direct product, abelian, monomial, 2-elementary
Aliases: C8×C40, SmallGroup(320,126)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C8×C40 |
| C1 — C8×C40 |
| C1 — C8×C40 |
Generators and relations for C8×C40
G = < a,b | a8=b40=1, ab=ba >
Subgroups: 74, all normal (8 characteristic)
C1, C2, C4, C22, C5, C8, C2×C4, C10, C42, C2×C8, C20, C2×C10, C4×C8, C40, C2×C20, C82, C4×C20, C2×C40, C4×C40, C8×C40
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C10, C42, C2×C8, C20, C2×C10, C4×C8, C40, C2×C20, C82, C4×C20, C2×C40, C4×C40, C8×C40
►Regular action on 320 points
Generators in S320
(1 213 115 131 168 74 265 308)(2 214 116 132 169 75 266 309)(3 215 117 133 170 76 267 310)(4 216 118 134 171 77 268 311)(5 217 119 135 172 78 269 312)(6 218 120 136 173 79 270 313)(7 219 81 137 174 80 271 314)(8 220 82 138 175 41 272 315)(9 221 83 139 176 42 273 316)(10 222 84 140 177 43 274 317)(11 223 85 141 178 44 275 318)(12 224 86 142 179 45 276 319)(13 225 87 143 180 46 277 320)(14 226 88 144 181 47 278 281)(15 227 89 145 182 48 279 282)(16 228 90 146 183 49 280 283)(17 229 91 147 184 50 241 284)(18 230 92 148 185 51 242 285)(19 231 93 149 186 52 243 286)(20 232 94 150 187 53 244 287)(21 233 95 151 188 54 245 288)(22 234 96 152 189 55 246 289)(23 235 97 153 190 56 247 290)(24 236 98 154 191 57 248 291)(25 237 99 155 192 58 249 292)(26 238 100 156 193 59 250 293)(27 239 101 157 194 60 251 294)(28 240 102 158 195 61 252 295)(29 201 103 159 196 62 253 296)(30 202 104 160 197 63 254 297)(31 203 105 121 198 64 255 298)(32 204 106 122 199 65 256 299)(33 205 107 123 200 66 257 300)(34 206 108 124 161 67 258 301)(35 207 109 125 162 68 259 302)(36 208 110 126 163 69 260 303)(37 209 111 127 164 70 261 304)(38 210 112 128 165 71 262 305)(39 211 113 129 166 72 263 306)(40 212 114 130 167 73 264 307)
(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)
G:=sub<Sym(320)| (1,213,115,131,168,74,265,308)(2,214,116,132,169,75,266,309)(3,215,117,133,170,76,267,310)(4,216,118,134,171,77,268,311)(5,217,119,135,172,78,269,312)(6,218,120,136,173,79,270,313)(7,219,81,137,174,80,271,314)(8,220,82,138,175,41,272,315)(9,221,83,139,176,42,273,316)(10,222,84,140,177,43,274,317)(11,223,85,141,178,44,275,318)(12,224,86,142,179,45,276,319)(13,225,87,143,180,46,277,320)(14,226,88,144,181,47,278,281)(15,227,89,145,182,48,279,282)(16,228,90,146,183,49,280,283)(17,229,91,147,184,50,241,284)(18,230,92,148,185,51,242,285)(19,231,93,149,186,52,243,286)(20,232,94,150,187,53,244,287)(21,233,95,151,188,54,245,288)(22,234,96,152,189,55,246,289)(23,235,97,153,190,56,247,290)(24,236,98,154,191,57,248,291)(25,237,99,155,192,58,249,292)(26,238,100,156,193,59,250,293)(27,239,101,157,194,60,251,294)(28,240,102,158,195,61,252,295)(29,201,103,159,196,62,253,296)(30,202,104,160,197,63,254,297)(31,203,105,121,198,64,255,298)(32,204,106,122,199,65,256,299)(33,205,107,123,200,66,257,300)(34,206,108,124,161,67,258,301)(35,207,109,125,162,68,259,302)(36,208,110,126,163,69,260,303)(37,209,111,127,164,70,261,304)(38,210,112,128,165,71,262,305)(39,211,113,129,166,72,263,306)(40,212,114,130,167,73,264,307), (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)>;
