metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: (C2×C20)⋊2C8, C10.9(C4⋊C8), C10.11(C4×C8), (C2×Dic5)⋊6C8, (C22×C4).9F5, C10.8(C8⋊C4), C2.2(C20⋊C8), C22.17(C4×F5), C23.56(C2×F5), C10.8(C22⋊C8), (C2×C10).14C42, (C22×C20).16C4, (C2×Dic5).32Q8, C22.21(C4⋊F5), C2.2(D10⋊C8), C22.7(C4.F5), Dic5.26(C4⋊C4), C22.11(D5⋊C8), (C2×Dic5).256D4, (C2×C10).20M4(2), C2.2(Dic5⋊C8), C2.2(D10.3Q8), C5⋊2(C22.7C42), (C22×Dic5).25C4, C22.39(C22⋊F5), C2.2(C23.2F5), C22.7(C22.F5), C2.2(C10.C42), Dic5.35(C22⋊C4), C10.18(C2.C42), (C22×Dic5).264C22, (C2×C5⋊C8)⋊4C4, C2.7(C4×C5⋊C8), (C2×C4)⋊2(C5⋊C8), (C22×C5⋊C8).1C2, C22.11(C2×C5⋊C8), (C2×C10).29(C2×C8), (C2×C4×Dic5).19C2, (C2×C10).17(C4⋊C4), (C22×C10).47(C2×C4), (C2×C10).31(C22⋊C4), (C2×Dic5).123(C2×C4), SmallGroup(320,256)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10.(C4⋊C8)
G = < a,b,c | a10=b4=c8=1, bab-1=a-1, cac-1=a3, cbc-1=a5b-1 >
Subgroups: 354 in 118 conjugacy classes, 60 normal (40 characteristic)
C1, C2, C4, C22, C5, C8, C2×C4, C2×C4, C23, C10, C42, C2×C8, C22×C4, C22×C4, Dic5, Dic5, C20, C2×C10, C2×C42, C22×C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C22.7C42, C4×Dic5, C2×C5⋊C8, C2×C5⋊C8, C22×Dic5, C22×C20, C2×C4×Dic5, C22×C5⋊C8, C10.(C4⋊C8)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), F5, C2.C42, C4×C8, C8⋊C4, C22⋊C8, C4⋊C8, C5⋊C8, C2×F5, C22.7C42, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C2×C5⋊C8, C22.F5, C22⋊F5, C4×C5⋊C8, C20⋊C8, C10.C42, D10⋊C8, Dic5⋊C8, D10.3Q8, C23.2F5, C10.(C4⋊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)
(1 152 28 171)(2 151 29 180)(3 160 30 179)(4 159 21 178)(5 158 22 177)(6 157 23 176)(7 156 24 175)(8 155 25 174)(9 154 26 173)(10 153 27 172)(11 146 58 124)(12 145 59 123)(13 144 60 122)(14 143 51 121)(15 142 52 130)(16 141 53 129)(17 150 54 128)(18 149 55 127)(19 148 56 126)(20 147 57 125)(31 197 318 213)(32 196 319 212)(33 195 320 211)(34 194 311 220)(35 193 312 219)(36 192 313 218)(37 191 314 217)(38 200 315 216)(39 199 316 215)(40 198 317 214)(41 135 63 117)(42 134 64 116)(43 133 65 115)(44 132 66 114)(45 131 67 113)(46 140 68 112)(47 139 69 111)(48 138 70 120)(49 137 61 119)(50 136 62 118)(71 109 98 86)(72 108 99 85)(73 107 100 84)(74 106 91 83)(75 105 92 82)(76 104 93 81)(77 103 94 90)(78 102 95 89)(79 101 96 88)(80 110 97 87)(161 289 182 303)(162 288 183 302)(163 287 184 301)(164 286 185 310)(165 285 186 309)(166 284 187 308)(167 283 188 307)(168 282 189 306)(169 281 190 305)(170 290 181 304)(201 280 225 296)(202 279 226 295)(203 278 227 294)(204 277 228 293)(205 276 229 292)(206 275 230 291)(207 274 221 300)(208 273 222 299)(209 272 223 298)(210 271 224 297)(231 243 260 264)(232 242 251 263)(233 241 252 262)(234 250 253 261)(235 249 254 270)(236 248 255 269)(237 247 256 268)(238 246 257 267)(239 245 258 266)(240 244 259 265)
