metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C20⋊3C16, C40.64D4, C8.28D20, C40.15Q8, C8.15Dic10, C42.8Dic5, C10.11M5(2), C20.51M4(2), C4⋊(C5⋊2C16), C5⋊3(C4⋊C16), (C4×C8).3D5, (C4×C40).20C2, (C4×C20).30C4, (C2×C40).34C4, (C2×C20).13C8, C20.69(C4⋊C4), C10.12(C4⋊C8), (C2×C8).5Dic5, C10.17(C2×C16), (C2×C8).331D10, C2.1(C20⋊3C8), C4.18(C4⋊Dic5), C2.2(C20.4C8), (C2×C40).396C22, C4.10(C4.Dic5), C2.3(C2×C5⋊2C16), (C2×C5⋊2C16).8C2, (C2×C10).55(C2×C8), (C2×C4).3(C5⋊2C8), C22.9(C2×C5⋊2C8), (C2×C20).484(C2×C4), (C2×C4).92(C2×Dic5), SmallGroup(320,20)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊3C16
G = < a,b | a20=b16=1, bab-1=a-1 >
Smallest permutation representation of C20⋊3C16
►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 89 55 274 39 175 226 185 73 293 248 213 315 108 152 130)(2 88 56 273 40 174 227 184 74 292 249 212 316 107 153 129)(3 87 57 272 21 173 228 183 75 291 250 211 317 106 154 128)(4 86 58 271 22 172 229 182 76 290 251 210 318 105 155 127)(5 85 59 270 23 171 230 181 77 289 252 209 319 104 156 126)(6 84 60 269 24 170 231 200 78 288 253 208 320 103 157 125)(7 83 41 268 25 169 232 199 79 287 254 207 301 102 158 124)(8 82 42 267 26 168 233 198 80 286 255 206 302 101 159 123)(9 81 43 266 27 167 234 197 61 285 256 205 303 120 160 122)(10 100 44 265 28 166 235 196 62 284 257 204 304 119 141 121)(11 99 45 264 29 165 236 195 63 283 258 203 305 118 142 140)(12 98 46 263 30 164 237 194 64 282 259 202 306 117 143 139)(13 97 47 262 31 163 238 193 65 281 260 201 307 116 144 138)(14 96 48 261 32 162 239 192 66 300 241 220 308 115 145 137)(15 95 49 280 33 161 240 191 67 299 242 219 309 114 146 136)(16 94 50 279 34 180 221 190 68 298 243 218 310 113 147 135)(17 93 51 278 35 179 222 189 69 297 244 217 311 112 148 134)(18 92 52 277 36 178 223 188 70 296 245 216 312 111 149 133)(19 91 53 276 37 177 224 187 71 295 246 215 313 110 150 132)(20 90 54 275 38 176 225 186 72 294 247 214 314 109 151 131)
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), (1,89,55,274,39,175,226,185,73,293,248,213,315,108,152,130)(2,88,56,273,40,174,227,184,74,292,249,212,316,107,153,129)(3,87,57,272,21,173,228,183,75,291,250,211,317,106,154,128)(4,86,58,271,22,172,229,182,76,290,251,210,318,105,155,127)(5,85,59,270,23,171,230,181,77,289,252,209,319,104,156,126)(6,84,60,269,24,170,231,200,78,288,253,208,320,103,157,125)(7,83,41,268,25,169,232,199,79,287,254,207,301,102,158,124)(8,82,42,267,26,168,233,198,80,286,255,206,302,101,159,123)(9,81,43,266,27,167,234,197,61,285,256,205,303,120,160,122)(10,100,44,265,28,166,235,196,62,284,257,204,304,119,141,121)(11,99,45,264,29,165,236,195,63,283,258,203,305,118,142,140)(12,98,46,263,30,164,237,194,64,282,259,202,306,117,143,139)(13,97,47,262,31,163,238,193,65,281,260,201,307,116,144,138)(14,96,48,261,32,162,239,192,66,300,241,220,308,115,145,137)(15,95,49,280,33,161,240,191,67,299,242,219,309,114,146,136)(16,94,50,279,34,180,221,190,68,298,243,218,310,113,147,135)(17,93,51,278,35,179,222,189,69,297,244,217,311,112,148,134)(18,92,52,277,36,178,223,188,70,296,245,216,312,111,149,133)(19,91,53,276,37,177,224,187,71,295,246,215,313,110,150,132)(20,90,54,275,38,176,225,186,72,294,247,214,314,109,151,131)>;
