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## G = C40⋊C8order 320 = 26·5

### 4th semidirect product of C40 and C8 acting via C8/C2=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C40⋊C8
 Chief series C1 — C5 — C10 — C2×C10 — C2×Dic5 — C4×Dic5 — C4×C5⋊C8 — C40⋊C8
 Lower central C5 — C10 — C40⋊C8
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C40⋊C8
G = < a,b | a40=b8=1, bab-1=a13 >

Subgroups: 178 in 66 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C5, C8, C8, C2×C4, C2×C4, C10, C10, C42, C2×C8, C2×C8, Dic5, C20, C2×C10, C4×C8, C52C8, C40, C5⋊C8, C2×Dic5, C2×C20, C8⋊C8, C2×C52C8, C4×Dic5, C2×C40, C2×C5⋊C8, C8×Dic5, C4×C5⋊C8, C40⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C42, C2×C8, M4(2), F5, C4×C8, C8⋊C4, C5⋊C8, C2×F5, C8⋊C8, D5⋊C8, C4×F5, C2×C5⋊C8, C8⋊F5, C4×C5⋊C8, C40⋊C8

Smallest permutation representation of C40⋊C8
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)
(1 234 48 187 282 269 121 97)(2 231 57 200 283 266 130 110)(3 228 66 173 284 263 139 83)(4 225 75 186 285 260 148 96)(5 222 44 199 286 257 157 109)(6 219 53 172 287 254 126 82)(7 216 62 185 288 251 135 95)(8 213 71 198 289 248 144 108)(9 210 80 171 290 245 153 81)(10 207 49 184 291 242 122 94)(11 204 58 197 292 279 131 107)(12 201 67 170 293 276 140 120)(13 238 76 183 294 273 149 93)(14 235 45 196 295 270 158 106)(15 232 54 169 296 267 127 119)(16 229 63 182 297 264 136 92)(17 226 72 195 298 261 145 105)(18 223 41 168 299 258 154 118)(19 220 50 181 300 255 123 91)(20 217 59 194 301 252 132 104)(21 214 68 167 302 249 141 117)(22 211 77 180 303 246 150 90)(23 208 46 193 304 243 159 103)(24 205 55 166 305 280 128 116)(25 202 64 179 306 277 137 89)(26 239 73 192 307 274 146 102)(27 236 42 165 308 271 155 115)(28 233 51 178 309 268 124 88)(29 230 60 191 310 265 133 101)(30 227 69 164 311 262 142 114)(31 224 78 177 312 259 151 87)(32 221 47 190 313 256 160 100)(33 218 56 163 314 253 129 113)(34 215 65 176 315 250 138 86)(35 212 74 189 316 247 147 99)(36 209 43 162 317 244 156 112)(37 206 52 175 318 241 125 85)(38 203 61 188 319 278 134 98)(39 240 70 161 320 275 143 111)(40 237 79 174 281 272 152 84)```

`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), (1,234,48,187,282,269,121,97)(2,231,57,200,283,266,130,110)(3,228,66,173,284,263,139,83)(4,225,75,186,285,260,148,96)(5,222,44,199,286,257,157,109)(6,219,53,172,287,254,126,82)(7,216,62,185,288,251,135,95)(8,213,71,198,289,248,144,108)(9,210,80,171,290,245,153,81)(10,207,49,184,291,242,122,94)(11,204,58,197,292,279,131,107)(12,201,67,170,293,276,140,120)(13,238,76,183,294,273,149,93)(14,235,45,196,295,270,158,106)(15,232,54,169,296,267,127,119)(16,229,63,182,297,264,136,92)(17,226,72,195,298,261,145,105)(18,223,41,168,299,258,154,118)(19,220,50,181,300,255,123,91)(20,217,59,194,301,252,132,104)(21,214,68,167,302,249,141,117)(22,211,77,180,303,246,150,90)(23,208,46,193,304,243,159,103)(24,205,55,166,305,280,128,116)(25,202,64,179,306,277,137,89)(26,239,73,192,307,274,146,102)(27,236,42,165,308,271,155,115)(28,233,51,178,309,268,124,88)(29,230,60,191,310,265,133,101)(30,227,69,164,311,262,142,114)(31,224,78,177,312,259,151,87)(32,221,47,190,313,256,160,100)(33,218,56,163,314,253,129,113)(34,215,65,176,315,250,138,86)(35,212,74,189,316,247,147,99)(36,209,43,162,317,244,156,112)(37,206,52,175,318,241,125,85)(38,203,61,188,319,278,134,98)(39,240,70,161,320,275,143,111)(40,237,79,174,281,272,152,84)>;`

