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G = C40.D5order 400 = 24·52

3rd non-split extension by C40 of D5 acting via D5/C5=C2

Aliases: C40.3D5, C525Q16, C51Dic20, C10.9D20, C20.48D10, C8.(C5⋊D5), (C5×C40).1C2, (C5×C10).24D4, C2.5(C20⋊D5), C524Q8.1C2, (C5×C20).34C22, C4.10(C2×C5⋊D5), SmallGroup(400,96)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C20 — C40.D5
 Chief series C1 — C5 — C52 — C5×C10 — C5×C20 — C52⋊4Q8 — C40.D5
 Lower central C52 — C5×C10 — C5×C20 — C40.D5
 Upper central C1 — C2 — C4 — C8

Generators and relations for C40.D5
G = < a,b,c | a40=b5=1, c2=a20, ab=ba, cac-1=a-1, cbc-1=b-1 >

Subgroups: 424 in 72 conjugacy classes, 35 normal (9 characteristic)
C1, C2, C4, C4, C5, C8, Q8, C10, Q16, Dic5, C20, C52, C40, Dic10, C5×C10, Dic20, C526C4, C5×C20, C5×C40, C524Q8, C40.D5
Quotients: C1, C2, C22, D4, D5, Q16, D10, D20, C5⋊D5, Dic20, C2×C5⋊D5, C20⋊D5, C40.D5

Smallest permutation representation of C40.D5
Regular action on 400 points
Generators in S400
```(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)(321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360)(361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400)
(1 279 377 141 78)(2 280 378 142 79)(3 241 379 143 80)(4 242 380 144 41)(5 243 381 145 42)(6 244 382 146 43)(7 245 383 147 44)(8 246 384 148 45)(9 247 385 149 46)(10 248 386 150 47)(11 249 387 151 48)(12 250 388 152 49)(13 251 389 153 50)(14 252 390 154 51)(15 253 391 155 52)(16 254 392 156 53)(17 255 393 157 54)(18 256 394 158 55)(19 257 395 159 56)(20 258 396 160 57)(21 259 397 121 58)(22 260 398 122 59)(23 261 399 123 60)(24 262 400 124 61)(25 263 361 125 62)(26 264 362 126 63)(27 265 363 127 64)(28 266 364 128 65)(29 267 365 129 66)(30 268 366 130 67)(31 269 367 131 68)(32 270 368 132 69)(33 271 369 133 70)(34 272 370 134 71)(35 273 371 135 72)(36 274 372 136 73)(37 275 373 137 74)(38 276 374 138 75)(39 277 375 139 76)(40 278 376 140 77)(81 354 309 226 188)(82 355 310 227 189)(83 356 311 228 190)(84 357 312 229 191)(85 358 313 230 192)(86 359 314 231 193)(87 360 315 232 194)(88 321 316 233 195)(89 322 317 234 196)(90 323 318 235 197)(91 324 319 236 198)(92 325 320 237 199)(93 326 281 238 200)(94 327 282 239 161)(95 328 283 240 162)(96 329 284 201 163)(97 330 285 202 164)(98 331 286 203 165)(99 332 287 204 166)(100 333 288 205 167)(101 334 289 206 168)(102 335 290 207 169)(103 336 291 208 170)(104 337 292 209 171)(105 338 293 210 172)(106 339 294 211 173)(107 340 295 212 174)(108 341 296 213 175)(109 342 297 214 176)(110 343 298 215 177)(111 344 299 216 178)(112 345 300 217 179)(113 346 301 218 180)(114 347 302 219 181)(115 348 303 220 182)(116 349 304 221 183)(117 350 305 222 184)(118 351 306 223 185)(119 352 307 224 186)(120 353 308 225 187)
