metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4.20D10, C40⋊8C4.8C2, (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(D4⋊2D5), (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, C5⋊2(C42.30C22), (C2×Dic10).70C22, C2.18(Dic5.5D4), (C2×C10).250(C2×D4), (C5×C4⋊C4).38C22, (C2×C5⋊2C8).32C22, (C5×Q8⋊C4).10C2, (C2×C4).344(C22×D5), SmallGroup(320,424)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C40⋊8C4.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, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C2×Q8, C2×Q8, Dic5, C20, C20, C2×C10, C8⋊C4, Q8⋊C4, Q8⋊C4, C42.C2, C4⋊Q8, C5⋊2C8, C40, Dic10, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C42.30C22, C2×C5⋊2C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, C5×C4⋊C4, C2×C40, C2×Dic10, Q8×C10, C10.Q16, C40⋊8C4, C20.44D4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, Dic5⋊Q8, C40⋊8C4.C2
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4.4D4, C8.C22, C22×D5, C42.30C22, C4○D20, D4×D5, D4⋊2D5, Dic5.5D4, SD16⋊D5, Q16⋊D5, C40⋊8C4.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)
(1 224 66 268)(2 213 67 257)(3 202 68 246)(4 231 69 275)(5 220 70 264)(6 209 71 253)(7 238 72 242)(8 227 73 271)(9 216 74 260)(10 205 75 249)(11 234 76 278)(12 223 77 267)(13 212 78 256)(14 201 79 245)(15 230 80 274)(16 219 41 263)(17 208 42 252)(18 237 43 241)(19 226 44 270)(20 215 45 259)(21 204 46 248)(22 233 47 277)(23 222 48 266)(24 211 49 255)(25 240 50 244)(26 229 51 273)(27 218 52 262)(28 207 53 251)(29 236 54 280)(30 225 55 269)(31 214 56 258)(32 203 57 247)(33 232 58 276)(34 221 59 265)(35 210 60 254)(36 239 61 243)(37 228 62 272)(38 217 63 261)(39 206 64 250)(40 235 65 279)(81 153 193 285)(82 142 194 314)(83 131 195 303)(84 160 196 292)(85 149 197 281)(86 138 198 310)(87 127 199 299)(88 156 200 288)(89 145 161 317)(90 134 162 306)(91 123 163 295)(92 152 164 284)(93 141 165 313)(94 130 166 302)(95 159 167 291)(96 148 168 320)(97 137 169 309)(98 126 170 298)(99 155 171 287)(100 144 172 316)(101 133 173 305)(102 122 174 294)(103 151 175 283)(104 140 176 312)(105 129 177 301)(106 158 178 290)(107 147 179 319)(108 136 180 308)(109 125 181 297)(110 154 182 286)(111 143 183 315)(112 132 184 304)(113 121 185 293)(114 150 186 282)(115 139 187 311)(116 128 188 300)(117 157 189 289)(118 146 190 318)(119 135 191 307)(120 124 192 296)
(1 191 21 171)(2 90 22 110)(3 173 23 193)(4 112 24 92)(5 195 25 175)(6 94 26 114)(7 177 27 197)(8 116 28 96)(9 199 29 179)(10 98 30 118)(11 181 31 161)(12 120 32 100)(13 163 33 183)(14 102 34 82)(15 185 35 165)(16 84 36 104)(17 167 37 187)(18 106 38 86)(19 189 39 169)(20 88 40 108)(41 196 61 176)(42 95 62 115)(43 178 63 198)(44 117 64 97)(45 200 65 180)(46 99 66 119)(47 182 67 162)(48 81 68 101)(49 164 69 184)(50 103 70 83)(51 186 71 166)(52 85 72 105)(53 168 73 188)(54 107 74 87)(55 190 75 170)(56 89 76 109)(57 172 77 192)(58 111 78 91)(59 194 79 174)(60 93 80 113)(121 210 141 230)(122 265 142 245)(123 232 143 212)(124 247 144 267)(125 214 145 234)(126 269 146 249)(127 236 147 216)(128 251 148 271)(129 218 149 238)(130 273 150 253)(131 240 151 220)(132 255 152 275)(133 222 153 202)(134 277 154 257)(135 204 155 224)(136 259 156 279)(137 226 157 206)(138 241 158 261)(139 208 159 228)(140 263 160 243)(201 294 221 314)(203 316 223 296)(205 298 225 318)(207 320 227 300)(209 302 229 282)(211 284 231 304)(213 306 233 286)(215 288 235 308)(217 310 237 290)(219 292 239 312)(242 301 262 281)(244 283 264 303)(246 305 266 285)(248 287 268 307)(250 309 270 289)(252 291 272 311)(254 313 274 293)(256 295 276 315)(258 317 278 297)(260 299 280 319)
