metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C40⋊6C8, C20.28SD16, C42.247D10, C20.45M4(2), C5⋊3(C8⋊2C8), C8⋊2(C5⋊2C8), (C4×C8).12D5, C20.69(C2×C8), (C2×C40).32C4, (C4×C40).14C2, C10.10(C4⋊C8), (C2×C20).45Q8, C20⋊3C8.2C2, (C2×C20).397D4, (C2×C4).159D20, (C2×C8).10Dic5, C2.3(C20⋊3C8), C2.1(C40⋊6C4), C4.15(C40⋊C2), (C2×C4).38Dic10, C10.10(C4.Q8), C4.4(C4.Dic5), C2.1(C40.6C4), C10.6(C8.C4), (C4×C20).318C22, C22.15(C4⋊Dic5), C4.6(C2×C5⋊2C8), (C2×C10).57(C4⋊C4), (C2×C20).464(C2×C4), (C2×C4).67(C2×Dic5), SmallGroup(320,15)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊6C8
G = < a,b | a40=b8=1, bab-1=a19 >
Smallest permutation representation of C40⋊6C8
►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)
(1 174 319 243 89 75 141 234)(2 193 320 262 90 54 142 213)(3 172 281 241 91 73 143 232)(4 191 282 260 92 52 144 211)(5 170 283 279 93 71 145 230)(6 189 284 258 94 50 146 209)(7 168 285 277 95 69 147 228)(8 187 286 256 96 48 148 207)(9 166 287 275 97 67 149 226)(10 185 288 254 98 46 150 205)(11 164 289 273 99 65 151 224)(12 183 290 252 100 44 152 203)(13 162 291 271 101 63 153 222)(14 181 292 250 102 42 154 201)(15 200 293 269 103 61 155 220)(16 179 294 248 104 80 156 239)(17 198 295 267 105 59 157 218)(18 177 296 246 106 78 158 237)(19 196 297 265 107 57 159 216)(20 175 298 244 108 76 160 235)(21 194 299 263 109 55 121 214)(22 173 300 242 110 74 122 233)(23 192 301 261 111 53 123 212)(24 171 302 280 112 72 124 231)(25 190 303 259 113 51 125 210)(26 169 304 278 114 70 126 229)(27 188 305 257 115 49 127 208)(28 167 306 276 116 68 128 227)(29 186 307 255 117 47 129 206)(30 165 308 274 118 66 130 225)(31 184 309 253 119 45 131 204)(32 163 310 272 120 64 132 223)(33 182 311 251 81 43 133 202)(34 161 312 270 82 62 134 221)(35 180 313 249 83 41 135 240)(36 199 314 268 84 60 136 219)(37 178 315 247 85 79 137 238)(38 197 316 266 86 58 138 217)(39 176 317 245 87 77 139 236)(40 195 318 264 88 56 140 215)
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), (1,174,319,243,89,75,141,234)(2,193,320,262,90,54,142,213)(3,172,281,241,91,73,143,232)(4,191,282,260,92,52,144,211)(5,170,283,279,93,71,145,230)(6,189,284,258,94,50,146,209)(7,168,285,277,95,69,147,228)(8,187,286,256,96,48,148,207)(9,166,287,275,97,67,149,226)(10,185,288,254,98,46,150,205)(11,164,289,273,99,65,151,224)(12,183,290,252,100,44,152,203)(13,162,291,271,101,63,153,222)(14,181,292,250,102,42,154,201)(15,200,293,269,103,61,155,220)(16,179,294,248,104,80,156,239)(17,198,295,267,105,59,157,218)(18,177,296,246,106,78,158,237)(19,196,297,265,107,57,159,216)(20,175,298,244,108,76,160,235)(21,194,299,263,109,55,121,214)(22,173,300,242,110,74,122,233)(23,192,301,261,111,53,123,212)(24,171,302,280,112,72,124,231)(25,190,303,259,113,51,125,210)(26,169,304,278,114,70,126,229)(27,188,305,257,115,49,127,208)(28,167,306,276,116,68,128,227)(29,186,307,255,117,47,129,206)(30,165,308,274,118,66,130,225)(31,184,309,253,119,45,131,204)(32,163,310,272,120,64,132,223)(33,182,311,251,81,43,133,202)(34,161,312,270,82,62,134,221)(35,180,313,249,83,41,135,240)(36,199,314,268,84,60,136,219)(37,178,315,247,85,79,137,238)(38,197,316,266,86,58,138,217)(39,176,317,245,87,77,139,236)(40,195,318,264,88,56,140,215)>;
