metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊1C8, Dic5.7Q16, Dic5.12D8, C20.9M4(2), C8⋊1(C5⋊C8), C5⋊1(C8⋊1C8), C10.4(C4⋊C8), (C2×C8).10F5, (C2×C40).10C4, C20.41(C2×C8), C20⋊C8.8C2, C4.6(C4.F5), C2.1(D5.D8), C2.4(C20⋊C8), C10.5(C2.D8), (C2×Dic5).28Q8, (C8×Dic5).19C2, C22.16(C4⋊F5), C2.1(D10.Q8), C10.2(C8.C4), (C2×Dic5).171D4, (C4×Dic5).341C22, C4.7(C2×C5⋊C8), (C2×C5⋊2C8).20C4, (C2×C10).9(C4⋊C4), (C2×C4).118(C2×F5), (C2×C20).115(C2×C4), SmallGroup(320,220)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊1C8
G = < a,b | a40=b8=1, bab-1=a23 >
Smallest permutation representation of C40⋊1C8
►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)
(1 287 129 200 251 216 70 95)(2 294 138 183 252 223 79 118)(3 301 147 166 253 230 48 101)(4 308 156 189 254 237 57 84)(5 315 125 172 255 204 66 107)(6 282 134 195 256 211 75 90)(7 289 143 178 257 218 44 113)(8 296 152 161 258 225 53 96)(9 303 121 184 259 232 62 119)(10 310 130 167 260 239 71 102)(11 317 139 190 261 206 80 85)(12 284 148 173 262 213 49 108)(13 291 157 196 263 220 58 91)(14 298 126 179 264 227 67 114)(15 305 135 162 265 234 76 97)(16 312 144 185 266 201 45 120)(17 319 153 168 267 208 54 103)(18 286 122 191 268 215 63 86)(19 293 131 174 269 222 72 109)(20 300 140 197 270 229 41 92)(21 307 149 180 271 236 50 115)(22 314 158 163 272 203 59 98)(23 281 127 186 273 210 68 81)(24 288 136 169 274 217 77 104)(25 295 145 192 275 224 46 87)(26 302 154 175 276 231 55 110)(27 309 123 198 277 238 64 93)(28 316 132 181 278 205 73 116)(29 283 141 164 279 212 42 99)(30 290 150 187 280 219 51 82)(31 297 159 170 241 226 60 105)(32 304 128 193 242 233 69 88)(33 311 137 176 243 240 78 111)(34 318 146 199 244 207 47 94)(35 285 155 182 245 214 56 117)(36 292 124 165 246 221 65 100)(37 299 133 188 247 228 74 83)(38 306 142 171 248 235 43 106)(39 313 151 194 249 202 52 89)(40 320 160 177 250 209 61 112)
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), (1,287,129,200,251,216,70,95)(2,294,138,183,252,223,79,118)(3,301,147,166,253,230,48,101)(4,308,156,189,254,237,57,84)(5,315,125,172,255,204,66,107)(6,282,134,195,256,211,75,90)(7,289,143,178,257,218,44,113)(8,296,152,161,258,225,53,96)(9,303,121,184,259,232,62,119)(10,310,130,167,260,239,71,102)(11,317,139,190,261,206,80,85)(12,284,148,173,262,213,49,108)(13,291,157,196,263,220,58,91)(14,298,126,179,264,227,67,114)(15,305,135,162,265,234,76,97)(16,312,144,185,266,201,45,120)(17,319,153,168,267,208,54,103)(18,286,122,191,268,215,63,86)(19,293,131,174,269,222,72,109)(20,300,140,197,270,229,41,92)(21,307,149,180,271,236,50,115)(22,314,158,163,272,203,59,98)(23,281,127,186,273,210,68,81)(24,288,136,169,274,217,77,104)(25,295,145,192,275,224,46,87)(26,302,154,175,276,231,55,110)(27,309,123,198,277,238,64,93)(28,316,132,181,278,205,73,116)(29,283,141,164,279,212,42,99)(30,290,150,187,280,219,51,82)(31,297,159,170,241,226,60,105)(32,304,128,193,242,233,69,88)(33,311,137,176,243,240,78,111)(34,318,146,199,244,207,47,94)(35,285,155,182,245,214,56,117)(36,292,124,165,246,221,65,100)(37,299,133,188,247,228,74,83)(38,306,142,171,248,235,43,106)(39,313,151,194,249,202,52,89)(40,320,160,177,250,209,61,112)>;
