direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C10×C8⋊C4, C42.5C20, C20.67C42, C8⋊9(C2×C20), (C2×C8)⋊8C20, C40⋊45(C2×C4), (C2×C40)⋊28C4, C4.8(C4×C20), (C4×C20).25C4, (C2×C42).6C10, (C22×C4).9C20, C22.9(C4×C20), (C22×C40).32C2, (C22×C8).14C10, (C2×C10).53C42, C4.32(C22×C20), C23.37(C2×C20), (C22×C20).45C4, C42.57(C2×C10), C10.53(C2×C42), C2.1(C10×M4(2)), (C2×C20).978C23, C20.249(C22×C4), (C4×C20).298C22, (C2×C40).444C22, (C2×C10).47M4(2), C10.79(C2×M4(2)), C22.8(C5×M4(2)), C22.16(C22×C20), (C22×C20).607C22, C2.5(C2×C4×C20), (C2×C4×C20).8C2, (C2×C4).58(C2×C20), (C2×C8).98(C2×C10), (C2×C20).460(C2×C4), (C2×C4).146(C22×C10), (C22×C4).134(C2×C10), (C2×C10).329(C22×C4), (C22×C10).214(C2×C4), SmallGroup(320,904)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C10×C8⋊C4
G = < a,b,c | a10=b8=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 162 in 146 conjugacy classes, 130 normal (18 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C8⋊C4, C2×C42, C22×C8, C40, C2×C20, C2×C20, C2×C20, C22×C10, C2×C8⋊C4, C4×C20, C2×C40, C22×C20, C22×C20, C5×C8⋊C4, C2×C4×C20, C22×C40, C10×C8⋊C4
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, M4(2), C22×C4, C20, C2×C10, C8⋊C4, C2×C42, C2×M4(2), C2×C20, C22×C10, C2×C8⋊C4, C4×C20, C5×M4(2), C22×C20, C5×C8⋊C4, C2×C4×C20, C10×M4(2), C10×C8⋊C4
►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 53 110 11 129 50 83 22)(2 54 101 12 130 41 84 23)(3 55 102 13 121 42 85 24)(4 56 103 14 122 43 86 25)(5 57 104 15 123 44 87 26)(6 58 105 16 124 45 88 27)(7 59 106 17 125 46 89 28)(8 60 107 18 126 47 90 29)(9 51 108 19 127 48 81 30)(10 52 109 20 128 49 82 21)(31 150 76 95 311 139 65 112)(32 141 77 96 312 140 66 113)(33 142 78 97 313 131 67 114)(34 143 79 98 314 132 68 115)(35 144 80 99 315 133 69 116)(36 145 71 100 316 134 70 117)(37 146 72 91 317 135 61 118)(38 147 73 92 318 136 62 119)(39 148 74 93 319 137 63 120)(40 149 75 94 320 138 64 111)(151 242 162 286 206 269 189 231)(152 243 163 287 207 270 190 232)(153 244 164 288 208 261 181 233)(154 245 165 289 209 262 182 234)(155 246 166 290 210 263 183 235)(156 247 167 281 201 264 184 236)(157 248 168 282 202 265 185 237)(158 249 169 283 203 266 186 238)(159 250 170 284 204 267 187 239)(160 241 161 285 205 268 188 240)(171 295 215 278 198 306 226 251)(172 296 216 279 199 307 227 252)(173 297 217 280 200 308 228 253)(174 298 218 271 191 309 229 254)(175 299 219 272 192 310 230 255)(176 300 220 273 193 301 221 256)(177 291 211 274 194 302 222 257)(178 292 212 275 195 303 223 258)(179 293 213 276 196 304 224 259)(180 294 214 277 197 305 225 260)
(1 157 141 215)(2 158 142 216)(3 159 143 217)(4 160 144 218)(5 151 145 219)(6 152 146 220)(7 153 147 211)(8 154 148 212)(9 155 149 213)(10 156 150 214)(11 237 312 295)(12 238 313 296)(13 239 314 297)(14 240 315 298)(15 231 316 299)(16 232 317 300)(17 233 318 291)(18 234 319 292)(19 235 320 293)(20 236 311 294)(21 281 31 305)(22 282 32 306)(23 283 33 307)(24 284 34 308)(25 285 35 309)(26 286 36 310)(27 287 37 301)(28 288 38 302)(29 289 39 303)(30 290 40 304)(41 249 67 279)(42 250 68 280)(43 241 69 271)(44 242 70 272)(45 243 61 273)(46 244 62 274)(47 245 63 275)(48 246 64 276)(49 247 65 277)(50 248 66 278)(51 263 75 