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G = C408C4.C2order 320 = 26·5

8th non-split extension by C408C4 of C2 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.20D10, C408C4.8C2, (C2×C8).174D10, Q8⋊C4.8D5, (C2×Q8).12D10, C4.30(C4○D20), C20.16(C4○D4), (C2×Dic5).35D4, Q8⋊Dic5.5C2, C10.Q16.3C2, C22.191(D4×D5), C4.Dic10.3C2, C4.56(D42D5), (C2×C20).237C23, (C2×C40).194C22, Dic5⋊Q8.6C2, C20.44D4.9C2, C4⋊Dic5.86C22, (Q8×C10).20C22, C2.10(Q16⋊D5), C10.28(C4.4D4), C2.16(SD16⋊D5), C10.55(C8.C22), (C4×Dic5).27C22, C52(C42.30C22), (C2×Dic10).70C22, C2.18(Dic5.5D4), (C2×C10).250(C2×D4), (C5×C4⋊C4).38C22, (C2×C52C8).32C22, (C5×Q8⋊C4).10C2, (C2×C4).344(C22×D5), SmallGroup(320,424)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C408C4.C2
C1C5C10C20C2×C20C4×Dic5Dic5⋊Q8 — C408C4.C2
C5C10C2×C20 — C408C4.C2
C1C22C2×C4Q8⋊C4

Generators and relations for C408C4.C2
 G = < a,b,c | a40=b4=1, c2=a20, bab-1=a29, cac-1=a11b2, cbc-1=b-1 >

Subgroups: 318 in 90 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C4 [×2], C4 [×6], C22, C5, C8 [×2], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], C42, C4⋊C4, C4⋊C4 [×5], C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C8⋊C4, Q8⋊C4, Q8⋊C4 [×3], C42.C2, C4⋊Q8, C52C8, C40, Dic10 [×2], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C42.30C22, C2×C52C8, C4×Dic5, C10.D4 [×3], C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.Q16, C408C4, C20.44D4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, Dic5⋊Q8, C408C4.C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C8.C22 [×2], C22×D5, C42.30C22, C4○D20, D4×D5, D42D5, Dic5.5D4, SD16⋊D5, Q16⋊D5, C408C4.C2