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), (1,287,129,200,251,216,70,95)(2,294,138,183,252,223,79,118)(3,301,147,166,253,230,48,101)(4,308,156,189,254,237,57,84)(5,315,125,172,255,204,66,107)(6,282,134,195,256,211,75,90)(7,289,143,178,257,218,44,113)(8,296,152,161,258,225,53,96)(9,303,121,184,259,232,62,119)(10,310,130,167,260,239,71,102)(11,317,139,190,261,206,80,85)(12,284,148,173,262,213,49,108)(13,291,157,196,263,220,58,91)(14,298,126,179,264,227,67,114)(15,305,135,162,265,234,76,97)(16,312,144,185,266,201,45,120)(17,319,153,168,267,208,54,103)(18,286,122,191,268,215,63,86)(19,293,131,174,269,222,72,109)(20,300,140,197,270,229,41,92)(21,307,149,180,271,236,50,115)(22,314,158,163,272,203,59,98)(23,281,127,186,273,210,68,81)(24,288,136,169,274,217,77,104)(25,295,145,192,275,224,46,87)(26,302,154,175,276,231,55,110)(27,309,123,198,277,238,64,93)(28,316,132,181,278,205,73,116)(29,283,141,164,279,212,42,99)(30,290,150,187,280,219,51,82)(31,297,159,170,241,226,60,105)(32,304,128,193,242,233,69,88)(33,311,137,176,243,240,78,111)(34,318,146,199,244,207,47,94)(35,285,155,182,245,214,56,117)(36,292,124,165,246,221,65,100)(37,299,133,188,247,228,74,83)(38,306,142,171,248,235,43,106)(39,313,151,194,249,202,52,89)(40,320,160,177,250,209,61,112) );
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)], [(1,287,129,200,251,216,70,95),(2,294,138,183,252,223,79,118),(3,301,147,166,253,230,48,101),(4,308,156,189,254,237,57,84),(5,315,125,172,255,204,66,107),(6,282,134,195,256,211,75,90),(7,289,143,178,257,218,44,113),(8,296,152,161,258,225,53,96),(9,303,121,184,259,232,62,119),(10,310,130,167,260,239,71,102),(11,317,139,190,261,206,80,85),(12,284,148,173,262,213,49,108),(13,291,157,196,263,220,58,91),(14,298,126,179,264,227,67,114),(15,305,135,162,265,234,76,97),(16,312,144,185,266,201,45,120),(17,319,153,168,267,208,54,103),(18,286,122,191,268,215,63,86),(19,293,131,174,269,222,72,109),(20,300,140,197,270,229,41,92),(21,307,149,180,271,236,50,115),(22,314,158,163,272,203,59,98),(23,281,127,186,273,210,68,81),(24,288,136,169,274,217,77,104),(25,295,145,192,275,224,46,87),(26,302,154,175,276,231,55,110),(27,309,123,198,277,238,64,93),(28,316,132,181,278,205,73,116),(29,283,141,164,279,212,42,99),(30,290,150,187,280,219,51,82),(31,297,159,170,241,226,60,105),(32,304,128,193,242,233,69,88),(33,311,137,176,243,240,78,111),(34,318,146,199,244,207,47,94),(35,285,155,182,245,214,56,117),(36,292,124,165,246,221,65,100),(37,299,133,188,247,228,74,83),(38,306,142,171,248,235,43,106),(39,313,151,194,249,202,52,89),(40,320,160,177,250,209,61,112)])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | ··· | 8P | 10A | 10B | 10C | 20A | 20B | 20C | 20D | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 4 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | D8 | Q16 | M4(2) | C8.C4 | F5 | C5⋊C8 | C2×F5 | C4.F5 | C4⋊F5 | D5.D8 | D10.Q8 |
| kernel | C40⋊1C8 | C8×Dic5 | C20⋊C8 | C2×C5⋊2C8 | C2×C40 | C40 | C2×Dic5 | C2×Dic5 | Dic5 | Dic5 | C20 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊1C8 ►in GL6(𝔽41)
| 40 | 2 | 0 | 0 | 0 | 0 |
| 40 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 37 | 27 | 37 |
| 0 | 0 | 4 | 4 | 0 | 31 |
| 0 | 0 | 10 | 14 | 14 | 10 |
| 0 | 0 | 31 | 0 | 4 | 4 |
| 0 | 10 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 26 | 6 | 25 |
| 0 | 0 | 21 | 40 | 15 | 9 |
| 0 | 0 | 16 | 10 | 1 | 22 |
| 0 | 0 | 32 | 12 | 31 | 6 |
G:=sub<GL(6,GF(41))| [40,40,0,0,0,0,2,1,0,0,0,0,0,0,0,4,10,31,0,0,37,4,14,0,0,0,27,0,14,4,0,0,37,31,10,4],[0,5,0,0,0,0,10,0,0,0,0,0,0,0,35,21,16,32,0,0,26,40,10,12,0,0,6,15,1,31,0,0,25,9,22,6] >;
C40⋊1C8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_1C_8 % in TeX
G:=Group("C40:1C8"); // GroupNames label
G:=SmallGroup(320,220);
// by ID
G=gap.SmallGroup(320,220);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,288,100,1123,136,6278,3156]);
// Polycyclic
G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^23>;
// generators/relations
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