metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊1C16, C42.8F5, C10.4M5(2), C20.12M4(2), C4⋊(C5⋊C16), C5⋊1(C4⋊C16), (C4×C20).8C4, (C2×C20).2C8, C10.1(C4⋊C8), C10.7(C2×C16), C5⋊2C8.36D4, C4.23(C4⋊F5), C20.23(C4⋊C4), C5⋊2C8.10Q8, C4.9(C4.F5), C2.1(C20⋊C8), C2.1(C20.C8), C2.3(C2×C5⋊C16), (C2×C5⋊C16).1C2, (C2×C4).4(C5⋊C8), C22.8(C2×C5⋊C8), (C2×C5⋊2C8).30C4, (C4×C5⋊2C8).18C2, (C2×C10).24(C2×C8), (C2×C4).148(C2×F5), (C2×C20).154(C2×C4), (C2×C5⋊2C8).341C22, SmallGroup(320,196)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊C16
G = < a,b | a20=b16=1, bab-1=a3 >
Smallest permutation representation of C20⋊C16
►Regular action on 320 points
Generators in S320
(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)
(1 238 93 210 48 302 270 183 29 130 171 117 157 258 63 286)(2 225 82 213 49 309 279 186 30 137 180 120 158 245 72 289)(3 232 91 216 50 316 268 189 31 124 169 103 159 252 61 292)(4 239 100 219 51 303 277 192 32 131 178 106 160 259 70 295)(5 226 89 202 52 310 266 195 33 138 167 109 141 246 79 298)(6 233 98 205 53 317 275 198 34 125 176 112 142 253 68 281)(7 240 87 208 54 304 264 181 35 132 165 115 143 260 77 284)(8 227 96 211 55 311 273 184 36 139 174 118 144 247 66 287)(9 234 85 214 56 318 262 187 37 126 163 101 145 254 75 290)(10 221 94 217 57 305 271 190 38 133 172 104 146 241 64 293)(11 228 83 220 58 312 280 193 39 140 161 107 147 248 73 296)(12 235 92 203 59 319 269 196 40 127 170 110 148 255 62 299)(13 222 81 206 60 306 278 199 21 134 179 113 149 242 71 282)(14 229 90 209 41 313 267 182 22 121 168 116 150 249 80 285)(15 236 99 212 42 320 276 185 23 128 177 119 151 256 69 288)(16 223 88 215 43 307 265 188 24 135 166 102 152 243 78 291)(17 230 97 218 44 314 274 191 25 122 175 105 153 250 67 294)(18 237 86 201 45 301 263 194 26 129 164 108 154 257 76 297)(19 224 95 204 46 308 272 197 27 136 173 111 155 244 65 300)(20 231 84 207 47 315 261 200 28 123 162 114 156 251 74 283)
G:=sub<Sym(320)| (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), (1,238,93,210,48,302,270,183,29,130,171,117,157,258,63,286)(2,225,82,213,49,309,279,186,30,137,180,120,158,245,72,289)(3,232,91,216,50,316,268,189,31,124,169,103,159,252,61,292)(4,239,100,219,51,303,277,192,32,131,178,106,160,259,70,295)(5,226,89,202,52,310,266,195,33,138,167,109,141,246,79,298)(6,233,98,205,53,317,275,198,34,125,176,112,142,253,68,281)(7,240,87,208,54,304,264,181,35,132,165,115,143,260,77,284)(8,227,96,211,55,311,273,184,36,139,174,118,144,247,66,287)(9,234,85,214,56,318,262,187,37,126,163,101,145,254,75,290)(10,221,94,217,57,305,271,190,38,133,172,104,146,241,64,293)(11,228,83,220,58,312,280,193,39,140,161,107,147,248,73,296)(12,235,92,203,59,319,269,196,40,127,170,110,148,255,62,299)(13,222,81,206,60,306,278,199,21,134,179,113,149,242,71,282)(14,229,90,209,41,313,267,182,22,121,168,116,150,249,80,285)(15,236,99,212,42,320,276,185,23,128,177,119,151,256,69,288)(16,223,88,215,43,307,265,188,24,135,166,102,152,243,78,291)(17,230,97,218,44,314,274,191,25,122,175,105,153,250,67,294)(18,237,86,201,45,301,263,194,26,129,164,108,154,257,76,297)(19,224,95,204,46,308,272,197,27,136,173,111,155,244,65,300)(20,231,84,207,47,315,261,200,28,123,162,114,156,251,74,283)>;
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), (1,238,93,210,48,302,270,183,29,130,171,117,157,258,63,286)(2,225,82,213,49,309,279,186,30,137,180,120,158,245,72,289)(3,232,91,216,50,316,268,189,31,124,169,103,159,252,61,292)(4,239,100,219,51,303,277,192,32,131,178,106,160,259,70,295)(5,226,89,202,52,310,266,195,33,138,167,109,141,246,79,298)(6,233,98,205,53,317,275,198,34,125,176,112,142,253,68,281)(7,240,87,208,54,304,264,181,35,132,165,115,143,260,77,284)(8,227,96,211,55,311,273,184,36,139,174,118,144,247,66,287)(9,234,85,214,56,318,262,187,37,126,163,101,145,254,75,290)(10,221,94,217,57