259)(52 264 76 260)(53 265 77 251)(54 266 78 252)(55 267 79 253)(56 268 80 254)(57 269 71 255)(58 270 72 256)(59 261 73 257)(60 262 74 258)(81 183 111 179)(82 184 112 180)(83 185 113 171)(84 186 114 172)(85 187 115 173)(86 188 116 174)(87 189 117 175)(88 190 118 176)(89 181 119 177)(90 182 120 178)(91 193 105 163)(92 194 106 164)(93 195 107 165)(94 196 108 166)(95 197 109 167)(96 198 110 168)(97 199 101 169)(98 200 102 170)(99 191 103 161)(100 192 104 162)(121 204 132 228)(122 205 133 229)(123 206 134 230)(124 207 135 221)(125 208 136 222)(126 209 137 223)(127 210 138 224)(128 201 139 225)(129 202 140 226)(130 203 131 227)
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), (1,53,110,11,129,50,83,22)(2,54,101,12,130,41,84,23)(3,55,102,13,121,42,85,24)(4,56,103,14,122,43,86,25)(5,57,104,15,123,44,87,26)(6,58,105,16,124,45,88,27)(7,59,106,17,125,46,89,28)(8,60,107,18,126,47,90,29)(9,51,108,19,127,48,81,30)(10,52,109,20,128,49,82,21)(31,150,76,95,311,139,65,112)(32,141,77,96,312,140,66,113)(33,142,78,97,313,131,67,114)(34,143,79,98,314,132,68,115)(35,144,80,99,315,133,69,116)(36,145,71,100,316,134,70,117)(37,146,72,91,317,135,61,118)(38,147,73,92,318,136,62,119)(39,148,74,93,319,137,63,120)(40,149,75,94,320,138,64,111)(151,242,162,286,206,269,189,231)(152,243,163,287,207,270,190,232)(153,244,164,288,208,261,181,233)(154,245,165,289,209,262,182,234)(155,246,166,290,210,263,183,235)(156,247,167,281,201,264,184,236)(157,248,168,282,202,265,185,237)(158,249,169,283,203,266,186,238)(159,250,170,284,204,267,187,239)(160,241,161,285,205,268,188,240)(171,295,215,278,198,306,226,251)(172,296,216,279,199,307,227,252)(173,297,217,280,200,308,228,253)(174,298,218,271,191,309,229,254)(175,299,219,272,192,310,230,255)(176,300,220,273,193,301,221,256)(177,291,211,274,194,302,222,257)(178,292,212,275,195,303,223,258)(179,293,213,276,196,304,224,259)(180,294,214,277,197,305,225,260), (1,157,141,215)(2,158,142,216)(3,159,143,217)(4,160,144,218)(5,151,145,219)(6,152,146,220)(7,153,147,211)(8,154,148,212)(9,155,149,213)(10,156,150,214)(11,237,312,295)(12,238,313,296)(13,239,314,297)(14,240,315,298)(15,231,316,299)(16,232,317,300)(17,233,318,291)(18,234,319,292)(19,235,320,293)(20,236,311,294)(21,281,31,305)(22,282,32,306)(23,283,33,307)(24,284,34,308)(25,285,35,309)(26,286,36,310)(27,287,37,301)(28,288,38,302)(29,289,39,303)(30,290,40,304)(41,249,67,279)(42,250,68,280)(43,241,69,271)(44,242,70,272)(45,243,61,273)(46,244,62,274)(47,245,63,275)(48,246,64,276)(49,247,65,277)(50,248,66,278)(51,263,75,259)(52,264,76,260)(53,265,77,251)(54,266,78,252)(55,267,79,253)(56,268,80,254)(57,269,71,255)(58,270,72,256)(59,261,73,257)(60,262,74,258)(81,183,111,179)(82,184,112,180)(83,185,113,171)(84,186,114,172)(85,187,115,173)(86,188,116,174)(87,189,117,175)(88,190,118,176)(89,181,119,177)(90,182,120,178)(91,193,105,163)(92,194,106,164)(93,195,107,165)(94,196,108,166)(95,197,109,167)(96,198,110,168)(97,199,101,169)(98,200,102,170)(99,191,103,161)(100,192,104,162)(121,204,132,228)(122,205,133,229)(123,206,134,230)(124,207,135,221)(125,208,136,222)(126,209,137,223)(127,210,138,224)(128,201,139,225)(129,202,140,226)(130,203,131,227)>;
