metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊5C8, C4.16D40, C20.34D8, C20.17Q16, C4.9Dic20, C42.248D10, C20.46M4(2), (C4×C8).6D5, C8⋊1(C5⋊2C8), C5⋊3(C8⋊1C8), (C4×C40).8C2, C20.70(C2×C8), (C2×C40).33C4, C10.11(C4⋊C8), (C2×C20).46Q8, C20⋊3C8.3C2, (C2×C8).8Dic5, (C2×C20).398D4, (C2×C4).160D20, C2.1(C40⋊5C4), C2.4(C20⋊3C8), (C2×C4).39Dic10, C10.13(C2.D8), C4.5(C4.Dic5), C2.2(C40.6C4), C10.7(C8.C4), (C4×C20).319C22, C22.16(C4⋊Dic5), C4.7(C2×C5⋊2C8), (C2×C10).58(C4⋊C4), (C2×C20).465(C2×C4), (C2×C4).68(C2×Dic5), SmallGroup(320,16)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊5C8
G = < a,b | a40=b8=1, bab-1=a-1 >
Smallest permutation representation of C40⋊5C8
►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)
(1 227 288 180 93 45 248 123)(2 226 289 179 94 44 249 122)(3 225 290 178 95 43 250 121)(4 224 291 177 96 42 251 160)(5 223 292 176 97 41 252 159)(6 222 293 175 98 80 253 158)(7 221 294 174 99 79 254 157)(8 220 295 173 100 78 255 156)(9 219 296 172 101 77 256 155)(10 218 297 171 102 76 257 154)(11 217 298 170 103 75 258 153)(12 216 299 169 104 74 259 152)(13 215 300 168 105 73 260 151)(14 214 301 167 106 72 261 150)(15 213 302 166 107 71 262 149)(16 212 303 165 108 70 263 148)(17 211 304 164 109 69 264 147)(18 210 305 163 110 68 265 146)(19 209 306 162 111 67 266 145)(20 208 307 161 112 66 267 144)(21 207 308 200 113 65 268 143)(22 206 309 199 114 64 269 142)(23 205 310 198 115 63 270 141)(24 204 311 197 116 62 271 140)(25 203 312 196 117 61 272 139)(26 202 313 195 118 60 273 138)(27 201 314 194 119 59 274 137)(28 240 315 193 120 58 275 136)(29 239 316 192 81 57 276 135)(30 238 317 191 82 56 277 134)(31 237 318 190 83 55 278 133)(32 236 319 189 84 54 279 132)(33 235 320 188 85 53 280 131)(34 234 281 187 86 52 241 130)(35 233 282 186 87 51 242 129)(36 232 283 185 88 50 243 128)(37 231 284 184 89 49 244 127)(38 230 285 183 90 48 245 126)(39 229 286 182 91 47 246 125)(40 228 287 181 92 46 247 124)
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), (1,227,288,180,93,45,248,123)(2,226,289,179,94,44,249,122)(3,225,290,178,95,43,250,121)(4,224,291,177,96,42,251,160)(5,223,292,176,97,41,252,159)(6,222,293,175,98,80,253,158)(7,221,294,174,99,79,254,157)(8,220,295,173,100,78,255,156)(9,219,296,172,101,77,256,155)(10,218,297,171,102,76,257,154)(11,217,298,170,103,75,258,153)(12,216,299,169,104,74,259,152)(13,215,300,168,105,73,260,151)(14,214,301,167,106,72,261,150)(15,213,302,166,107,71,262,149)(16,212,303,165,108,70,263,148)(17,211,304,164,109,69,264,147)(18,210,305,163,110,68,265,146)(19,209,306,162,111,67,266,145)(20,208,307,161,112,66,267,144)(21,207,308,200,113,65,268,143)(22,206,309,199,114,64,269,142)(23,205,310,198,115,63,270,141)(24,204,311,197,116,62,271,140)(25,203,312,196,117,61,272,139)(26,202,313,195,118,60,273,138)(27,201,314,194,119,59,274,137)(28,240,315,193,120,58,275,136)(29,239,316,192,81,57,276,135)(30,238,317,191,82,56,277,134)(31,237,318,190,83,55,278,133)(32,236,319,189,84,54,279,132)(33,235,320,188,85,53,280,131)(34,234,281,187,86,52,241,130)(35,233,282,186,87,51,242,129)(36,232,283,185,88,50,243,128)(37,231,284,184,89,49,244,127)(38,230,285,183,90,48,245,126)(39,229,286,182,91,47,246,125)(40,228,287,181,92,46,247,124)>;