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), (1,89,55,274,39,175,226,185,73,293,248,213,315,108,152,130)(2,88,56,273,40,174,227,184,74,292,249,212,316,107,153,129)(3,87,57,272,21,173,228,183,75,291,250,211,317,106,154,128)(4,86,58,271,22,172,229,182,76,290,251,210,318,105,155,127)(5,85,59,270,23,171,230,181,77,289,252,209,319,104,156,126)(6,84,60,269,24,170,231,200,78,288,253,208,320,103,157,125)(7,83,41,268,25,169,232,199,79,287,254,207,301,102,158,124)(8,82,42,267,26,168,233,198,80,286,255,206,302,101,159,123)(9,81,43,266,27,167,234,197,61,285,256,205,303,120,160,122)(10,100,44,265,28,166,235,196,62,284,257,204,304,119,141,121)(11,99,45,264,29,165,236,195,63,283,258,203,305,118,142,140)(12,98,46,263,30,164,237,194,64,282,259,202,306,117,143,139)(13,97,47,262,31,163,238,193,65,281,260,201,307,116,144,138)(14,96,48,261,32,162,239,192,66,300,241,220,308,115,145,137)(15,95,49,280,33,161,240,191,67,299,242,219,309,114,146,136)(16,94,50,279,34,180,221,190,68,298,243,218,310,113,147,135)(17,93,51,278,35,179,222,189,69,297,244,217,311,112,148,134)(18,92,52,277,36,178,223,188,70,296,245,216,312,111,149,133)(19,91,53,276,37,177,224,187,71,295,246,215,313,110,150,132)(20,90,54,275,38,176,225,186,72,294,247,214,314,109,151,131) );
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)], [(1,89,55,274,39,175,226,185,73,293,248,213,315,108,152,130),(2,88,56,273,40,174,227,184,74,292,249,212,316,107,153,129),(3,87,57,272,21,173,228,183,75,291,250,211,317,106,154,128),(4,86,58,271,22,172,229,182,76,290,251,210,318,105,155,127),(5,85,59,270,23,171,230,181,77,289,252,209,319,104,156,126),(6,84,60,269,24,170,231,200,78,288,253,208,320,103,157,125),(7,83,41,268,25,169,232,199,79,287,254,207,301,102,158,124),(8,82,42,267,26,168,233,198,80,286,255,206,302,101,159,123),(9,81,43,266,27,167,234,197,61,285,256,205,303,120,160,122),(10,100,44,265,28,166,235,196,62,284,257,204,304,119,141,121),(11,99,45,264,29,165,236,195,63,283,258,203,305,118,142,140),(12,98,46,263,30,164,237,194,64,282,259,202,306,117,143,139),(13,97,47,262,31,163,238,193,65,281,260,201,307,116,144,138),(14,96,48,261,32,162,239,192,66,300,241,220,308,115,145,137),(15,95,49,280,33,161,240,191,67,299,242,219,309,114,146,136),(16,94,50,279,34,180,221,190,68,298,243,218,310,113,147,135),(17,93,51,278,35,179,222,189,69,297,244,217,311,112,148,134),(18,92,52,277,36,178,223,188,70,296,245,216,312,111,149,133),(19,91,53,276,37,177,224,187,71,295,246,215,313,110,150,132),(20,90,54,275,38,176,225,186,72,294,247,214,314,109,151,131)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 10A | ··· | 10F | 16A | ··· | 16P | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 16 | ··· | 16 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | C16 | D4 | Q8 | D5 | M4(2) | Dic5 | Dic5 | D10 | M5(2) | Dic10 | D20 | C5⋊2C8 | C5⋊2C16 | C4.Dic5 | C20.4C8 |
| kernel | C20⋊3C16 | C2×C5⋊2C16 | C4×C40 | C4×C20 | C2×C40 | C2×C20 | C20 | C40 | C40 | C4×C8 | C20 | C42 | C2×C8 | C2×C8 | C10 | C8 | C8 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 16 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 16 | 8 | 16 |
Matrix representation of C20⋊3C16 ►in GL3(𝔽241) generated by
| 1 | 0 | 0 |
| 0 | 119 | 44 |
| 0 | 78 | 41 |
| 44 | 0 | 0 |
| 0 | 14 | 168 |
| 0 | 6 | 227 |
G:=sub<GL(3,GF(241))| [1,0,0,0,119,78,0,44,41],[44,0,0,0,14,6,0,168,227] >;
C20⋊3C16 in GAP, Magma, Sage, TeX
C_{20}\rtimes_3C_{16} % in TeX
G:=Group("C20:3C16"); // GroupNames label
G:=SmallGroup(320,20);
// by ID
G=gap.SmallGroup(320,20);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,102,12550]);
// Polycyclic
G:=Group<a,b|a^20=b^16=1,b*a*b^-1=a^-1>;
// generators/relations
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