(1 196 21 176)(2 195 22 175)(3 194 23 174)(4 193 24 173)(5 192 25 172)(6 191 26 171)(7 190 27 170)(8 189 28 169)(9 188 29 168)(10 187 30 167)(11 186 31 166)(12 185 32 165)(13 184 33 164)(14 183 34 163)(15 182 35 162)(16 181 36 161)(17 180 37 200)(18 179 38 199)(19 178 39 198)(20 177 40 197)(41 86 61 106)(42 85 62 105)(43 84 63 104)(44 83 64 103)(45 82 65 102)(46 81 66 101)(47 120 67 100)(48 119 68 99)(49 118 69 98)(50 117 70 97)(51 116 71 96)(52 115 72 95)(53 114 73 94)(54 113 74 93)(55 112 75 92)(56 111 76 91)(57 110 77 90)(58 109 78 89)(59 108 79 88)(60 107 80 87)(121 342 141 322)(122 341 142 321)(123 340 143 360)(124 339 144 359)(125 338 145 358)(126 337 146 357)(127 336 147 356)(128 335 148 355)(129 334 149 354)(130 333 150 353)(131 332 151 352)(132 331 152 351)(133 330 153 350)(134 329 154 349)(135 328 155 348)(136 327 156 347)(137 326 157 346)(138 325 158 345)(139 324 159 344)(140 323 160 343)(201 252 221 272)(202 251 222 271)(203 250 223 270)(204 249 224 269)(205 248 225 268)(206 247 226 267)(207 246 227 266)(208 245 228 265)(209 244 229 264)(210 243 230 263)(211 242 231 262)(212 241 232 261)(213 280 233 260)(214 279 234 259)(215 278 235 258)(216 277 236 257)(217 276 237 256)(218 275 238 255)(219 274 239 254)(220 273 240 253)(281 393 301 373)(282 392 302 372)(283 391 303 371)(284 390 304 370)(285 389 305 369)(286 388 306 368)(287 387 307 367)(288 386 308 366)(289 385 309 365)(290 384 310 364)(291 383 311 363)(292 382 312 362)(293 381 313 361)(294 380 314 400)(295 379 315 399)(296 378 316 398)(297 377 317 397)(298 376 318 396)(299 375 319 395)(300 374 320 394)```

`G:=sub<Sym(400)| (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)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400), (1,279,377,141,78)(2,280,378,142,79)(3,241,379,143,80)(4,242,380,144,41)(5,243,381,145,42)(6,244,382,146,43)(7,245,383,147,44)(8,246,384,148,45)(9,247,385,149,46)(10,248,386,150,47)(11,249,387,151,48)(12,250,388,152,49)(13,251,389,153,50)(14,252,390,154,51)(15,253,391,155,52)(16,254,392,156,53)(17,255,393,157,54)(18,256,394,158,55)(19,257,395,159,56)(20,258,396,160,57)(21,259,397,121,58)(22,260,398,122,59)(23,261,399,123,60)(24,262,400,124,61)(25,263,361,125,62)(26,264,362,126,63)(27,265,363,127,64)(28,266,364,128,65)(29,267,365,129,66)(30,268,366,130,67)(31,269,367,131,68)(32,270,368,132,69)(33,271,369,133,70)(34,272,370,134,71)(35,273,371,135,72)(36,274,372,136,73)(37,275,373,137,74)(38,276,374,138,75)(39,277,375,139,76)(40,278,376,140,77)(81,354,309,226,188)(82,355,310,227,189)(83,356,311,228,190)(84,357,312,229,191)(85,358,313,230,192)(86,359,314,231,193)(87,360,315,232,194)(88,321,316,233,195)(89,322,317,234,196)(90,323,318,235,197)(91,324,319,236,198)(92,325,320,237,199)(93,326,281,238,200)(94,327,282,239,161)(95,328,283,240,162)(96,329,284,201,163)(97,330,285,202,164)(98,331,286,203,165)(99,332,287,204,166)(100,333,288,205,167)(101,334,289,206,168)(102,335,290,207,169)(103,336,291,208,170)(104,337,292,209,171)(105,338,293,210,172)(106,339,294,211,173)(107,340,295,212,174)(108,341,296,213,175)(109,342,297,214,176)(110,343,298,215,177)(111,344,299,216,178)(112,345,300,217,179)(113,346,301,218,180)(114,347,302,219,181)(115,348,303,220,182)(116,349,304,221,183)(117,350,305,222,184)(118,351,306,223,185)(119,352,307,224,186)(120,353,308,225,187), (1,196,21,176)(2,195,22,175)(3,194,23,174)(4,193,24,173)(5,192,25,172)(6,191,26,171)(7,190,27,170)(8,189,28,169)(9,188,29,168)(10,187,30,167)(11,186,31,166)(12,185,32,165)(13,184,33,164)(14,183,34,163)(15,182,35,162)(16,181,36,161)(17,180,37,200)(18,179,38,199)(19