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), (1,224,66,268)(2,213,67,257)(3,202,68,246)(4,231,69,275)(5,220,70,264)(6,209,71,253)(7,238,72,242)(8,227,73,271)(9,216,74,260)(10,205,75,249)(11,234,76,278)(12,223,77,267)(13,212,78,256)(14,201,79,245)(15,230,80,274)(16,219,41,263)(17,208,42,252)(18,237,43,241)(19,226,44,270)(20,215,45,259)(21,204,46,248)(22,233,47,277)(23,222,48,266)(24,211,49,255)(25,240,50,244)(26,229,51,273)(27,218,52,262)(28,207,53,251)(29,236,54,280)(30,225,55,269)(31,214,56,258)(32,203,57,247)(33,232,58,276)(34,221,59,265)(35,210,60,254)(36,239,61,243)(37,228,62,272)(38,217,63,261)(39,206,64,250)(40,235,65,279)(81,153,193,285)(82,142,194,314)(83,131,195,303)(84,160,196,292)(85,149,197,281)(86,138,198,310)(87,127,199,299)(88,156,200,288)(89,145,161,317)(90,134,162,306)(91,123,163,295)(92,152,164,284)(93,141,165,313)(94,130,166,302)(95,159,167,291)(96,148,168,320)(97,137,169,309)(98,126,170,298)(99,155,171,287)(100,144,172,316)(101,133,173,305)(102,122,174,294)(103,151,175,283)(104,140,176,312)(105,129,177,301)(106,158,178,290)(107,147,179,319)(108,136,180,308)(109,125,181,297)(110,154,182,286)(111,143,183,315)(112,132,184,304)(113,121,185,293)(114,150,186,282)(115,139,187,311)(116,128,188,300)(117,157,189,289)(118,146,190,318)(119,135,191,307)(120,124,192,296), (1,191,21,171)(2,90,22,110)(3,173,23,193)(4,112,24,92)(5,195,25,175)(6,94,26,114)(7,177,27,197)(8,116,28,96)(9,199,29,179)(10,98,30,118)(11,181,31,161)(12,120,32,100)(13,163,33,183)(14,102,34,82)(15,185,35,165)(16,84,36,104)(17,167,37,187)(18,106,38,86)(19,189,39,169)(20,88,40,108)(41,196,61,176)(42,95,62,115)(43,178,63,198)(44,117,64,97)(45,200,65,180)(46,99,66,119)(47,182,67,162)(48,81,68,101)(49,164,69,184)(50,103,70,83)(51,186,71,166)(52,85,72,105)(53,168,73,188)(54,107,74,87)(55,190,75,170)(56,89,76,109)(57,172,77,192)(58,111,78,91)(59,194,79,174)(60,93,80,113)(121,210,141,230)(122,265,142,245)(123,232,143,212)(124,247,144,267)(125,214,145,234)(126,269,146,249)(127,236,147,216)(128,251,148,271)(129,218,149,238)(130,273,150,253)(131,240,151,220)(132,255,152,275)(133,222,153,202)(134,277,154,257)(135,204,155,224)(136,259,156,279)(137,226,157,206)(138,241,158,261)(139,208,159,228)(140,263,160,243)(201,294,221,314)(203,316,223,296)(205,298,225,318)(207,320,227,300)(209,302,229,282)(211,284,231,304)(213,306,233,286)(215,288,235,308)(217,310,237,290)(219,292,239,312)(242,301,262,281)(244,283,264,303)(246,305,266,285)(248,287,268,307)(250,309,270,289)(252,291,272,311)(254,313,274,293)(256,295,276,315)(258,317,278,297)(260,299,280,319)>;