(1 226 62 184 52 255 99 220)(2 223 61 187 53 252 98 213)(3 230 70 190 54 259 97 216)(4 227 69 183 55 256 96 219)(5 224 68 186 56 253 95 212)(6 221 67 189 57 260 94 215)(7 228 66 182 58 257 93 218)(8 225 65 185 59 254 92 211)(9 222 64 188 60 251 91 214)(10 229 63 181 51 258 100 217)(11 238 76 192 24 204 44 161)(12 235 75 195 25 201 43 164)(13 232 74 198 26 208 42 167)(14 239 73 191 27 205 41 170)(15 236 72 194 28 202 50 163)(16 233 71 197 29 209 49 166)(17 240 80 200 30 206 48 169)(18 237 79 193 21 203 47 162)(19 234 78 196 22 210 46 165)(20 231 77 199 23 207 45 168)(31 175 298 132 308 146 262 104)(32 172 297 135 309 143 261 107)(33 179 296 138 310 150 270 110)(34 176 295 131 301 147 269 103)(35 173 294 134 302 144 268 106)(36 180 293 137 303 141 267 109)(37 177 292 140 304 148 266 102)(38 174 291 133 305 145 265 105)(39 171 300 136 306 142 264 108)(40 178 299 139 307 149 263 101)(81 318 156 272 114 284 124 241)(82 315 155 275 115 281 123 244)(83 312 154 278 116 288 122 247)(84 319 153 271 117 285 121 250)(85 316 152 274 118 282 130 243)(86 313 151 277 119 289 129 246)(87 320 160 280 120 286 128 249)(88 317 159 273 111 283 127 242)(89 314 158 276 112 290 126 245)(90 311 157 279 113 287 125 248)
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), (1,152,28,171)(2,151,29,180)(3,160,30,179)(4,159,21,178)(5,158,22,177)(6,157,23,176)(7,156,24,175)(8,155,25,174)(9,154,26,173)(10,153,27,172)(11,146,58,124)(12,145,59,123)(13,144,60,122)(14,143,51,121)(15,142,52,130)(16,141,53,129)(17,150,54,128)(18,149,55,127)(19,148,56,126)(20,147,57,125)(31,197,318,213)(32,196,319,212)(33,195,320,211)(34,194,311,220)(35,193,312,219)(36,192,313,218)(37,191,314,217)(38,200,315,216)(39,199,316,215)(40,198,317,214)(41,135,63,117)(42,134,64,116)(43,133,65,115)(44,132,66,114)(45,131,67,113)(46,140,68,112)(47,139,69,111)(48,138,70,120)(49,137,61,119)(50,136,62,118)(71,109,98,86)(72,108,99,85)(73,107,100,84)(74,106,91,83)(75,105,92,82)(76,104,93,81)(77,103,94,90)(78,102,95,89)(79,101,96,88)(80,110,97,87)(161,289,182,303)(162,288,183,302)(163,287,184,301)(164,286,185,310)(165,285,186,309)(166,284,187,308)(167,283,188,307)(168,282,189,306)(169,281,190,305)(170,290,181,304)(201,280,225,296)(202,279,226,295)(203,278,227,294)(204,277,228,293)(205,276,229,292)(206,275,230,291)(207,274,221,300)(208,273,222,299)(209,272,223,298)(210,271,224,297)(231,243,260,264)(232,242,251,263)(233,241,252,262)(234,250,253,261)(235,249,254,270)(236,248,255,269)(237,247,256,268)(238,246,257,267)(239,245,258,266)(240,244,259,265), (1,226,62,184,52,255,99,220)(2,223,61,187,53,252,98,213)(3,230,70,190,54,259,97,216)(4,227,69,183,55,256,96,219)(5,224,68,186,56,253,95,212)(6,221,67,189,