,178,39,198)(20,177,40,197)(41,86,61,106)(42,85,62,105)(43,84,63,104)(44,83,64,103)(45,82,65,102)(46,81,66,101)(47,120,67,100)(48,119,68,99)(49,118,69,98)(50,117,70,97)(51,116,71,96)(52,115,72,95)(53,114,73,94)(54,113,74,93)(55,112,75,92)(56,111,76,91)(57,110,77,90)(58,109,78,89)(59,108,79,88)(60,107,80,87)(121,342,141,322)(122,341,142,321)(123,340,143,360)(124,339,144,359)(125,338,145,358)(126,337,146,357)(127,336,147,356)(128,335,148,355)(129,334,149,354)(130,333,150,353)(131,332,151,352)(132,331,152,351)(133,330,153,350)(134,329,154,349)(135,328,155,348)(136,327,156,347)(137,326,157,346)(138,325,158,345)(139,324,159,344)(140,323,160,343)(201,252,221,272)(202,251,222,271)(203,250,223,270)(204,249,224,269)(205,248,225,268)(206,247,226,267)(207,246,227,266)(208,245,228,265)(209,244,229,264)(210,243,230,263)(211,242,231,262)(212,241,232,261)(213,280,233,260)(214,279,234,259)(215,278,235,258)(216,277,236,257)(217,276,237,256)(218,275,238,255)(219,274,239,254)(220,273,240,253)(281,393,301,373)(282,392,302,372)(283,391,303,371)(284,390,304,370)(285,389,305,369)(286,388,306,368)(287,387,307,367)(288,386,308,366)(289,385,309,365)(290,384,310,364)(291,383,311,363)(292,382,312,362)(293,381,313,361)(294,380,314,400)(295,379,315,399)(296,378,316,398)(297,377,317,397)(298,376,318,396)(299,375,319,395)(300,374,320,394)>;`

`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)(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360)(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400), (1,279,377,141,78)(2,280,378,142,79)(3,241,379,143,80)(4,242,380,144,41)(5,243,381,145,42)(6,244,382,146,43)(7,245,383,147,44)(8,246,384,148,45)(9,247,385,149,46)(10,248,386,150,47)(11,249,387,151,48)(12,250,388,152,49)(13,251,389,153,50)(14,252,390,154,51)(15,253,391,155,52)(16,254,392,156,53)(17,255,393,157,54)(18,256,394,158,55)(19,257,395,159,56)(20,258,396,160,57)(21,259,397,121,58)(22,260,398,122,59)(23,261,399,123,60)(24,262,400,124,61)(25,263,361,125,62)(26,264,362,126,63)(27,265,363,127,64)(28,266,364,128,65)(29,267,365,129,66)(30,268,366,130,67)(31,269,367,131,68)(32,270,368,132,69)(33,271,369,133,70)(34,272,370,134,71)(35,273,371,135,72)(36,274,372,136,73)(37,275,373,137,74)(38,276,374,138,75)(39,277,375,139,76)(40,278,376,140,77)(81,354,309,226,188)(82,355,310,227,189)(83,356,311,228,190)(84,357,312,229,191)(85,358,313,230,192)(86,359,314,231,193)(87,360,315,232,194)(88,321,316,233,195)(89,322,317,234,196)(90,323,318,235,197)(91,324,319,236,198)(92,325,320,237,199)(93,326,281,238,200)(94,327,282,239,161)(95,328,283,240,162)(96,329,284,201,163)(97,330,285,202,164)(98,331,286,203,165)(99,332,287,204,166)(100,333,288,205,167)(101,334,289,206,168)(102,335,290,207,169)(103,336,291,208,170)(104,337,292,209,171)(105,338,293,210,172)(106,339,294,211,173)(107,340,295,212,174)(108,341,296,213,175)(109,342,297,214,176)(110,343,298,215,177)(111,344,299,216,178)(112,345,300,217,179)(113,346,301,218,180)(114,347,302,219,181)(115,348,303,220,182)(116,349,304,221,183)(117,350,305,222,184)(118,351,306,223,185)(119,352,307,224,186)(120,353,308,225,187), (1,196,21,176)(2,195,22,175)(3,194,23,174)(4,193,24,173)(5,192,25,172)(6,191,26