Smallest permutation representation of C408C4.C2
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)
(1 262 284 117)(2 251 285 106)(3 280 286 95)(4 269 287 84)(5 258 288 113)(6 247 289 102)(7 276 290 91)(8 265 291 120)(9 254 292 109)(10 243 293 98)(11 272 294 87)(12 261 295 116)(13 250 296 105)(14 279 297 94)(15 268 298 83)(16 257 299 112)(17 246 300 101)(18 275 301 90)(19 264 302 119)(20 253 303 108)(21 242 304 97)(22 271 305 86)(23 260 306 115)(24 249 307 104)(25 278 308 93)(26 267 309 82)(27 256 310 111)(28 245 311 100)(29 274 312 89)(30 263 313 118)(31 252 314 107)(32 241 315 96)(33 270 316 85)(34 259 317 114)(35 248 318 103)(36 277 319 92)(37 266 320 81)(38 255 281 110)(39 244 282 99)(40 273 283 88)(41 201 135 181)(42 230 136 170)(43 219 137 199)(44 208 138 188)(45 237 139 177)(46 226 140 166)(47 215 141 195)(48 204 142 184)(49 233 143 173)(50 222 144 162)(51 211 145 191)(52 240 146 180)(53 229 147 169)(54 218 148 198)(55 207 149 187)(56 236 150 176)(57 225 151 165)(58 214 152 194)(59 203 153 183)(60 232 154 172)(61 221 155 161)(62 210 156 190)(63 239 157 179)(64 228 158 168)(65 217 159 197)(66 206 160 186)(67 235 121 175)(68 224 122 164)(69 213 123 193)(70 202 124 182)(71 231 125 171)(72 220 126 200)(73 209 127 189)(74 238 128 178)(75 227 129 167)(76 216 130 196)(77 205 131 185)(78 234 132 174)(79 223 133 163)(80 212 134 192)
(1 49 21 69)(2 154 22 134)(3 71 23 51)(4 136 24 156)(5 53 25 73)(6 158 26 138)(7 75 27 55)(8 140 28 160)(9 57 29 77)(10 122 30 142)(11 79 31 59)(12 144 32 124)(13 61 33 41)(14 126 34 146)(15 43 35 63)(16 148 36 128)(17 65 37 45)(18 130 38 150)(19 47 39 67)(20 152 40 132)(42 307 62 287)(44 289 64 309)(46 311 66 291)(48 293 68 313)(50 315 70 295)(52 297 72 317)(54 319 74 299)(56 301 76 281)(58 283 78 303)(60 305 80 285)(81 237 101 217)(82 188 102 168)(83 219 103 239)(84 170 104 190)(85 201 105 221)(86 192 106 172)(87 223 107 203)(88 174 108 194)(89 205 109 225)(90 196 110 176)(91 227 111 207)(92 178 112 198)(93 209 113 229)(94 200 114 180)(95 231 115 211)(96 182 116 162)(97 213 117 233)(98 164 118 184)(99 235 119 215)(100 186 120 166)(121 302 141 282)(123 284 143 304)(125 306 145 286)(127 288 147 308)(129 310 149 290)(131 292 151 312)(133 314 153 294)(135 296 155 316)(137 318 157 298)(139 300 159 320)(161 270 181 250)(163 252 183 272)(165 274 185 254)(167 256 187 276)(169 278 189 258)(171 260 191 280)(173 242 193 262)(175 264 195 244)(177 246 197 266)(179 268 199 248)(202 261 222 241)(204 243 224 263)(206 265 226 245)(208 247 228 267)(210 269 230 249)(212 251 232 271)(214 273 234 253)(216 255 236 275)(218 277 238 257)(220 259 240 279)

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

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

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

44 conjugacy classes

class 1 2A2B2C4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12224444444455888810···102020202020···2040···40
size1111228820204040224420202···244448···84···4

44 irreducible representations

dim11111111222222244444
type+++++++++++++--+-
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C4○D20C8.C22D42D5D4×D5SD16⋊D5Q16⋊D5
kernelC408C4.C2C10.Q16C408C4C20.44D4Q8⋊Dic5C5×Q8⋊C4C4.Dic10Dic5⋊Q8C2×Dic5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C4C10C4C22C2C2
# reps11111111224222822244

Matrix representation of C408C4.C2 in GL6(𝔽41)

13220000
11280000
00300011
00503036
00002615
002215264
,
670000
24350000
008261024
004091214
001912915
0023213815
,
11190000
39300000
0014221040
002625113
004028619
0021313237

G:=sub<GL(6,GF(41))| [13,11,0,0,0,0,22,28,0,0,0,0,0,0,30,5,0,22,0,0,0,0,0,15,0,0,0,30,26,26,0,0,11,36,15,4],[6,24,0,0,0,0,7,35,0,0,0,0,0,0,8,40,19,23,0,0,26,9,12,21,0,0,10,12,9,38,0,0,24,14,15,15],[11,39,0,0,0,0,19,30,0,0,0,0,0,0,14,26,40,21,0,0,22,25,28,31,0,0,10,11,6,32,0,0,40,3,19,37] >;

C408C4.C2 in GAP, Magma, Sage, TeX

C_{40}\rtimes_8C_4.C_2
% in TeX

G:=Group("C40:8C4.C2");
// GroupNames label

G:=SmallGroup(320,424);
// by ID

G=gap.SmallGroup(320,424);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,232,1094,135,100,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^40=b^4=1,c^2=a^20,b*a*b^-1=a^29,c*a*c^-1=a^11*b^2,c*b*c^-1=b^-1>;
// generators/relations

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