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), (1,227,288,180,93,45,248,123)(2,226,289,179,94,44,249,122)(3,225,290,178,95,43,250,121)(4,224,291,177,96,42,251,160)(5,223,292,176,97,41,252,159)(6,222,293,175,98,80,253,158)(7,221,294,174,99,79,254,157)(8,220,295,173,100,78,255,156)(9,219,296,172,101,77,256,155)(10,218,297,171,102,76,257,154)(11,217,298,170,103,75,258,153)(12,216,299,169,104,74,259,152)(13,215,300,168,105,73,260,151)(14,214,301,167,106,72,261,150)(15,213,302,166,107,71,262,149)(16,212,303,165,108,70,263,148)(17,211,304,164,109,69,264,147)(18,210,305,163,110,68,265,146)(19,209,306,162,111,67,266,145)(20,208,307,161,112,66,267,144)(21,207,308,200,113,65,268,143)(22,206,309,199,114,64,269,142)(23,205,310,198,115,63,270,141)(24,204,311,197,116,62,271,140)(25,203,312,196,117,61,272,139)(26,202,313,195,118,60,273,138)(27,201,314,194,119,59,274,137)(28,240,315,193,120,58,275,136)(29,239,316,192,81,57,276,135)(30,238,317,191,82,56,277,134)(31,237,318,190,83,55,278,133)(32,236,319,189,84,54,279,132)(33,235,320,188,85,53,280,131)(34,234,281,187,86,52,241,130)(35,233,282,186,87,51,242,129)(36,232,283,185,88,50,243,128)(37,231,284,184,89,49,244,127)(38,230,285,183,90,48,245,126)(39,229,286,182,91,47,246,125)(40,228,287,181,92,46,247,124) );
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)], [(1,227,288,180,93,45,248,123),(2,226,289,179,94,44,249,122),(3,225,290,178,95,43,250,121),(4,224,291,177,96,42,251,160),(5,223,292,176,97,41,252,159),(6,222,293,175,98,80,253,158),(7,221,294,174,99,79,254,157),(8,220,295,173,100,78,255,156),(9,219,296,172,101,77,256,155),(10,218,297,171,102,76,257,154),(11,217,298,170,103,75,258,153),(12,216,299,169,104,74,259,152),(13,215,300,168,105,73,260,151),(14,214,301,167,106,72,261,150),(15,213,302,166,107,71,262,149),(16,212,303,165,108,70,263,148),(17,211,304,164,109,69,264,147),(18,210,305,163,110,68,265,146),(19,209,306,162,111,67,266,145),(20,208,307,161,112,66,267,144),(21,207,308,200,113,65,268,143),(22,206,309,199,114,64,269,142),(23,205,310,198,115,63,270,141),(24,204,311,197,116,62,271,140),(25,203,312,196,117,61,272,139),(26,202,313,195,118,60,273,138),(27,201,314,194,119,59,274,137),(28,240,315,193,120,58,275,136),(29,239,316,192,81,57,276,135),(30,238,317,191,82,56,277,134),(31,237,318,190,83,55,278,133),(32,236,319,189,84,54,279,132),(33,235,320,188,85,53,280,131),(34,234,281,187,86,52,241,130),(35,233,282,186,87,51,242,129),(36,232,283,185,88,50,243,128),(37,231,284,184,89,49,244,127),(38,230,285,183,90,48,245,126),(39,229,286,182,91,47,246,125),(40,228,287,181,92,46,247,124)]])
92 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 20 | ··· | 20 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
92 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | + | - | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | D5 | M4(2) | D8 | Q16 | D10 | Dic5 | C8.C4 | C5⋊2C8 | Dic10 | D20 | D40 | Dic20 | C4.Dic5 | C40.6C4 |
| kernel | C40⋊5C8 | C20⋊3C8 | C4×C40 | C2×C40 | C40 | C2×C20 | C2×C20 | C4×C8 | C20 | C20 | C20 | C42 | C2×C8 | C10 | C8 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C40⋊5C8 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 21 | 39 |
| 0 | 2 | 35 |
| 38 | 0 | 0 |
| 0 | 7 | 36 |
| 0 | 28 | 34 |
G:=sub<GL(3,GF(41))| [1,0,0,0,21,2,0,39,35],[38,0,0,0,7,28,0,36,34] >;
C40⋊5C8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_5C_8 % in TeX
G:=Group("C40:5C8"); // GroupNames label
G:=SmallGroup(320,16);
// by ID
G=gap.SmallGroup(320,16);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,288,100,1123,136,12550]);
// Polycyclic
G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^-1>;
// generators/relations
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