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), (1,224,66,268)(2,213,67,257)(3,202,68,246)(4,231,69,275)(5,220,70,264)(6,209,71,253)(7,238,72,242)(8,227,73,271)(9,216,74,260)(10,205,75,249)(11,234,76,278)(12,223,77,267)(13,212,78,256)(14,201,79,245)(15,230,80,274)(16,219,41,263)(17,208,42,252)(18,237,43,241)(19,226,44,270)(20,215,45,259)(21,204,46,248)(22,233,47,277)(23,222,48,266)(24,211,49,255)(25,240,50,244)(26,229,51,273)(27,218,52,262)(28,207,53,251)(29,236,54,280)(30,225,55,269)(31,214,56,258)(32,203,57,247)(33,232,58,276)(34,221,59,265)(35,210,60,254)(36,239,61,243)(37,228,62,272)(38,217,63,261)(39,206,64,250)(40,235,65,279)(81,153,193,285)(82,142,194,314)(83,131,195,303)(84,160,196,292)(85,149,197,281)(86,138,198,310)(87,127,199,299)(88,156,200,288)(89,145,161,317)(90,134,162,306)(91,123,163,295)(92,152,164,284)(93,141,165,313)(94,130,166,302)(95,159,167,291)(96,148,168,320)(97,137,169,309)(98,126,170,298)(99,155,171,287)(100,144,172,316)(101,133,173,305)(102,122,174,294)(103,151,175,283)(104,140,176,312)(105,129,177,301)(106,158,178,290)(107,147,179,319)(108,136,180,308)(109,125,181,297)(110,154,182,286)(111,143,183,315)(112,132,184,304)(113,121,185,293)(114,150,186,282)(115,139,187,311)(116,128,188,300)(117,157,189,289)(118,146,190,318)(119,135,191,307)(120,124,192,296), (1,191,21,171)(2,90,22,110)(3,173,23,193)(4,112,24,92)(5,195,25,175)(6,94,26,114)(7,177,27,197)(8,116,28,96)(9,199,29,179)(10,98,30,118)(11,181,31,161)(12,120,32,100)(13,163,33,183)(14,102,34,82)(15,185,35,165)(16,84,36,104)(17,167,37,187)(18,106,38,86)(19,189,39,169)(20,88,40,108)(41,196,61,176)(42,95,62,115)(43,178,63,198)(44,117,64,97)(45,200,65,180)(46,99,66,119)(47,182,67,162)(48,81,68,101)(49,164,69,184)(50,103,70,83)(51,186,71,166)(52,85,72,105)(53,168,73,188)(54,107,74,87)(55,190,75,170)(56,89,76,109)(57,172,77,192)(58,111,78,91)(59,194,79,174)(60,93,80,113)(121,210,141,230)(122,265,142,245)(123,232,143,212)(124,247,144,267)(125,214,145,234)(126,269,146,249)(127,236,147,216)(128,251,148,271)(129,218,149,238)(130,273,150,253)(131,240,151,220)(132,255,152,275)(133,222,153,202)(134,277,154,257)(135,204,155,224)(136,259,156,279)(137,226,157,206)(138,241,158,261)(139,208,159,228)(140,263,160,243)(201,294,221,314)(203,316,223,296)(205,298,225,318)(207,320,227,300)(209,302,229,282)(211,284,231,304)(213,306,233,286)(215,288,235,308)(217,310,237,290)(219,292,239,312)(242,301,262,281)(244,283,264,303)(246,305,266,285)(248,287,268,307)(250,309,270,289)(252,291,272,311)(254,313,274,293)(256,295,276,315)(258,317,278,297)(260,299,280,319) );
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)], [(1,224,66,268),(2,213,67,257),(3,202,68,246),(4,231,69,275),(5,220,70,264),(6,209,71,253),(7,238,72,242),(8,227,73,271),(9,216,74,260),(10,205,75,249),(11,234,76,278),(12,223,77,267),(13,212,78,256),(14,201,79,245),(15,230,80,274),(16,219,41,263),(17,208,42,252),(18,237,43,241),(19,226,44,270),(20,215,45,259),(21,204,46,248),(22,233,47,277),(23,222,48,266),(24,211,49,255),(25,240,50,244),(26,229,51,273),(27,218,52,262),(28,207,53,251),(29,236,54,280),(30,225,55,269),(31,214,56,258),(32,203,57,247),(33,232,58,276),(34,221,59,265),(35,210,60,254),(36,239,61,243),(37,228,62,272),(38,217,63,261),(39,206,64,250),(40,235,65,279),(81,153,193,285),(82,142,194,314),(83,131,195,303),(84,160,196,292),(85,149,197,281),(86,138,198,