metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊8C8, C42.280D10, C20.38M4(2), C5⋊4(C8⋊C8), C8⋊3(C5⋊2C8), (C4×C8).14D5, (C2×C40).45C4, (C4×C40).17C2, C20.73(C2×C8), C10.13(C4×C8), (C2×C8).12Dic5, C2.2(C40⋊8C4), C4.13(C8⋊D5), (C2×C10).37C42, C10.12(C8⋊C4), (C4×C20).336C22, C22.15(C4×Dic5), C2.3(C4×C5⋊2C8), C4.11(C2×C5⋊2C8), (C4×C5⋊2C8).17C2, (C2×C5⋊2C8).14C4, (C2×C4).165(C4×D5), (C2×C20).412(C2×C4), (C2×C4).89(C2×Dic5), SmallGroup(320,13)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8C8
G = < a,b | a40=b8=1, bab-1=a29 >
Subgroups: 134 in 66 conjugacy classes, 51 normal (13 characteristic)
C1, C2, C2, C4, C22, C5, C8, C8, C2×C4, C2×C4, C10, C10, C42, C2×C8, C2×C8, C20, C2×C10, C4×C8, C4×C8, C5⋊2C8, C40, C2×C20, C2×C20, C8⋊C8, C2×C5⋊2C8, C4×C20, C2×C40, C4×C5⋊2C8, C4×C40, C40⋊8C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D5, C42, C2×C8, M4(2), Dic5, D10, C4×C8, C8⋊C4, C5⋊2C8, C4×D5, C2×Dic5, C8⋊C8, C8⋊D5, C2×C5⋊2C8, C4×Dic5, C4×C5⋊2C8, C40⋊8C4, C40⋊8C8
►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)
(1 257 144 301 74 226 183 120)(2 246 145 290 75 215 184 109)(3 275 146 319 76 204 185 98)(4 264 147 308 77 233 186 87)(5 253 148 297 78 222 187 116)(6 242 149 286 79 211 188 105)(7 271 150 315 80 240 189 94)(8 260 151 304 41 229 190 83)(9 249 152 293 42 218 191 112)(10 278 153 282 43 207 192 101)(11 267 154 311 44 236 193 90)(12 256 155 300 45 225 194 119)(13 245 156 289 46 214 195 108)(14 274 157 318 47 203 196 97)(15 263 158 307 48 232 197 86)(16 252 159 296 49 221 198 115)(17 241 160 285 50 210 199 104)(18 270 121 314 51 239 200 93)(19 259 122 303 52 228 161 82)(20 248 123 292 53 217 162 111)(21 277 124 281 54 206 163 100)(22 266 125 310 55 235 164 89)(23 255 126 299 56 224 165 118)(24 244 127 288 57 213 166 107)(25 273 128 317 58 202 167 96)(26 262 129 306 59 231 168 85)(27 251 130 295 60 220 169 114)(28 280 131 284 61 209 170 103)(29 269 132 313 62 238 171 92)(30 258 133 302 63 227 172 81)(31 247 134 291 64 216 173 110)(32 276 135 320 65 205 174 99)(33 265 136 309 66 234 175 88)(34 254 137 298 67 223 176 117)(35 243 138 287 68 212 177 106)(36 272 139 316 69 201 178 95)(37 261 140 305 70 230 179 84)(38 250 141 294 71 219 180 113)(39 279 142 283 72 208 181 102)(40 268 143 312 73 237 182 91)
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), (1,257,144,301,74,226,183,120)(2,246,145,290,75,215,184,109)(3,275,146,319,76,204,185,98)(4,264,147,308,77,233,186,87)(5,253,148,297,78,222,187,116)(6,242,149,286,79,211,188,105)(7,271,150,315,80,240,189,94)(8,260,151,304,41,229,190,83)(9,249,152,293,42,218,191,112)(10,278,153,282,43,207,192,101)(11,267,154,311,44,236,193,90)(12,256,155,300,45,225,194,119)(13,245,156,289,46,214,195,108)(14,274,157,318,47,203,196,97)(15,263,158,307,48,232,197,86)(16,252,159,296,49,221,198,115)(17,241,160,285,50,210,199,104)(18,270,121,314,51,239,200,93)(19,259,122,303,52,228,161,82)(20,248,123,292,53,217,162,111)(21,277,124,281,54,206,163,100)(22,266,125,310,55,235,164,89)(23,255,126,299,56,224,165,118)(24,244,127,288,57,213,166,107)(25,273,128,317,58,202,167,96)(26,262,129,306,59,231,168,85)(27,251,130,295,60,220,169,114)(28,280,131,284,61,209,170,103)(29,269,132,313,62,238,171,92)(30,258,133,302,63,227,172,81)(31,247,134,291,64,216,173,110)(32,276,135,320,65,205,174,99)(33,265,136,309,66,234,175,88)(34,254,137,298,67,223,176,117)(35,243,138,287,68,212,177,106)(36,272,139,316,69,201,178,95)(37,261,140,305,70,230,179,84)(38,250,141,294,71,219,180,113)(39,279,142,283,72,208,181,102)(40,268,143,312,73,237,182,91)>;