57,260,94,215)(7,228,66,182,58,257,93,218)(8,225,65,185,59,254,92,211)(9,222,64,188,60,251,91,214)(10,229,63,181,51,258,100,217)(11,238,76,192,24,204,44,161)(12,235,75,195,25,201,43,164)(13,232,74,198,26,208,42,167)(14,239,73,191,27,205,41,170)(15,236,72,194,28,202,50,163)(16,233,71,197,29,209,49,166)(17,240,80,200,30,206,48,169)(18,237,79,193,21,203,47,162)(19,234,78,196,22,210,46,165)(20,231,77,199,23,207,45,168)(31,175,298,132,308,146,262,104)(32,172,297,135,309,143,261,107)(33,179,296,138,310,150,270,110)(34,176,295,131,301,147,269,103)(35,173,294,134,302,144,268,106)(36,180,293,137,303,141,267,109)(37,177,292,140,304,148,266,102)(38,174,291,133,305,145,265,105)(39,171,300,136,306,142,264,108)(40,178,299,139,307,149,263,101)(81,318,156,272,114,284,124,241)(82,315,155,275,115,281,123,244)(83,312,154,278,116,288,122,247)(84,319,153,271,117,285,121,250)(85,316,152,274,118,282,130,243)(86,313,151,277,119,289,129,246)(87,320,160,280,120,286,128,249)(88,317,159,273,111,283,127,242)(89,314,158,276,112,290,126,245)(90,311,157,279,113,287,125,248)>;
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), (1,152,28,171)(2,151,29,180)(3,160,30,179)(4,159,21,178)(5,158,22,177)(6,157,23,176)(7,156,24,175)(8,155,25,174)(9,154,26,173)(10,153,27,172)(11,146,58,124)(12,145,59,123)(13,144,60,122)(14,143,51,121)(15,142,52,130)(16,141,53,129)(17,150,54,128)(18,149,55,127)(19,148,56,126)(20,147,57,125)(31,197,318,213)(32,196,319,212)(33,195,320,211)(34,194,311,220)(35,193,312,219)(36,192,313,218)(37,191,314,217)(38,200,315,216)(39,199,316,215)(40,198,317,214)(41,135,63,117)(42,134,64,116)(43,133,65,115)(44,132,66,114)(45,131,67,113)(46,140,68,112)(47,139,69,111)(48,138,70,120)(49,137,61,119)(50,136,62,118)(71,109,98,86)(72,108,99,85)(73,107,100,84)(74,106,91,83)(75,105,92,82)(76,104,93,81)(77,103,94,90)(78,102,95,89)(79,101,96,88)(80,110,97,87)(161,289,182,303)(162,288,183,302)(163,287,184,301)(164,286,185,310)(165,285,186,309)(166,284,187,308)(167,283,188,307)(168,282,189,306)(169,281,190,305)(170,290,181,304)(201,280,225,296)(202,279,226,295)(203,278,227,294)(204,277,228,293)(205,276,229,292)(206,275,230,291)(207,274,221,300)(208,273,222,299)(209,272,223,298)(210,271,224,297)(231,243,260,264)(232,242,251,263)(233,241,252,262)(234,250,253,261)(235,249,254,270)(236,248,255,269)(237,247,256,268)(238,246,257,267)(239,245,258,266)(240,244,259,265), (1,226,62,184,52,255,99,220)(2,223,61,187,53,252,98,213)(3,230,70,190,54,259,97,216)(4,227,69,183,55,256,96,219)(5,224,68,186,56,253,95,212)(6,221,67,189,57,260,94,215)(7,228,66,182,58,257,93,218)(8,225,65,185,59,254,92,211)(9,222,64,188,60,251,91,214)(10,229,63,181,51,258,100,217)(11,238,76,192,24,204,44,161)(12,235,75,195,25,201,43,164)(13,232,74,198,26,208,42,167