310),(87,127,199,299),(88,156,200,288),(89,145,161,317),(90,134,162,306),(91,123,163,295),(92,152,164,284),(93,141,165,313),(94,130,166,302),(95,159,167,291),(96,148,168,320),(97,137,169,309),(98,126,170,298),(99,155,171,287),(100,144,172,316),(101,133,173,305),(102,122,174,294),(103,151,175,283),(104,140,176,312),(105,129,177,301),(106,158,178,290),(107,147,179,319),(108,136,180,308),(109,125,181,297),(110,154,182,286),(111,143,183,315),(112,132,184,304),(113,121,185,293),(114,150,186,282),(115,139,187,311),(116,128,188,300),(117,157,189,289),(118,146,190,318),(119,135,191,307),(120,124,192,296)], [(1,191,21,171),(2,90,22,110),(3,173,23,193),(4,112,24,92),(5,195,25,175),(6,94,26,114),(7,177,27,197),(8,116,28,96),(9,199,29,179),(10,98,30,118),(11,181,31,161),(12,120,32,100),(13,163,33,183),(14,102,34,82),(15,185,35,165),(16,84,36,104),(17,167,37,187),(18,106,38,86),(19,189,39,169),(20,88,40,108),(41,196,61,176),(42,95,62,115),(43,178,63,198),(44,117,64,97),(45,200,65,180),(46,99,66,119),(47,182,67,162),(48,81,68,101),(49,164,69,184),(50,103,70,83),(51,186,71,166),(52,85,72,105),(53,168,73,188),(54,107,74,87),(55,190,75,170),(56,89,76,109),(57,172,77,192),(58,111,78,91),(59,194,79,174),(60,93,80,113),(121,210,141,230),(122,265,142,245),(123,232,143,212),(124,247,144,267),(125,214,145,234),(126,269,146,249),(127,236,147,216),(128,251,148,271),(129,218,149,238),(130,273,150,253),(131,240,151,220),(132,255,152,275),(133,222,153,202),(134,277,154,257),(135,204,155,224),(136,259,156,279),(137,226,157,206),(138,241,158,261),(139,208,159,228),(140,263,160,243),(201,294,221,314),(203,316,223,296),(205,298,225,318),(207,320,227,300),(209,302,229,282),(211,284,231,304),(213,306,233,286),(215,288,235,308),(217,310,237,290),(219,292,239,312),(242,301,262,281),(244,283,264,303),(246,305,266,285),(248,287,268,307),(250,309,270,289),(252,291,272,311),(254,313,274,293),(256,295,276,315),(258,317,278,297),(260,299,280,319)]])
44 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 40A | ··· | 40H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 8 | 20 | 20 | 40 | 40 | 2 | 2 | 4 | 4 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C4○D20 | C8.C22 | D4⋊2D5 | D4×D5 | SD16⋊D5 | Q16⋊D5 |
| kernel | C40⋊8C4.C2 | C10.Q16 | C40⋊8C4 | C20.44D4 | Q8⋊Dic5 | C5×Q8⋊C4 | C4.Dic10 | Dic5⋊Q8 | C2×Dic5 | Q8⋊C4 | C20 | C4⋊C4 | C2×C8 | C2×Q8 | C4 | C10 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 4 |
Matrix representation of C40⋊8C4.C2 ►in GL6(𝔽41)
| 13 | 22 | 0 | 0 | 0 | 0 |
| 11 | 28 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 0 | 0 | 11 |
| 0 | 0 | 5 | 0 | 30 | 36 |
| 0 | 0 | 0 | 0 | 26 | 15 |
| 0 | 0 | 22 | 15 | 26 | 4 |
| 6 | 7 | 0 | 0 | 0 | 0 |
| 24 | 35 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 26 | 10 | 24 |
| 0 | 0 | 40 | 9 | 12 | 14 |
| 0 | 0 | 19 | 12 | 9 | 15 |
| 0 | 0 | 23 | 21 | 38 | 15 |
| 11 | 19 | 0 | 0 | 0 | 0 |
| 39 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 14 | 22 | 10 | 40 |
| 0 | 0 | 26 | 25 | 11 | 3 |
| 0 | 0 | 40 | 28 | 6 | 19 |
| 0 | 0 | 21 | 31 | 32 | 37 |
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] >;
C40⋊8C4.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