317,148,250,82,294,109,191,41,311,129,212,60,269,89,187)(18,316,149,249,83,293,110,190,42,310,130,211,61,268,90,186)(19,336,150,248,84,292,111,210,22,309,131,231,62,267,91,185)(20,335,151,247,64,291,112,209,23,308,132,230,63,266,92,184)(21,334,152,246,65,290,113,208,24,307,133,229,43,265,93,183)>;
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), (1,333,153,245,66,289,114,207,25,306,134,228,44,264,94,182)(2,332,154,244,67,288,115,206,26,305,135,227,45,263,95,181)(3,331,155,243,68,287,116,205,27,304,136,226,46,262,96,180)(4,330,156,242,69,286,117,204,28,303,137,225,47,261,97,179)(5,329,157,241,70,285,118,203,29,302,138,224,48,260,98,178)(6,328,158,240,71,284,119,202,30,301,139,223,49,259,99,177)(7,327,159,239,72,283,120,201,31,300,140,222,50,258,100,176)(8,326,160,238,73,282,121,200,32,299,141,221,51,257,101,175)(9,325,161,237,74,281,122,199,33,298,142,220,52,256,102,174)(10,324,162,236,75,280,123,198,34,297,143,219,53,255,103,173)(11,323,163,235,76,279,124,197,35,296,144,218,54,254,104,172)(12,322,164,234,77,278,125,196,36,295,145,217,55,253,105,171)(13,321,165,233,78,277,126,195,37,315,146,216,56,273,85,170)(14,320,166,232,79,276,106,194,38,314,147,215,57,272,86,169)(15,319,167,252,80,275,107,193,39,313,127,214,58,271,87,189)(16,318,168,251,81,274,108,192,40,312,128,213,59,270,88,188)(17,317,148,250,82,294,109,191,41,311,129,212,60,269,89,187)(18,316,149,249,83,293,110,190,42,310,130,211,61,268,90,186)(19,336,150,248,84,292,111,210,22,309,131,231,62,267,91,185)(20,335,151,247,64,291,112,209,23,308,132,230,63,266,92,184)(21,334,152,246,65,290,113,208,24,307,133,229,43,265,93,183) );
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)], [(1,333,153,245,66,289,114,207,25,306,134,228,44,264,94,182),(2,332,154,244,67,288,115,206,26,305,135,227,45,263,95,181),(3,331,155,243,68,287,116,205,27,304,136,226,46,262,96,180),(4,330,156,242,69,286,117,204,28,303,137,225,47,261,97,179),(5,329,157,241,70,285,118,203,29,302,138,224,48,260,98,178),(6,328,158,240,71,284,119,202,30,301,139,223,49,259,99,177),(7,327,159,239,72,283,120,201,31,300,140,222,50,258,100,176),(8,326,160,238,73,282,121,200,32,299,141,221,51,257,101,175),(9,325,161,237,74,281,122,199,33,298,142,220,52,256,102,174),(10,324,162,236,75,280,123,198,34,297,143,219,53,255,103,173),(11,323,163,235,76,279,124,197,35,296,144,218,54,254,104,172),(12,322,164,234,77,278,125,196,36,295,145,217,55,253,105,171),(13,321,165,233,78,277,126,195,37,315,146,216,56,273,85,170),(14,320,166,232,79,276,106,194,38,314,147,215,57,272,86,169),(15,319,167,252,80,275,107,193,39,313,127,214,58,271,87,189),(16,318,168,251,81,274,108,192,40,312,128,213,59,270,88,188),(17,317,148,250,82,294,109,191,41,311,129,212,60,269,89,187),(18,316,149,249,83,293,110,190,42,310,130,211,61,268,90,186),(19,336,150,248,84,292,111,210,22,309,131,231,62,267,91,185),(20,335,151,247,64,291,112,209,23,308,132,230,63,266,92,184),(21,334,152,246,65,290,113,208,24,307,133,229,43,265,93,183)]])
96 conjugacy classes
| class | 1 | 2 | 3 | 4A | 4B | 6 | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 12A | 12B | 14A | 14B | 14C | 16A | ··· | 16H | 21A | ··· | 21F | 24A | 24B | 24C | 24D | 28A | ··· | 28F | 42A | ··· | 42F | 56A | ··· | 56L | 84A | ··· | 84L | 168A | ··· | 168X |
| order | 1 | 2 | 3 | 4 | 4 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 14 | 14 | 14 | 16 | ··· | 16 | 21 | ··· | 21 | 24 | 24 | 24 | 24 | 28 | ··· | 28 | 42 | ··· | 42 | 56 | ··· | 56 | 84 | ··· | 84 | 168 | ··· | 168 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 21 | ··· | 21 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
96 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | - | + | - | + | - | |||||||||
| image | C1 | C2 | C4 | C8 | C16 | S3 | Dic3 | D7 | C3⋊C8 | Dic7 | D21 | C3⋊C16 | C7⋊C8 | Dic21 | C7⋊C16 | C21⋊C8 | C21⋊C16 |
| kernel | C21⋊C16 | C168 | C84 | C42 | C21 | C56 | C28 | C24 | C14 | C12 | C8 | C7 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 4 | 8 | 1 | 1 | 3 | 2 | 3 | 6 | 4 | 6 | 6 | 12 | 12 | 24 |
Matrix representation of C21⋊C16 ►in GL2(𝔽41) generated by
| 38 | 16 |
| 22 | 19 |
| 0 | 3 |
| 1 | 0 |
G:=sub<GL(2,GF(41))| [38,22,16,19],[0,1,3,0] >;
C21⋊C16 in GAP, Magma, Sage, TeX
C_{21}\rtimes C_{16} % in TeX
G:=Group("C21:C16"); // GroupNames label
G:=SmallGroup(336,5);
// by ID
G=gap.SmallGroup(336,5);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,12,31,50,964,10373]);
// Polycyclic
G:=Group<a,b|a^21=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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