,171)(7,190,27,170)(8,189,28,169)(9,188,29,168)(10,187,30,167)(11,186,31,166)(12,185,32,165)(13,184,33,164)(14,183,34,163)(15,182,35,162)(16,181,36,161)(17,180,37,200)(18,179,38,199)(19,178,39,198)(20,177,40,197)(41,86,61,106)(42,85,62,105)(43,84,63,104)(44,83,64,103)(45,82,65,102)(46,81,66,101)(47,120,67,100)(48,119,68,99)(49,118,69,98)(50,117,70,97)(51,116,71,96)(52,115,72,95)(53,114,73,94)(54,113,74,93)(55,112,75,92)(56,111,76,91)(57,110,77,90)(58,109,78,89)(59,108,79,88)(60,107,80,87)(121,342,141,322)(122,341,142,321)(123,340,143,360)(124,339,144,359)(125,338,145,358)(126,337,146,357)(127,336,147,356)(128,335,148,355)(129,334,149,354)(130,333,150,353)(131,332,151,352)(132,331,152,351)(133,330,153,350)(134,329,154,349)(135,328,155,348)(136,327,156,347)(137,326,157,346)(138,325,158,345)(139,324,159,344)(140,323,160,343)(201,252,221,272)(202,251,222,271)(203,250,223,270)(204,249,224,269)(205,248,225,268)(206,247,226,267)(207,246,227,266)(208,245,228,265)(209,244,229,264)(210,243,230,263)(211,242,231,262)(212,241,232,261)(213,280,233,260)(214,279,234,259)(215,278,235,258)(216,277,236,257)(217,276,237,256)(218,275,238,255)(219,274,239,254)(220,273,240,253)(281,393,301,373)(282,392,302,372)(283,391,303,371)(284,390,304,370)(285,389,305,369)(286,388,306,368)(287,387,307,367)(288,386,308,366)(289,385,309,365)(290,384,310,364)(291,383,311,363)(292,382,312,362)(293,381,313,361)(294,380,314,400)(295,379,315,399)(296,378,316,398)(297,377,317,397)(298,376,318,396)(299,375,319,395)(300,374,320,394) );`

`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),(321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360),(361,362,363,364,365,366,367,368,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,391,392,393,394,395,396,397,398,399,400)], [(1,279,377,141,78),(2,280,378,142,79),(3,241,379,143,80),(4,242,380,144,41),(5,243,381,145,42),(6,244,382,146,43),(7,245,383,147,44),(8,246,384,148,45),(9,247,385,149,46),(10,248,386,150,47),(11,249,387,151,48),(12,250,388,152,49),(13,251,389,153,50),(14,252,390,154,51),(15,253,391,155,52),(16,254,392,156,53),(17,255,393,157,54),(18,256,394,158,55),(19,257,395,159,56),(20,258,396,160,57),(21,259,397,121,58),(22,260,398,122,59),(23,261,399,123,60),(24,262,400,124,61),(25,263,361,125,62),(26,264,362,126,63),(27,265,363,127,64),(28,266,364,128,65),(29,267,365,129,66),(30,268,366,130,67),(31,269,367,131,68),(32,270,368,132,69),(33,271,369,133,70),(34,272,370,134,71),(35,273,371,135,72),(36,274,372,136,73),(37,275,373,137,74),(38,276,374,138,75),(39,277,375,139,76),(40,278,376,140,77),(81,354,309,226,188),(82,355,310,227,189),(83,356,311,228,190),(84,357,312,229,191),(85,358,313,230,192),(86,359,314,231,193),(87,360,315,232,194),(88,321,316,233,195),(89,322,317,234,196),(90,323,318,235,197),(91,324,319,236,198),(92,325,320,237,199),(93,326,281,238,200),(94,327,282,239,161),(95,328,283,240,162),(96,329,284,201,163),(97,330,285,202,164),(98,331,286,203,165),(99,332,287,204,166),(100,333,288,205,167),(101,334,289,206,168),(102,335,290,207,169),(103,336,291,208,170),(104,337,292,209,171),(105,338,293,210,172),(106,339,294,211,173),(107,340,295,212,174),(108,341,296,213,175),(109