G:=Group( (1,213,115,131,168,74,265,308)(2,214,116,132,169,75,266,309)(3,215,117,133,170,76,267,310)(4,216,118,134,171,77,268,311)(5,217,119,135,172,78,269,312)(6,218,120,136,173,79,270,313)(7,219,81,137,174,80,271,314)(8,220,82,138,175,41,272,315)(9,221,83,139,176,42,273,316)(10,222,84,140,177,43,274,317)(11,223,85,141,178,44,275,318)(12,224,86,142,179,45,276,319)(13,225,87,143,180,46,277,320)(14,226,88,144,181,47,278,281)(15,227,89,145,182,48,279,282)(16,228,90,146,183,49,280,283)(17,229,91,147,184,50,241,284)(18,230,92,148,185,51,242,285)(19,231,93,149,186,52,243,286)(20,232,94,150,187,53,244,287)(21,233,95,151,188,54,245,288)(22,234,96,152,189,55,246,289)(23,235,97,153,190,56,247,290)(24,236,98,154,191,57,248,291)(25,237,99,155,192,58,249,292)(26,238,100,156,193,59,250,293)(27,239,101,157,194,60,251,294)(28,240,102,158,195,61,252,295)(29,201,103,159,196,62,253,296)(30,202,104,160,197,63,254,297)(31,203,105,121,198,64,255,298)(32,204,106,122,199,65,256,299)(33,205,107,123,200,66,257,300)(34,206,108,124,161,67,258,301)(35,207,109,125,162,68,259,302)(36,208,110,126,163,69,260,303)(37,209,111,127,164,70,261,304)(38,210,112,128,165,71,262,305)(39,211,113,129,166,72,263,306)(40,212,114,130,167,73,264,307), (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) );
G=PermutationGroup([[(1,213,115,131,168,74,265,308),(2,214,116,132,169,75,266,309),(3,215,117,133,170,76,267,310),(4,216,118,134,171,77,268,311),(5,217,119,135,172,78,269,312),(6,218,120,136,173,79,270,313),(7,219,81,137,174,80,271,314),(8,220,82,138,175,41,272,315),(9,221,83,139,176,42,273,316),(10,222,84,140,177,43,274,317),(11,223,85,141,178,44,275,318),(12,224,86,142,179,45,276,319),(13,225,87,143,180,46,277,320),(14,226,88,144,181,47,278,281),(15,227,89,145,182,48,279,282),(16,228,90,146,183,49,280,283),(17,229,91,147,184,50,241,284),(18,230,92,148,185,51,242,285),(19,231,93,149,186,52,243,286),(20,232,94,150,187,53,244,287),(21,233,95,151,188,54,245,288),(22,234,96,152,189,55,246,289),(23,235,97,153,190,56,247,290),(24,236,98,154,191,57,248,291),(25,237,99,155,192,58,249,292),(26,238,100,156,193,59,250,293),(27,239,101,157,194,60,251,294),(28,240,102,158,195,61,252,295),(29,201,103,159,196,62,253,296),(30,202,104,160,197,63,254,297),(31,203,105,121,198,64,255,298),(32,204,106,122,199,65,256,299),(33,205,107,123,200,66,257,300),(34,206,108,124,161,67,258,301),(35,207,109,125,162,68,259,302),(36,208,110,126,163,69,260,303),(37,209,111,127,164,70,261,304),(38,210,112,128,165,71,262,305),(39,211,113,129,166,72,263,306),(40,212,114,130,167,73,264,307)], [(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)]])
320 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8AV | 10A | ··· | 10L | 20A | ··· | 20AV | 40A | ··· | 40GJ |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
320 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | ||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 |
| kernel | C8×C40 | C4×C40 | C2×C40 | C82 | C40 | C4×C8 | C2×C8 | C8 |
| # reps | 1 | 3 | 12 | 4 | 48 | 12 | 48 | 192 |
Matrix representation of C8×C40 ►in GL2(𝔽41) generated by
| 38 | 0 |
| 0 | 1 |
| 12 | 0 |
| 0 | 13 |
G:=sub<GL(2,GF(41))| [38,0,0,1],[12,0,0,13] >;
C8×C40 in GAP, Magma, Sage, TeX
C_8\times C_{40} % in TeX
G:=Group("C8xC40"); // GroupNames label
G:=SmallGroup(320,126);
// by ID
G=gap.SmallGroup(320,126);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,288,136,172]);
// Polycyclic
G:=Group<a,b|a^8=b^40=1,a*b=b*a>;
// generators/relations