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G = C40⋊C8order 320 = 26·5

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

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C404C8, Dic5.6M4(2), C83(C5⋊C8), C52C87C8, C51(C8⋊C8), C10.8(C4×C8), (C2×C8).17F5, C20.21(C2×C8), (C2×C40).19C4, C4.16(D5⋊C8), (C2×C10).9C42, C2.1(C8⋊F5), C10.3(C8⋊C4), C22.14(C4×F5), (C8×Dic5).24C2, (C4×Dic5).351C22, C2.5(C4×C5⋊C8), (C2×C5⋊C8).1C4, (C4×C5⋊C8).6C2, C4.13(C2×C5⋊C8), (C2×C52C8).33C4, (C2×C4).151(C2×F5), (C2×C20).157(C2×C4), (C2×Dic5).120(C2×C4), SmallGroup(320,217)

Series: Derived Chief Lower central Upper central

C1C10 — C40⋊C8
C1C5C10C2×C10C2×Dic5C4×Dic5C4×C5⋊C8 — C40⋊C8
C5C10 — C40⋊C8
C1C2×C4C2×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 [×2], C4 [×2], C4 [×4], C22, C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×2], C10, C10 [×2], C42, C2×C8, C2×C8 [×5], Dic5 [×4], C20 [×2], C2×C10, C4×C8 [×3], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C2×Dic5 [×2], C2×C20, C8⋊C8, C2×C52C8, C4×Dic5, C2×C40, C2×C5⋊C8 [×4], C8×Dic5, C4×C5⋊C8 [×2], C40⋊C8
Quotients: C1, C2 [×3], C4 [×6], C22, C8 [×4], C2×C4 [×3], C42, C2×C8 [×2], M4(2) [×4], F5, C4×C8, C8⋊C4 [×2], C5⋊C8 [×2], C2×F5, C8⋊C8, D5⋊C8, C4×F5, C2×C5⋊C8, C8⋊F5 [×2], 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)
(1 240 90 44 164 276 146 319)(2 237 99 57 165 273 155 292)(3 234 108 70 166 270 124 305)(4 231 117 43 167 267 133 318)(5 228 86 56 168 264 142 291)(6 225 95 69 169 261 151 304)(7 222 104 42 170 258 160 317)(8 219 113 55 171 255 129 290)(9 216 82 68 172 252 138 303)(10 213 91 41 173 249 147 316)(11 210 100 54 174 246 156 289)(12 207 109 67 175 243 125 302)(13 204 118 80 176 280 134 315)(14 201 87 53 177 277 143 288)(15 238 96 66 178 274 152 301)(16 235 105 79 179 271 121 314)(17 232 114 52 180 268 130 287)(18 229 83 65 181 265 139 300)(19 226 92 78 182 262 148 313)(20 223 101 51 183 259 157 286)(21 220 110 64 184 256 126 299)(22 217 119 77 185 253 135 312)(23 214 88 50 186 250 144 285)(24 211 97 63 187 247 153 298)(25 208 106 76 188 244 122 311)(26 205 115 49 189 241 131 284)(27 202 84 62 190 278 140 297)(28 239 93 75 191 275 149 310)(29 236 102 48 192 272 158 283)(30 233 111 61 193 269 127 296)(31 230 120 74 194 266 136 309)(32 227 89 47 195 263 145 282)(33 224 98 60 196 260 154 295)(34 221 107 73 197 257 123 308)(35 218 116 46 198 254 132 281)(36 215 85 59 199 251 141 294)(37 212 94 72 200 248 150 307)(38 209 103 45 161 245 159 320)(39 206 112 58 162 242 128 293)(40 203 81 71 163 279 137 306)

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

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

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

56 conjugacy classes

class 1 2A2B2C4A4B4C4D4E···4L 5 8A8B8C8D8E···8X10A10B10C20A20B20C20D40A···40H
order122244444···4588888···81010102020202040···40
size111111115···54222210···1044444444···4

56 irreducible representations

dim111111112444444
type++++-+
imageC1C2C2C4C4C4C8C8M4(2)F5C5⋊C8C2×F5D5⋊C8C4×F5C8⋊F5
kernelC40⋊C8C8×Dic5C4×C5⋊C8C2×C52C8C2×C40C2×C5⋊C8C52C8C40Dic5C2×C8C8C2×C4C4C22C2
# reps112228888121228

Matrix representation of C40⋊C8 in GL6(𝔽41)

9390000
4320000
0003290
0003209
0003200
0093200
,
30390000
20110000
002930425
0033143913
002722817
001661211

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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