)(14,239,73,191,27,205,41,170)(15,236,72,194,28,202,50,163)(16,233,71,197,29,209,49,166)(17,240,80,200,30,206,48,169)(18,237,79,193,21,203,47,162)(19,234,78,196,22,210,46,165)(20,231,77,199,23,207,45,168)(31,175,298,132,308,146,262,104)(32,172,297,135,309,143,261,107)(33,179,296,138,310,150,270,110)(34,176,295,131,301,147,269,103)(35,173,294,134,302,144,268,106)(36,180,293,137,303,141,267,109)(37,177,292,140,304,148,266,102)(38,174,291,133,305,145,265,105)(39,171,300,136,306,142,264,108)(40,178,299,139,307,149,263,101)(81,318,156,272,114,284,124,241)(82,315,155,275,115,281,123,244)(83,312,154,278,116,288,122,247)(84,319,153,271,117,285,121,250)(85,316,152,274,118,282,130,243)(86,313,151,277,119,289,129,246)(87,320,160,280,120,286,128,249)(88,317,159,273,111,283,127,242)(89,314,158,276,112,290,126,245)(90,311,157,279,113,287,125,248) );
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)], [(1,152,28,171),(2,151,29,180),(3,160,30,179),(4,159,21,178),(5,158,22,177),(6,157,23,176),(7,156,24,175),(8,155,25,174),(9,154,26,173),(10,153,27,172),(11,146,58,124),(12,145,59,123),(13,144,60,122),(14,143,51,121),(15,142,52,130),(16,141,53,129),(17,150,54,128),(18,149,55,127),(19,148,56,126),(20,147,57,125),(31,197,318,213),(32,196,319,212),(33,195,320,211),(34,194,311,220),(35,193,312,219),(36,192,313,218),(37,191,314,217),(38,200,315,216),(39,199,316,215),(40,198,317,214),(41,135,63,117),(42,134,64,116),(43,133,65,115),(44,132,66,114),(45,131,67,113),(46,140,68,112),(47,139,69,111),(48,138,70,120),(49,137,61,119),(50,136,62,118),(71,109,98,86),(72,108,99,85),(73,107,100,84),(74,106,91,83),(75,105,92,82),(76,104,93,81),(77,103,94,90),(78,102,95,89),(79,101,96,88),(80,110,97,87),(161,289,182,303),(162,288,183,302),(163,287,184,301),(164,286,185,310),(165,285,186,309),(166,284,187,308),(167,283,188,307),(168,282,189,306),(169,281,190,305),(170,290,181,304),(201,280,225,296),(202,279,226,295),(203,278,227,294),(204,277,228,293),(205,276,229,292),(206,275,230,291),(207,274,221,300),(208,273,222,299),(209,272,223,298),(210,271,224,297),(231,243,260,264),(232,242,251,263),(233,241,252,262),(234,250,253,261),(235,249,254,270),(236,248,255,269),(237,247,256,268),(238,246,257,267),(239,245,258,266),(240,244,259,265)], [(1,226,62,184,52,255,99,220),(2,223,61,187,53,252,98,213),(3,230,70,190,54,259,97,216),(4,227,69,183,55,256,96,219),(5,224,68,186,56,253,95,212),(6,221,67,189,57,260,94,215),(7,228,66,182,58,257,93,218),(8,225,65,185,59,254,92,211),(9,222,64,188,60,251,91,214),(10,229,63,181,51,258,100,217),(11,238,76,192,24,204,44,161),(12,235,75,195,25,201,43,164),(13,232,74,198,26,208,42,167),(14,239,73,191,27,205,41,170),(15,236,72,194,28,202,50,163),(16,233,71,197,29,209,49,166),(17,240,80,200,30,206,48