,305,271,190,38,133,172,104,146,241,64,293)(11,228,83,220,58,312,280,193,39,140,161,107,147,248,73,296)(12,235,92,203,59,319,269,196,40,127,170,110,148,255,62,299)(13,222,81,206,60,306,278,199,21,134,179,113,149,242,71,282)(14,229,90,209,41,313,267,182,22,121,168,116,150,249,80,285)(15,236,99,212,42,320,276,185,23,128,177,119,151,256,69,288)(16,223,88,215,43,307,265,188,24,135,166,102,152,243,78,291)(17,230,97,218,44,314,274,191,25,122,175,105,153,250,67,294)(18,237,86,201,45,301,263,194,26,129,164,108,154,257,76,297)(19,224,95,204,46,308,272,197,27,136,173,111,155,244,65,300)(20,231,84,207,47,315,261,200,28,123,162,114,156,251,74,283) );
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)], [(1,238,93,210,48,302,270,183,29,130,171,117,157,258,63,286),(2,225,82,213,49,309,279,186,30,137,180,120,158,245,72,289),(3,232,91,216,50,316,268,189,31,124,169,103,159,252,61,292),(4,239,100,219,51,303,277,192,32,131,178,106,160,259,70,295),(5,226,89,202,52,310,266,195,33,138,167,109,141,246,79,298),(6,233,98,205,53,317,275,198,34,125,176,112,142,253,68,281),(7,240,87,208,54,304,264,181,35,132,165,115,143,260,77,284),(8,227,96,211,55,311,273,184,36,139,174,118,144,247,66,287),(9,234,85,214,56,318,262,187,37,126,163,101,145,254,75,290),(10,221,94,217,57,305,271,190,38,133,172,104,146,241,64,293),(11,228,83,220,58,312,280,193,39,140,161,107,147,248,73,296),(12,235,92,203,59,319,269,196,40,127,170,110,148,255,62,299),(13,222,81,206,60,306,278,199,21,134,179,113,149,242,71,282),(14,229,90,209,41,313,267,182,22,121,168,116,150,249,80,285),(15,236,99,212,42,320,276,185,23,128,177,119,151,256,69,288),(16,223,88,215,43,307,265,188,24,135,166,102,152,243,78,291),(17,230,97,218,44,314,274,191,25,122,175,105,153,250,67,294),(18,237,86,201,45,301,263,194,26,129,164,108,154,257,76,297),(19,224,95,204,46,308,272,197,27,136,173,111,155,244,65,300),(20,231,84,207,47,315,261,200,28,123,162,114,156,251,74,283)]])
56 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | 10B | 10C | 16A | ··· | 16P | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 16 | ··· | 16 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 10 | ··· | 10 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | Q8 | M4(2) | M5(2) | F5 | C5⋊C8 | C2×F5 | C5⋊C16 | C4.F5 | C4⋊F5 | C20.C8 |
| kernel | C20⋊C16 | C4×C5⋊2C8 | C2×C5⋊C16 | C2×C5⋊2C8 | C4×C20 | C2×C20 | C20 | C5⋊2C8 | C5⋊2C8 | C20 | C10 | C42 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 16 | 1 | 1 | 2 | 4 | 1 | 2 | 1 | 4 | 2 | 2 | 4 |
Matrix representation of C20⋊C16 ►in GL8(𝔽241)
| 177 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 239 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 240 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 240 |
| 18 | 134 | 0 | 0 | 0 | 0 | 0 | 0 |
| 188 | 223 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 234 | 19 | 0 | 0 | 0 | 0 |
| 0 | 0 | 137 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 155 | 128 | 136 |
| 0 | 0 | 0 | 0 | 139 | 50 | 180 | 147 |
| 0 | 0 | 0 | 0 | 191 | 61 | 94 | 34 |
| 0 | 0 | 0 | 0 | 105 | 189 | 230 | 86 |
G:=sub<GL(8,GF(241))| [177,0,0,0,0,0,0,0,2,64,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,239,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,240,240,240,240],[18,188,0,0,0,0,0,0,134,223,0,0,0,0,0,0,0,0,234,137,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,11,139,191,105,0,0,0,0,155,50,61,189,0,0,0,0,128,180,94,230,0,0,0,0,136,147,34,86] >;
C20⋊C16 in GAP, Magma, Sage, TeX
C_{20}\rtimes C_{16} % in TeX
G:=Group("C20:C16"); // GroupNames label
G:=SmallGroup(320,196);
// by ID
G=gap.SmallGroup(320,196);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,102,6278,3156]);
// Polycyclic
G:=Group<a,b|a^20=b^16=1,b*a*b^-1=a^3>;
// generators/relations
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