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), (1,257,144,301,74,226,183,120)(2,246,145,290,75,215,184,109)(3,275,146,319,76,204,185,98)(4,264,147,308,77,233,186,87)(5,253,148,297,78,222,187,116)(6,242,149,286,79,211,188,105)(7,271,150,315,80,240,189,94)(8,260,151,304,41,229,190,83)(9,249,152,293,42,218,191,112)(10,278,153,282,43,207,192,101)(11,267,154,311,44,236,193,90)(12,256,155,300,45,225,194,119)(13,245,156,289,46,214,195,108)(14,274,157,318,47,203,196,97)(15,263,158,307,48,232,197,86)(16,252,159,296,49,221,198,115)(17,241,160,285,50,210,199,104)(18,270,121,314,51,239,200,93)(19,259,122,303,52,228,161,82)(20,248,123,292,53,217,162,111)(21,277,124,281,54,206,163,100)(22,266,125,310,55,235,164,89)(23,255,126,299,56,224,165,118)(24,244,127,288,57,213,166,107)(25,273,128,317,58,202,167,96)(26,262,129,306,59,231,168,85)(27,251,130,295,60,220,169,114)(28,280,131,284,61,209,170,103)(29,269,132,313,62,238,171,92)(30,258,133,302,63,227,172,81)(31,247,134,291,64,216,173,110)(32,276,135,320,65,205,174,99)(33,265,136,309,66,234,175,88)(34,254,137,298,67,223,176,117)(35,243,138,287,68,212,177,106)(36,272,139,316,69,201,178,95)(37,261,140,305,70,230,179,84)(38,250,141,294,71,219,180,113)(39,279,142,283,72,208,181,102)(40,268,143,312,73,237,182,91) );
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)], [(1,257,144,301,74,226,183,120),(2,246,145,290,75,215,184,109),(3,275,146,319,76,204,185,98),(4,264,147,308,77,233,186,87),(5,253,148,297,78,222,187,116),(6,242,149,286,79,211,188,105),(7,271,150,315,80,240,189,94),(8,260,151,304,41,229,190,83),(9,249,152,293,42,218,191,112),(10,278,153,282,43,207,192,101),(11,267,154,311,44,236,193,90),(12,256,155,300,45,225,194,119),(13,245,156,289,46,214,195,108),(14,274,157,318,47,203,196,97),(15,263,158,307,48,232,197,86),(16,252,159,296,49,221,198,115),(17,241,160,285,50,210,199,104),(18,270,121,314,51,239,200,93),(19,259,122,303,52,228,161,82),(20,248,123,292,53,217,162,111),(21,277,124,281,54,206,163,100),(22,266,125,310,55,235,164,89),(23,255,126,299,56,224,165,118),(24,244,127,288,57,213,166,107),(25,273,128,317,58,202,167,96),(26,262,129,306,59,231,168,85),(27,251,130,295,60,220,169,114),(28,280,131,284,61,209,170,103),(29,269,132,313,62,238,171,92),(30,258,133,302,63,227,172,81),(31,247,134,291,64,216,173,110),(32,276,135,320,65,205,174,99),(33,265,136,309,66,234,175,88),(34,254,137,298,67,223,176,117),(35,243,138,287,68,212,177,106),(36,272,139,316,69,201,178,95),(37,261,140,305,70,230,179,84),(38,250,141,294,71,219,180,113),(39,279,142,283,72,208,181,102),(40,268,143,312,73,237,182,91)]])
104 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 8A | ··· | 8H | 8I | ··· | 8X | 10A | ··· | 10F | 20A | ··· | 20X | 40A | ··· | 40AF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
104 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D5 | M4(2) | D10 | Dic5 | C5⋊2C8 | C4×D5 | C8⋊D5 |
| kernel | C40⋊8C8 | C4×C5⋊2C8 | C4×C40 | C2×C5⋊2C8 | C2×C40 | C40 | C4×C8 | C20 | C42 | C2×C8 | C8 | C2×C4 | C4 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 2 | 8 | 2 | 4 | 16 | 8 | 32 |
Matrix representation of C40⋊8C8 ►in GL3(𝔽41) generated by
| 9 | 0 | 0 |
| 0 | 33 | 40 |
| 0 | 1 | 40 |
| 14 | 0 | 0 |
| 0 | 6 | 19 |
| 0 | 18 | 35 |
G:=sub<GL(3,GF(41))| [9,0,0,0,33,1,0,40,40],[14,0,0,0,6,18,0,19,35] >;
C40⋊8C8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_8C_8 % in TeX
G:=Group("C40:8C8"); // GroupNames label
G:=SmallGroup(320,13);
// by ID
G=gap.SmallGroup(320,13);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,100,136,12550]);
// Polycyclic
G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^29>;
// generators/relations