,169),(18,237,79,193,21,203,47,162),(19,234,78,196,22,210,46,165),(20,231,77,199,23,207,45,168),(31,175,298,132,308,146,262,104),(32,172,297,135,309,143,261,107),(33,179,296,138,310,150,270,110),(34,176,295,131,301,147,269,103),(35,173,294,134,302,144,268,106),(36,180,293,137,303,141,267,109),(37,177,292,140,304,148,266,102),(38,174,291,133,305,145,265,105),(39,171,300,136,306,142,264,108),(40,178,299,139,307,149,263,101),(81,318,156,272,114,284,124,241),(82,315,155,275,115,281,123,244),(83,312,154,278,116,288,122,247),(84,319,153,271,117,285,121,250),(85,316,152,274,118,282,130,243),(86,313,151,277,119,289,129,246),(87,320,160,280,120,286,128,249),(88,317,159,273,111,283,127,242),(89,314,158,276,112,290,126,245),(90,311,157,279,113,287,125,248)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5 | 8A | ··· | 8P | 10A | ··· | 10G | 20A | ··· | 20H |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 5 | ··· | 5 | 10 | 10 | 10 | 10 | 4 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | ||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C4 | C8 | C8 | D4 | Q8 | M4(2) | F5 | C5⋊C8 | C2×F5 | D5⋊C8 | C4.F5 | C4×F5 | C4⋊F5 | C22.F5 | C22⋊F5 |
| kernel | C10.(C4⋊C8) | C2×C4×Dic5 | C22×C5⋊C8 | C2×C5⋊C8 | C22×Dic5 | C22×C20 | C2×Dic5 | C2×C20 | C2×Dic5 | C2×Dic5 | C2×C10 | C22×C4 | C2×C4 | C23 | C22 | C22 | C22 | C22 | C22 | C22 |
| # reps | 1 | 1 | 2 | 8 | 2 | 2 | 8 | 8 | 3 | 1 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
Matrix representation of C10.(C4⋊C8) ►in GL8(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 | 1 | 0 |
| 9 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 15 | 3 | 23 | 38 |
| 0 | 0 | 0 | 0 | 18 | 26 | 20 | 0 |
| 0 | 0 | 0 | 0 | 0 | 23 | 23 | 15 |
| 0 | 0 | 0 | 0 | 38 | 26 | 38 | 18 |
| 24 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 34 | 17 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 22 | 39 | 23 |
| 0 | 0 | 0 | 0 | 25 | 4 | 10 | 9 |
| 0 | 0 | 0 | 0 | 37 | 31 | 32 | 7 |
| 0 | 0 | 0 | 0 | 18 | 29 | 14 | 19 |
G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1,0,0,0,0,40,0,0,0],[9,0,0,0,0,0,0,0,9,32,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,15,18,0,38,0,0,0,0,3,26,23,26,0,0,0,0,23,20,23,38,0,0,0,0,38,0,15,18],[24,34,0,0,0,0,0,0,40,17,0,0,0,0,0,0,0,0,40,11,0,0,0,0,0,0,11,1,0,0,0,0,0,0,0,0,27,25,37,18,0,0,0,0,22,4,31,29,0,0,0,0,39,10,32,14,0,0,0,0,23,9,7,19] >;
C10.(C4⋊C8) in GAP, Magma, Sage, TeX
C_{10}.(C_4\rtimes C_8) % in TeX
G:=Group("C10.(C4:C8)"); // GroupNames label
G:=SmallGroup(320,256);
// by ID
G=gap.SmallGroup(320,256);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,184,6278,3156]);
// Polycyclic
G:=Group<a,b,c|a^10=b^4=c^8=1,b*a*b^-1=a^-1,c*a*c^-1=a^3,c*b*c^-1=a^5*b^-1>;
// generators/relations