`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), (1,234,48,187,282,269,121,97)(2,231,57,200,283,266,130,110)(3,228,66,173,284,263,139,83)(4,225,75,186,285,260,148,96)(5,222,44,199,286,257,157,109)(6,219,53,172,287,254,126,82)(7,216,62,185,288,251,135,95)(8,213,71,198,289,248,144,108)(9,210,80,171,290,245,153,81)(10,207,49,184,291,242,122,94)(11,204,58,197,292,279,131,107)(12,201,67,170,293,276,140,120)(13,238,76,183,294,273,149,93)(14,235,45,196,295,270,158,106)(15,232,54,169,296,267,127,119)(16,229,63,182,297,264,136,92)(17,226,72,195,298,261,145,105)(18,223,41,168,299,258,154,118)(19,220,50,181,300,255,123,91)(20,217,59,194,301,252,132,104)(21,214,68,167,302,249,141,117)(22,211,77,180,303,246,150,90)(23,208,46,193,304,243,159,103)(24,205,55,166,305,280,128,116)(25,202,64,179,306,277,137,89)(26,239,73,192,307,274,146,102)(27,236,42,165,308,271,155,115)(28,233,51,178,309,268,124,88)(29,230,60,191,310,265,133,101)(30,227,69,164,311,262,142,114)(31,224,78,177,312,259,151,87)(32,221,47,190,313,256,160,100)(33,218,56,163,314,253,129,113)(34,215,65,176,315,250,138,86)(35,212,74,189,316,247,147,99)(36,209,43,162,317,244,156,112)(37,206,52,175,318,241,125,85)(38,203,61,188,319,278,134,98)(39,240,70,161,320,275,143,111)(40,237,79,174,281,272,152,84) );`

`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)], [(1,234,48,187,282,269,121,97),(2,231,57,200,283,266,130,110),(3,228,66,173,284,263,139,83),(4,225,75,186,285,260,148,96),(5,222,44,199,286,257,157,109),(6,219,53,172,287,254,126,82),(7,216,62,185,288,251,135,95),(8,213,71,198,289,248,144,108),(9,210,80,171,290,245,153,81),(10,207,49,184,291,242,122,94),(11,204,58,197,292,279,131,107),(12,201,67,170,293,276,140,120),(13,238,76,183,294,273,149,93),(14,235,45,196,295,270,158,106),(15,232,54,169,296,267,127,119),(16,229,63,182,297,264,136,92),(17,226,72,195,298,261,145,105),(18,223,41,168,299,258,154,118),(19,220,50,181,300,255,123,91),(20,217,59,194,301,252,132,104),(21,214,68,167,302,249,141,117),(22,211,77,180,303,246,150,90),(23,208,46,193,304,243,159,103),(24,205,55,166,305,280,128,116),(25,202,64,179,306,277,137,89),(26,239,73,192,307,274,146,102),(27,236,42,165,308,271,155,115),(28,233,51,178,309,268,124,88),(29,230,60,191,310,265,133,101),(30,227,69,164,311,262,142,114),(31,224,78,177,312,259,151,87),(32,221,47,190,313,256,160,100),(33,218,56,163,314,253,129,113),(34,215,65,176,315,250,138,86),(35,212,74,189,316,247,147,99),(36,209,43,162,317,244,156,112),(37,206,52,175,318,241,125,85),(38,203,61,188,319,278,134,98),(39,240,70,161,320,275,143,111),(40,237,79,174,281,272,152,84)]])`

56 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4L 5 8A 8B 8C 8D 8E ··· 8X 10A 10B 10C 20A 20B 20C 20D 40A ··· 40H order 1 2 2 2 4 4 4 4 4 ··· 4 5 8 8 8 8 8 ··· 8 10 10 10 20 20 20 20 40 ··· 40 size 1 1 1 1 1 1 1 1 5 ··· 5 4 2 2 2 2 10 ··· 10 4 4 4 4 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 4 4 type + + + + - + image C1 C2 C2 C4 C4 C4 C8 C8 M4(2) F5 C5⋊C8 C2×F5 D5⋊C8 C4×F5 C8⋊F5 kernel C40⋊C8 C8×Dic5 C4×C5⋊C8 C2×C5⋊2C8 C2×C40 C2×C5⋊C8 C5⋊2C8 C40 Dic5 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 1 2 2 2 8 8 8 8 1 2 1 2 2 8

Matrix representation of C40⋊C8 in GL6(𝔽41)

 9 39 0 0 0 0 4 32 0 0 0 0 0 0 0 32 9 0 0 0 0 32 0 9 0 0 0 32 0 0 0 0 9 32 0 0
,
 30 39 0 0 0 0 20 11 0 0 0 0 0 0 29 30 4 25 0 0 33 14 39 13 0 0 27 2 28 17 0 0 16 6 12 11

`G:=sub<GL(6,GF(41))| [9,4,0,0,0,0,39,32,0,0,0,0,0,0,0,0,0,9,0,0,32,32,32,32,0,0,9,0,0,0,0,0,0,9,0,0],[30,20,0,0,0,0,39,11,0,0,0,0,0,0,29,33,27,16,0,0,30,14,2,6,0,0,4,39,28,12,0,0,25,13,17,11] >;`

C40⋊C8 in GAP, Magma, Sage, TeX

`C_{40}\rtimes C_8`
`% in TeX`

`G:=Group("C40:C8");`
`// GroupNames label`

`G:=SmallGroup(320,217);`
`// by ID`

`G=gap.SmallGroup(320,217);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,477,64,100,136,6278,3156]);`
`// Polycyclic`

`G:=Group<a,b|a^40=b^8=1,b*a*b^-1=a^13>;`
`// generators/relations`

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