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), (1,53,110,11,129,50,83,22)(2,54,101,12,130,41,84,23)(3,55,102,13,121,42,85,24)(4,56,103,14,122,43,86,25)(5,57,104,15,123,44,87,26)(6,58,105,16,124,45,88,27)(7,59,106,17,125,46,89,28)(8,60,107,18,126,47,90,29)(9,51,108,19,127,48,81,30)(10,52,109,20,128,49,82,21)(31,150,76,95,311,139,65,112)(32,141,77,96,312,140,66,113)(33,142,78,97,313,131,67,114)(34,143,79,98,314,132,68,115)(35,144,80,99,315,133,69,116)(36,145,71,100,316,134,70,117)(37,146,72,91,317,135,61,118)(38,147,73,92,318,136,62,119)(39,148,74,93,319,137,63,120)(40,149,75,94,320,138,64,111)(151,242,162,286,206,269,189,231)(152,243,163,287,207,270,190,232)(153,244,164,288,208,261,181,233)(154,245,165,289,209,262,182,234)(155,246,166,290,210,263,183,235)(156,247,167,281,201,264,184,236)(157,248,168,282,202,265,185,237)(158,249,169,283,203,266,186,238)(159,250,170,284,204,267,187,239)(160,241,161,285,205,268,188,240)(171,295,215,278,198,306,226,251)(172,296,216,279,199,307,227,252)(173,297,217,280,200,308,228,253)(174,298,218,271,191,309,229,254)(175,299,219,272,192,310,230,255)(176,300,220,273,193,301,221,256)(177,291,211,274,194,302,222,257)(178,292,212,275,195,303,223,258)(179,293,213,276,196,304,224,259)(180,294,214,277,197,305,225,260), (1,157,141,215)(2,158,142,216)(3,159,143,217)(4,160,144,218)(5,151,145,219)(6,152,146,220)(7,153,147,211)(8,154,148,212)(9,155,149,213)(10,156,150,214)(11,237,312,295)(12,238,313,296)(13,239,314,297)(14,240,315,298)(15,231,316,299)(16,232,317,300)(17,233,318,291)(18,234,319,292)(19,235,320,293)(20,236,311,294)(21,281,31,305)(22,282,32,306)(23,283,33,307)(24,284,34,308)(25,285,35,309)(26,286,36,310)(27,287,37,301)(28,288,38,302)(29,289,39,303)(30,290,40,304)(41,249,67,279)(42,250,68,280)(43,241,69,271)(44,242,70,272)(45,243,61,273)(46,244,62,274)(47,245,63,275)(48,246,64,276)(49,247,65,277)(50,248,66,278)(51,263,75,259)(52,264,76,260)(53,265,77,251)(54,266,78,252)(55,267,79,253)(56,268,80,254)(57,269,71,255)(58,270,72,256)(59,261,73,257)(60,262,74,258)(81,183,111,179)(82,184,112,180)(83,185,113,171)(84,186,114,172)(85,187,115,173)(86,188,116,174)(87,189,117,175)(88,190,118,176)(89,181,119,177)(90,182,120,178)(91,193,105,163)(92,194,106,164)(93,195,107,165)(94,196,108,166)(95,197,109,167)(96,198,110,168)(97,199,101,169)(98,200,102,170)(99,191,103,161)(100,192,104,162)(121,204,132,228)(122,205,133,229)(123,206,134,230)(124,207,135,221)(125,208,136,222)(126,209,137,223)(127,210,138,224)(128,201,139,225)(129,202,140,226)(130,203,131,227) );
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)], [(1,53,110,11,129,50,83,22),(2,54,101,12,130,41,84,23),(3,55,102,13,121,42,85,24),(4,56,103,14,122,43,86,25),(5,57,104,15,123,44,87,26),(6,58,105,16,124,45,88,27),(7,59,106,17,125,46,89,28),(8,60,107,18,126,47,90,29),(9,51,108,19,127,48,81,30),(10,52,109,20,128,49,82,21),(31,150,76,95,311,139,65,112),(32,141,77,96,312,140,66,113),(33,142,78,97,313,131,67,114),(34,143,79,98,314,132,68,115),(35,144,80,99,315,133,69,116),(36,145,71,100,316,134,70,117),(37,146,72,91,317,135,61,118),(38,147,73,92,318,136,62,119),(39,148,74,93,319,137,63,120),(40,149,75,94,320,138,64,111),(151,242,162,286,206,269,189,231),(152,243,163,287,207,270,190,232),(153,244,164