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), (1,174,319,243,89,75,141,234)(2,193,320,262,90,54,142,213)(3,172,281,241,91,73,143,232)(4,191,282,260,92,52,144,211)(5,170,283,279,93,71,145,230)(6,189,284,258,94,50,146,209)(7,168,285,277,95,69,147,228)(8,187,286,256,96,48,148,207)(9,166,287,275,97,67,149,226)(10,185,288,254,98,46,150,205)(11,164,289,273,99,65,151,224)(12,183,290,252,100,44,152,203)(13,162,291,271,101,63,153,222)(14,181,292,250,102,42,154,201)(15,200,293,269,103,61,155,220)(16,179,294,248,104,80,156,239)(17,198,295,267,105,59,157,218)(18,177,296,246,106,78,158,237)(19,196,297,265,107,57,159,216)(20,175,298,244,108,76,160,235)(21,194,299,263,109,55,121,214)(22,173,300,242,110,74,122,233)(23,192,301,261,111,53,123,212)(24,171,302,280,112,72,124,231)(25,190,303,259,113,51,125,210)(26,169,304,278,114,70,126,229)(27,188,305,257,115,49,127,208)(28,167,306,276,116,68,128,227)(29,186,307,255,117,47,129,206)(30,165,308,274,118,66,130,225)(31,184,309,253,119,45,131,204)(32,163,310,272,120,64,132,223)(33,182,311,251,81,43,133,202)(34,161,312,270,82,62,134,221)(35,180,313,249,83,41,135,240)(36,199,314,268,84,60,136,219)(37,178,315,247,85,79,137,238)(38,197,316,266,86,58,138,217)(39,176,317,245,87,77,139,236)(40,195,318,264,88,56,140,215) );
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)], [(1,174,319,243,89,75,141,234),(2,193,320,262,90,54,142,213),(3,172,281,241,91,73,143,232),(4,191,282,260,92,52,144,211),(5,170,283,279,93,71,145,230),(6,189,284,258,94,50,146,209),(7,168,285,277,95,69,147,228),(8,187,286,256,96,48,148,207),(9,166,287,275,97,67,149,226),(10,185,288,254,98,46,150,205),(11,164,289,273,99,65,151,224),(12,183,290,252,100,44,152,203),(13,162,291,271,101,63,153,222),(14,181,292,250,102,42,154,201),(15,200,293,269,103,61,155,220),(16,179,294,248,104,80,156,239),(17,198,295,267,105,59,157,218),(18,177,296,246,106,78,158,237),(19,196,297,265,107,57,159,216),(20,175,298,244,108,76,160,235),(21,194,299,263,109,55,121,214),(22,173,300,242,110,74,122,233),(23,192,301,261,111,53,123,212),(24,171,302,280,112,72,124,231),(25,190,303,259,113,51,125,210),(26,169,304,278,114,70,126,229),(27,188,305,257,115,49,127,208),(28,167,306,276,116,68,128,227),(29,186,307,255,117,47,129,206),(30,165,308,274,118,66,130,225),(31,184,309,253,119,45,131,204),(32,163,310,272,120,64,132,223),(33,182,311,251,81,43,133,202),(34,161,312,270,82,62,134,221),(35,180,313,249,83,41,135,240),(36,199,314,268,84,60,136,219),(37,178,315,247,85,79,137,238),(38,197,316,266,86,58,138,217),(39,176,317,245,87,77,139,236),(40,195,318,264,88,56,140,215)]])
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 |
| type | + | + | + | + | - | + | + | - | - | + | |||||||||
| image | C1 | C2 | C2 | C4 | C8 | D4 | Q8 | D5 | M4(2) | SD16 | D10 | Dic5 | C8.C4 | C5⋊2C8 | Dic10 | D20 | C40⋊C2 | C4.Dic5 | C40.6C4 |
| kernel | C40⋊6C8 | C20⋊3C8 | C4×C40 | C2×C40 | C40 | C2×C20 | C2×C20 | C4×C8 | C20 | C20 | C42 | C2×C8 | C10 | C8 | C2×C4 | C2×C4 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 4 | 4 | 16 | 8 | 16 |
Matrix representation of C40⋊6C8 ►in GL4(𝔽41) generated by
| 1 | 35 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 25 | 23 |
| 0 | 0 | 18 | 28 |
| 24 | 15 | 0 | 0 |
| 36 | 17 | 0 | 0 |
| 0 | 0 | 3 | 31 |
| 0 | 0 | 10 | 38 |
G:=sub<GL(4,GF(41))| [1,6,0,0,35,6,0,0,0,0,25,18,0,0,23,28],[24,36,0,0,15,17,0,0,0,0,3,10,0,0,31,38] >;
C40⋊6C8 in GAP, Magma, Sage, TeX
C_{40}\rtimes_6C_8 % in TeX
G:=Group("C40:6C8"); // GroupNames label
G:=SmallGroup(320,15);
// by ID
G=gap.SmallGroup(320,15);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,64,100,1123,136,12550]);
// Polycyclic
G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^19>;
// generators/relations
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