,342,297,214,176),(110,343,298,215,177),(111,344,299,216,178),(112,345,300,217,179),(113,346,301,218,180),(114,347,302,219,181),(115,348,303,220,182),(116,349,304,221,183),(117,350,305,222,184),(118,351,306,223,185),(119,352,307,224,186),(120,353,308,225,187)], [(1,196,21,176),(2,195,22,175),(3,194,23,174),(4,193,24,173),(5,192,25,172),(6,191,26,171),(7,190,27,170),(8,189,28,169),(9,188,29,168),(10,187,30,167),(11,186,31,166),(12,185,32,165),(13,184,33,164),(14,183,34,163),(15,182,35,162),(16,181,36,161),(17,180,37,200),(18,179,38,199),(19,178,39,198),(20,177,40,197),(41,86,61,106),(42,85,62,105),(43,84,63,104),(44,83,64,103),(45,82,65,102),(46,81,66,101),(47,120,67,100),(48,119,68,99),(49,118,69,98),(50,117,70,97),(51,116,71,96),(52,115,72,95),(53,114,73,94),(54,113,74,93),(55,112,75,92),(56,111,76,91),(57,110,77,90),(58,109,78,89),(59,108,79,88),(60,107,80,87),(121,342,141,322),(122,341,142,321),(123,340,143,360),(124,339,144,359),(125,338,145,358),(126,337,146,357),(127,336,147,356),(128,335,148,355),(129,334,149,354),(130,333,150,353),(131,332,151,352),(132,331,152,351),(133,330,153,350),(134,329,154,349),(135,328,155,348),(136,327,156,347),(137,326,157,346),(138,325,158,345),(139,324,159,344),(140,323,160,343),(201,252,221,272),(202,251,222,271),(203,250,223,270),(204,249,224,269),(205,248,225,268),(206,247,226,267),(207,246,227,266),(208,245,228,265),(209,244,229,264),(210,243,230,263),(211,242,231,262),(212,241,232,261),(213,280,233,260),(214,279,234,259),(215,278,235,258),(216,277,236,257),(217,276,237,256),(218,275,238,255),(219,274,239,254),(220,273,240,253),(281,393,301,373),(282,392,302,372),(283,391,303,371),(284,390,304,370),(285,389,305,369),(286,388,306,368),(287,387,307,367),(288,386,308,366),(289,385,309,365),(290,384,310,364),(291,383,311,363),(292,382,312,362),(293,381,313,361),(294,380,314,400),(295,379,315,399),(296,378,316,398),(297,377,317,397),(298,376,318,396),(299,375,319,395),(300,374,320,394)]])`

103 conjugacy classes

 class 1 2 4A 4B 4C 5A ··· 5L 8A 8B 10A ··· 10L 20A ··· 20X 40A ··· 40AV order 1 2 4 4 4 5 ··· 5 8 8 10 ··· 10 20 ··· 20 40 ··· 40 size 1 1 2 100 100 2 ··· 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

103 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 type + + + + + - + + - image C1 C2 C2 D4 D5 Q16 D10 D20 Dic20 kernel C40.D5 C5×C40 C52⋊4Q8 C5×C10 C40 C52 C20 C10 C5 # reps 1 1 2 1 12 2 12 24 48

Matrix representation of C40.D5 in GL4(𝔽41) generated by

 5 3 0 0 38 23 0 0 0 0 26 36 0 0 37 15
,
 6 40 0 0 1 0 0 0 0 0 5 37 0 0 5 29
,
 10 26 0 0 4 31 0 0 0 0 5 28 0 0 5 36
`G:=sub<GL(4,GF(41))| [5,38,0,0,3,23,0,0,0,0,26,37,0,0,36,15],[6,1,0,0,40,0,0,0,0,0,5,5,0,0,37,29],[10,4,0,0,26,31,0,0,0,0,5,5,0,0,28,36] >;`

C40.D5 in GAP, Magma, Sage, TeX

`C_{40}.D_5`
`% in TeX`

`G:=Group("C40.D5");`
`// GroupNames label`

`G:=SmallGroup(400,96);`
`// by ID`

`G=gap.SmallGroup(400,96);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,73,79,218,50,1924,11525]);`
`// Polycyclic`

`G:=Group<a,b,c|a^40=b^5=1,c^2=a^20,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;`
`// generators/relations`

׿
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