metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊2C8, C20.8M4(2), Dic5.11SD16, C8⋊2(C5⋊C8), C5⋊1(C8⋊2C8), (C2×C8).14F5, C10.3(C4⋊C8), C20.40(C2×C8), (C2×C40).14C4, C20⋊C8.7C2, C4.5(C4.F5), C2.1(C40⋊C4), C2.3(C20⋊C8), C10.3(C4.Q8), (C8×Dic5).21C2, (C2×Dic5).27Q8, C22.15(C4⋊F5), C2.1(C40.C4), C10.1(C8.C4), (C2×Dic5).170D4, (C4×Dic5).340C22, C4.6(C2×C5⋊C8), (C2×C5⋊2C8).19C4, (C2×C10).8(C4⋊C4), (C2×C4).117(C2×F5), (C2×C20).114(C2×C4), SmallGroup(320,219)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊2C8
G = < a,b | a40=b8=1, bab-1=a3 >
Smallest permutation representation of C40⋊2C8
►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)
(1 172 258 302 116 156 56 218)(2 199 267 305 117 143 65 221)(3 186 276 308 118 130 74 224)(4 173 245 311 119 157 43 227)(5 200 254 314 120 144 52 230)(6 187 263 317 81 131 61 233)(7 174 272 320 82 158 70 236)(8 161 241 283 83 145 79 239)(9 188 250 286 84 132 48 202)(10 175 259 289 85 159 57 205)(11 162 268 292 86 146 66 208)(12 189 277 295 87 133 75 211)(13 176 246 298 88 160 44 214)(14 163 255 301 89 147 53 217)(15 190 264 304 90 134 62 220)(16 177 273 307 91 121 71 223)(17 164 242 310 92 148 80 226)(18 191 251 313 93 135 49 229)(19 178 260 316 94 122 58 232)(20 165 269 319 95 149 67 235)(21 192 278 282 96 136 76 238)(22 179 247 285 97 123 45 201)(23 166 256 288 98 150 54 204)(24 193 265 291 99 137 63 207)(25 180 274 294 100 124 72 210)(26 167 243 297 101 151 41 213)(27 194 252 300 102 138 50 216)(28 181 261 303 103 125 59 219)(29 168 270 306 104 152 68 222)(30 195 279 309 105 139 77 225)(31 182 248 312 106 126 46 228)(32 169 257 315 107 153 55 231)(33 196 266 318 108 140 64 234)(34 183 275 281 109 127 73 237)(35 170 244 284 110 154 42 240)(36 197 253 287 111 141 51 203)(37 184 262 290 112 128 60 206)(38 171 271 293 113 155 69 209)(39 198 280 296 114 142 78 212)(40 185 249 299 115 129 47 215)
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), (1,172,258,302,116,156,56,218)(2,199,267,305,117,143,65,221)(3,186,276,308,118,130,74,224)(4,173,245,311,119,157,43,227)(5,200,254,314,120,144,52,230)(6,187,263,317,81,131,61,233)(7,174,272,320,82,158,70,236)(8,161,241,283,83,145,79,239)(9,188,250,286,84,132,48,202)(10,175,259,289,85,159,57,205)(11,162,268,292,86,146,66,208)(12,189,277,295,87,133,75,211)(13,176,246,298,88,160,44,214)(14,163,255,301,89,147,53,217)(15,190,264,304,90,134,62,220)(16,177,273,307,91,121,71,223)(17,164,242,310,92,148,80,226)(18,191,251,313,93,135,49,229)(19,178,260,316,94,122,58,232)(20,165,269,319,95,149,67,235)(21,192,278,282,96,136,76,238)(22,179,247,285,97,123,45,201)(23,166,256,288,98,150,54,204)(24,193,265,291,99,137,63,207)(25,180,274,294,100,124,72,210)(26,167,243,297,101,151,41,213)(27,194,252,300,102,138,50,216)(28,181,261,303,103,125,59,219)(29,168,270,306,104,152,68,222)(30,195,279,309,105,139,77,225)(31,182,248,312,106,126,46,228)(32,169,257,315,107,153,55,231)(33,196,266,318,108,140,64,234)(34,183,275,281,109,127,73,237)(35,170,244,284,110,154,42,240)(36,197,253,287,111,141,51,203)(37,184,262,290,112,128,60,206)(38,171,271,293,113,155,69,209)(39,198,280,296,114,142,78,212)(40,185,249,299,115,129,47,215)>;
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), (1,172,258,302,116,156,56,218)(2,199,267,305,117,143,65,221)(3,186,276,308,118,130,74,224)(4,173,245,311,119,157,43,227)(5,200,254,314,120,144,52,230)(6,187,263,317,81,131,61,233)(7,174,272,320,82,158,70,236)(8,161,241,283,83,145,79,239)(9,188,250,286,84,132,48,202)(10,175,259,289,85,159,57,205)(11,162,268,292,86,146,66,208)(12,189,277,295,87,133,75,211)(13,176,246,298,88,160,44,214)(14,163,255,301,89,147,53,217)(15,190,264,304,90,134,62,220)(16,177,273,307,91,121,71,223)(17,164,242,310,92,148,80,226)(18,191,251,313,93,135,49,229)(19,178,260,316,94,122,58,232)(20,165,269,319,95,149,67,235)(21,192,278,282,96,136,76,238)(22,179,247,285,97,123,45,201)(23,166,256,288,98,150,54,204)(24,193,265,291,99,137,63,207)(25,180,274,294,100,124,72,210)(26,167,243,297,101,151,41,213)(27,194,252,300,102,138,50,216)(28,181,261,303,103,125,59,219)(29,168,270,306,104,152,68,222)(30,195,279,309,105,139,77,225)(31,182,248,312,106,126,46,228)(32,169,257,315,107,153,55,231)(33,196,266,318,108,140,64,234)(34,183,275,281,109,127,73,237)(35,170,244,284,110,154,42,240)(36,197,253,287,111,141,51,203)(37,184,262,290,112,128,60,206)(38,171,271,293,113,155,69,209)(39,198,280,296,114,142,78,212)(40,185,249,299,115,129,47,215) );
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)], [(1,172,258,302,116,156,56,218),(2,199,267,305,117,143,65,221),(3,186,276,308,118,130,74,224),(4,173,245,311,119,157,43,227),(5,200,254,314,120,144,52,230),(6,187,263,317,81,131,61,233),(7,174,272,320,82,158,70,236),(8,161,241,283,83,145,79,239),(9,188,250,286,84,132,48,202),(10,175,259,289,85,159,57,205),(11,162,268,292,86,146,66,208),(12,189,277,295,87,133,75,211),(13,176,246,298,88,160,44,214),(14,163,255,301,89,147,53,217),(15,190,264,304,90,134,62,220),(16,177,273,307,91,121,71,223),(17,164,242,310,92,148,80,226),(18,191,251,313,93,135,49,229),(19,178,260,316,94,122,58,232),(20,165,269,319,95,149,67,235),(21,192,278,282,96,136,76,238),(22,179,247,285,97,123,45,201),(23,166,256,288,98,150,54,204),(24,193,265,291,99,137,63,207),(25,180,274,294,100,124,72,210),(26,167,243,297,101,151,41,213),(27,194,252,300,102,138,50,216),(28,181,261,303,103,125,59,219),(29,168,270,306,104,152,68,222),(30,195,279,309,105,139,77,225),(31,182,248,312,106,126,46,228),(32,169,257,315,107,153,55,231),(33,196,266,318,108,140,64,234),(34,183,275,281,109,127,73,237),(35,170,244,284,110,154,42,240),(36,197,253,287,111,141,51,203),(37,184,262,290,112,128,60,206),(38,171,271,293,113,155,69,209),(39,198,280,296,114,142,78,212),(40,185,249,299,115,129,47,215)]])
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 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | SD16 | M4(2) | C8.C4 | F5 | C5⋊C8 | C2×F5 | C4.F5 | C4⋊F5 | C40⋊C4 | C40.C4 |
| kernel | C40⋊2C8 | C8×Dic5 | C20⋊C8 | C2×C5⋊2C8 | C2×C40 | C40 | C2×Dic5 | C2×Dic5 | Dic5 | C20 | C10 | C2×C8 | C8 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 4 | 2 | 4 | 1 | 2 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊2C8 ►in GL6(𝔽41)
| 28 | 31 | 0 | 0 | 0 | 0 |
| 17 | 13 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 38 | 5 |
| 0 | 0 | 36 | 36 | 0 | 33 |
| 0 | 0 | 8 | 3 | 3 | 8 |
| 0 | 0 | 33 | 0 | 36 | 36 |
| 16 | 2 | 0 | 0 | 0 | 0 |
| 11 | 25 | 0 | 0 | 0 | 0 |
| 0 | 0 | 19 | 18 | 2 | 39 |
| 0 | 0 | 25 | 21 | 23 | 1 |
| 0 | 0 | 2 | 21 | 20 | 4 |
| 0 | 0 | 40 | 24 | 20 | 22 |
G:=sub<GL(6,GF(41))| [28,17,0,0,0,0,31,13,0,0,0,0,0,0,0,36,8,33,0,0,5,36,3,0,0,0,38,0,3,36,0,0,5,33,8,36],[16,11,0,0,0,0,2,25,0,0,0,0,0,0,19,25,2,40,0,0,18,21,21,24,0,0,2,23,20,20,0,0,39,1,4,22] >;
C40⋊2C8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_2C_8 % in TeX
G:=Group("C40:2C8"); // GroupNames label
G:=SmallGroup(320,219);
// by ID
G=gap.SmallGroup(320,219);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,1123,136,6278,3156]);
// Polycyclic
G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^3>;
// generators/relations
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