,288,208,261,181,233),(154,245,165,289,209,262,182,234),(155,246,166,290,210,263,183,235),(156,247,167,281,201,264,184,236),(157,248,168,282,202,265,185,237),(158,249,169,283,203,266,186,238),(159,250,170,284,204,267,187,239),(160,241,161,285,205,268,188,240),(171,295,215,278,198,306,226,251),(172,296,216,279,199,307,227,252),(173,297,217,280,200,308,228,253),(174,298,218,271,191,309,229,254),(175,299,219,272,192,310,230,255),(176,300,220,273,193,301,221,256),(177,291,211,274,194,302,222,257),(178,292,212,275,195,303,223,258),(179,293,213,276,196,304,224,259),(180,294,214,277,197,305,225,260)], [(1,157,141,215),(2,158,142,216),(3,159,143,217),(4,160,144,218),(5,151,145,219),(6,152,146,220),(7,153,147,211),(8,154,148,212),(9,155,149,213),(10,156,150,214),(11,237,312,295),(12,238,313,296),(13,239,314,297),(14,240,315,298),(15,231,316,299),(16,232,317,300),(17,233,318,291),(18,234,319,292),(19,235,320,293),(20,236,311,294),(21,281,31,305),(22,282,32,306),(23,283,33,307),(24,284,34,308),(25,285,35,309),(26,286,36,310),(27,287,37,301),(28,288,38,302),(29,289,39,303),(30,290,40,304),(41,249,67,279),(42,250,68,280),(43,241,69,271),(44,242,70,272),(45,243,61,273),(46,244,62,274),(47,245,63,275),(48,246,64,276),(49,247,65,277),(50,248,66,278),(51,263,75,259),(52,264,76,260),(53,265,77,251),(54,266,78,252),(55,267,79,253),(56,268,80,254),(57,269,71,255),(58,270,72,256),(59,261,73,257),(60,262,74,258),(81,183,111,179),(82,184,112,180),(83,185,113,171),(84,186,114,172),(85,187,115,173),(86,188,116,174),(87,189,117,175),(88,190,118,176),(89,181,119,177),(90,182,120,178),(91,193,105,163),(92,194,106,164),(93,195,107,165),(94,196,108,166),(95,197,109,167),(96,198,110,168),(97,199,101,169),(98,200,102,170),(99,191,103,161),(100,192,104,162),(121,204,132,228),(122,205,133,229),(123,206,134,230),(124,207,135,221),(125,208,136,222),(126,209,137,223),(127,210,138,224),(128,201,139,225),(129,202,140,226),(130,203,131,227)]])
200 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | ··· | 4H | 4I | ··· | 4P | 5A | 5B | 5C | 5D | 8A | ··· | 8P | 10A | ··· | 10AB | 20A | ··· | 20AF | 20AG | ··· | 20BL | 40A | ··· | 40BL |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C5 | C10 | C10 | C10 | C20 | C20 | C20 | M4(2) | C5×M4(2) |
| kernel | C10×C8⋊C4 | C5×C8⋊C4 | C2×C4×C20 | C22×C40 | C4×C20 | C2×C40 | C22×C20 | C2×C8⋊C4 | C8⋊C4 | C2×C42 | C22×C8 | C42 | C2×C8 | C22×C4 | C2×C10 | C22 |
| # reps | 1 | 4 | 1 | 2 | 4 | 16 | 4 | 4 | 16 | 4 | 8 | 16 | 64 | 16 | 8 | 32 |
Matrix representation of C10×C8⋊C4 ►in GL4(𝔽41) generated by
| 31 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 39 | 35 |
| 0 | 0 | 6 | 2 |
| 40 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 40 |
| 0 | 0 | 40 | 0 |
G:=sub<GL(4,GF(41))| [31,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,39,6,0,0,35,2],[40,0,0,0,0,32,0,0,0,0,0,40,0,0,40,0] >;
C10×C8⋊C4 in GAP, Magma, Sage, TeX
C_{10}\times C_8\rtimes C_4 % in TeX
G:=Group("C10xC8:C4"); // GroupNames label
G:=SmallGroup(320,904);
// by ID
G=gap.SmallGroup(320,904);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,2269,568,172]);
// Polycyclic
G:=Group<a,b,c|a^10=b^8=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations