direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C8⋊1C8, C8⋊1C40, C40⋊9C8, C20.68D8, C20.31Q16, C20.55M4(2), C4⋊C8.5C10, (C2×C8).8C20, C4.7(C2×C40), (C4×C8).5C10, C4.17(C5×D8), C4.9(C5×Q16), C20.81(C2×C8), (C2×C40).38C4, (C4×C40).23C2, C10.21(C4⋊C8), (C2×C20).48Q8, (C2×C20).531D4, C4.5(C5×M4(2)), C42.65(C2×C10), C10.18(C2.D8), (C4×C20).349C22, C10.13(C8.C4), C2.4(C5×C4⋊C8), (C5×C4⋊C8).11C2, (C2×C4).9(C5×Q8), C2.1(C5×C2.D8), (C2×C4).62(C2×C20), (C2×C4).140(C5×D4), C2.2(C5×C8.C4), C22.14(C5×C4⋊C4), (C2×C10).85(C4⋊C4), (C2×C20).496(C2×C4), SmallGroup(320,140)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊1C8
G = < a,b,c | a5=b8=c8=1, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C5×C8⋊1C8
►Regular action on 320 points
Generators in S320
(1 124 33 108 17)(2 125 34 109 18)(3 126 35 110 19)(4 127 36 111 20)(5 128 37 112 21)(6 121 38 105 22)(7 122 39 106 23)(8 123 40 107 24)(9 231 312 215 90)(10 232 305 216 91)(11 225 306 209 92)(12 226 307 210 93)(13 227 308 211 94)(14 228 309 212 95)(15 229 310 213 96)(16 230 311 214 89)(25 104 136 41 120)(26 97 129 42 113)(27 98 130 43 114)(28 99 131 44 115)(29 100 132 45 116)(30 101 133 46 117)(31 102 134 47 118)(32 103 135 48 119)(49 169 156 65 140)(50 170 157 66 141)(51 171 158 67 142)(52 172 159 68 143)(53 173 160 69 144)(54 174 153 70 137)(55 175 154 71 138)(56 176 155 72 139)(57 161 242 73 152)(58 162 243 74 145)(59 163 244 75 146)(60 164 245 76 147)(61 165 246 77 148)(62 166 247 78 149)(63 167 248 79 150)(64 168 241 80 151)(81 317 234 301 218)(82 318 235 302 219)(83 319 236 303 220)(84 320 237 304 221)(85 313 238 297 222)(86 314 239 298 223)(87 315 240 299 224)(88 316 233 300 217)(177 257 256 292 195)(178 258 249 293 196)(179 259 250 294 197)(180 260 251 295 198)(181 261 252 296 199)(182 262 253 289 200)(183 263 254 290 193)(184 264 255 291 194)(185 279 284 201 268)(186 280 285 202 269)(187 273 286 203 270)(188 274 287 204 271)(189 275 288 205 272)(190 276 281 206 265)(191 277 282 207 266)(192 278 283 208 267)
(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)
(1 220 104 196 50 212 58 192)(2 219 97 195 51 211 59 191)(3 218 98 194 52 210 60 190)(4 217 99 193 53 209 61 189)(5 224 100 200 54 216 62 188)(6 223 101 199 55 215 63 187)(7 222 102 198 56 214 64 186)(8 221 103 197 49 213 57 185)(9 248 286 38 314 46 261 154)(10 247 287 37 315 45 262 153)(11 246 288 36 316 44 263 160)(12 245 281 35 317 43 264 159)(13 244 282 34 318 42 257 158)(14 243 283 33 319 41 258 157)(15 242 284 40 320 48 259 156)(16 241 285 39 313 47 260 155)(17 303 25 293 141 309 145 267)(18 302 26 292 142 308 146 266)(19 301 27 291 143 307 147 265)(20 300 28 290 144 306 148 272)(21 299 29 289 137 305 149 271)(22 298 30 296 138 312 150 270)(23 297 31 295 139 311 151 269)(24 304 32 294 140 310 152 268)(65 229 73 201 107 237 119 250)(66 228 74 208 108 236 120 249)(67 227 75 207 109 235 113 256)(68 226 76 206 110 234 114 255)(69 225 77 205 111 233 115 254)(70 232 78 204 112 240 116 253)(71 231 79 203 105 239 117 252)(72 230 80 202 106 238 118 251)(81 130 184 172 93 164 276 126)(82 129 177 171 94 163 277 125)(83 136 178 170 95 162 278 124)(84 135 179 169 96 161 279 123)(85 134 180 176 89 168 280 122)(86 133 181 175 90 167 273 121)(87 132 182 174 91 166 274 128)(88 131 183 173 92 165 275 127)
G:=sub<Sym(320)| (1,124,33,108,17)(2,125,34,109,18)(3,126,35,110,19)(4,127,36,111,20)(5,128,37,112,21)(6,121,38,105,22)(7,122,39,106,23)(8,123,40,107,24)(9,231,312,215,90)(10,232,305,216,91)(11,225,306,209,92)(12,226,307,210,93)(13,227,308,211,94)(14,228,309,212,95)(15,229,310,213,96)(16,230,311,214,89)(25,104,136,41,120)(26,97,129,42,113)(27,98,130,43,114)(28,99,131,44,115)(29,100,132,45,116)(30,101,133,46,117)(31,102,134,47,118)(32,103,135,48,119)(49,169,156,65,140)(50,170,157,66,141)(51,171,158,67,142)(52,172,159,68,143)(53,173,160,69,144)(54,174,153,70,137)(55,175,154,71,138)(56,176,155,72,139)(57,161,242,73,152)(58,162,243,74,145)(59,163,244,75,146)(60,164,245,76,147)(61,165,246,77,148)(62,166,247,78,149)(63,167,248,79,150)(64,168,241,80,151)(81,317,234,301,218)(82,318,235,302,219)(83,319,236,303,220)(84,320,237,304,221)(85,313,238,297,222)(86,314,239,298,223)(87,315,240,299,224)(88,316,233,300,217)(177,257,256,292,195)(178,258,249,293,196)(179,259,250,294,197)(180,260,251,295,198)(181,261,252,296,199)(182,262,253,289,200)(183,263,254,290,193)(184,264,255,291,194)(185,279,284,201,268)(186,280,285,202,269)(187,273,286,203,270)(188,274,287,204,271)(189,275,288,205,272)(190,276,281,206,265)(191,277,282,207,266)(192,278,283,208,267), (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), (1,220,104,196,50,212,58,192)(2,219,97,195,51,211,59,191)(3,218,98,194,52,210,60,190)(4,217,99,193,53,209,61,189)(5,224,100,200,54,216,62,188)(6,223,101,199,55,215,63,187)(7,222,102,198,56,214,64,186)(8,221,103,197,49,213,57,185)(9,248,286,38,314,46,261,154)(10,247,287,37,315,45,262,153)(11,246,288,36,316,44,263,160)(12,245,281,35,317,43,264,159)(13,244,282,34,318,42,257,158)(14,243,283,33,319,41,258,157)(15,242,284,40,320,48,259,156)(16,241,285,39,313,47,260,155)(17,303,25,293,141,309,145,267)(18,302,26,292,142,308,146,266)(19,301,27,291,143,307,147,265)(20,300,28,290,144,306,148,272)(21,299,29,289,137,305,149,271)(22,298,30,296,138,312,150,270)(23,297,31,295,139,311,151,269)(24,304,32,294,140,310,152,268)(65,229,73,201,107,237,119,250)(66,228,74,208,108,236,120,249)(67,227,75,207,109,235,113,256)(68,226,76,206,110,234,114,255)(69,225,77,205,111,233,115,254)(70,232,78,204,112,240,116,253)(71,231,79,203,105,239,117,252)(72,230,80,202,106,238,118,251)(81,130,184,172,93,164,276,126)(82,129,177,171,94,163,277,125)(83,136,178,170,95,162,278,124)(84,135,179,169,96,161,279,123)(85,134,180,176,89,168,280,122)(86,133,181,175,90,167,273,121)(87,132,182,174,91,166,274,128)(88,131,183,173,92,165,275,127)>;
G:=Group( (1,124,33,108,17)(2,125,34,109,18)(3,126,35,110,19)(4,127,36,111,20)(5,128,37,112,21)(6,121,38,105,22)(7,122,39,106,23)(8,123,40,107,24)(9,231,312,215,90)(10,232,305,216,91)(11,225,306,209,92)(12,226,307,210,93)(13,227,308,211,94)(14,228,309,212,95)(15,229,310,213,96)(16,230,311,214,89)(25,104,136,41,120)(26,97,129,42,113)(27,98,130,43,114)(28,99,131,44,115)(29,100,132,45,116)(30,101,133,46,117)(31,102,134,47,118)(32,103,135,48,119)(49,169,156,65,140)(50,170,157,66,141)(51,171,158,67,142)(52,172,159,68,143)(53,173,160,69,144)(54,174,153,70,137)(55,175,154,71,138)(56,176,155,72,139)(57,161,242,73,152)(58,162,243,74,145)(59,163,244,75,146)(60,164,245,76,147)(61,165,246,77,148)(62,166,247,78,149)(63,167,248,79,150)(64,168,241,80,151)(81,317,234,301,218)(82,318,235,302,219)(83,319,236,303,220)(84,320,237,304,221)(85,313,238,297,222)(86,314,239,298,223)(87,315,240,299,224)(88,316,233,300,217)(177,257,256,292,195)(178,258,249,293,196)(179,259,250,294,197)(180,260,251,295,198)(181,261,252,296,199)(182,262,253,289,200)(183,263,254,290,193)(184,264,255,291,194)(185,279,284,201,268)(186,280,285,202,269)(187,273,286,203,270)(188,274,287,204,271)(189,275,288,205,272)(190,276,281,206,265)(191,277,282,207,266)(192,278,283,208,267), (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), (1,220,104,196,50,212,58,192)(2,219,97,195,51,211,59,191)(3,218,98,194,52,210,60,190)(4,217,99,193,53,209,61,189)(5,224,100,200,54,216,62,188)(6,223,101,199,55,215,63,187)(7,222,102,198,56,214,64,186)(8,221,103,197,49,213,57,185)(9,248,286,38,314,46,261,154)(10,247,287,37,315,45,262,153)(11,246,288,36,316,44,263,160)(12,245,281,35,317,43,264,159)(13,244,282,34,318,42,257,158)(14,243,283,33,319,41,258,157)(15,242,284,40,320,48,259,156)(16,241,285,39,313,47,260,155)(17,303,25,293,141,309,145,267)(18,302,26,292,142,308,146,266)(19,301,27,291,143,307,147,265)(20,300,28,290,144,306,148,272)(21,299,29,289,137,305,149,271)(22,298,30,296,138,312,150,270)(23,297,31,295,139,311,151,269)(24,304,32,294,140,310,152,268)(65,229,73,201,107,237,119,250)(66,228,74,208,108,236,120,249)(67,227,75,207,109,235,113,256)(68,226,76,206,110,234,114,255)(69,225,77,205,111,233,115,254)(70,232,78,204,112,240,116,253)(71,231,79,203,105,239,117,252)(72,230,80,202,106,238,118,251)(81,130,184,172,93,164,276,126)(82,129,177,171,94,163,277,125)(83,136,178,170,95,162,278,124)(84,135,179,169,96,161,279,123)(85,134,180,176,89,168,280,122)(86,133,181,175,90,167,273,121)(87,132,182,174,91,166,274,128)(88,131,183,173,92,165,275,127) );
G=PermutationGroup([[(1,124,33,108,17),(2,125,34,109,18),(3,126,35,110,19),(4,127,36,111,20),(5,128,37,112,21),(6,121,38,105,22),(7,122,39,106,23),(8,123,40,107,24),(9,231,312,215,90),(10,232,305,216,91),(11,225,306,209,92),(12,226,307,210,93),(13,227,308,211,94),(14,228,309,212,95),(15,229,310,213,96),(16,230,311,214,89),(25,104,136,41,120),(26,97,129,42,113),(27,98,130,43,114),(28,99,131,44,115),(29,100,132,45,116),(30,101,133,46,117),(31,102,134,47,118),(32,103,135,48,119),(49,169,156,65,140),(50,170,157,66,141),(51,171,158,67,142),(52,172,159,68,143),(53,173,160,69,144),(54,174,153,70,137),(55,175,154,71,138),(56,176,155,72,139),(57,161,242,73,152),(58,162,243,74,145),(59,163,244,75,146),(60,164,245,76,147),(61,165,246,77,148),(62,166,247,78,149),(63,167,248,79,150),(64,168,241,80,151),(81,317,234,301,218),(82,318,235,302,219),(83,319,236,303,220),(84,320,237,304,221),(85,313,238,297,222),(86,314,239,298,223),(87,315,240,299,224),(88,316,233,300,217),(177,257,256,292,195),(178,258,249,293,196),(179,259,250,294,197),(180,260,251,295,198),(181,261,252,296,199),(182,262,253,289,200),(183,263,254,290,193),(184,264,255,291,194),(185,279,284,201,268),(186,280,285,202,269),(187,273,286,203,270),(188,274,287,204,271),(189,275,288,205,272),(190,276,281,206,265),(191,277,282,207,266),(192,278,283,208,267)], [(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)], [(1,220,104,196,50,212,58,192),(2,219,97,195,51,211,59,191),(3,218,98,194,52,210,60,190),(4,217,99,193,53,209,61,189),(5,224,100,200,54,216,62,188),(6,223,101,199,55,215,63,187),(7,222,102,198,56,214,64,186),(8,221,103,197,49,213,57,185),(9,248,286,38,314,46,261,154),(10,247,287,37,315,45,262,153),(11,246,288,36,316,44,263,160),(12,245,281,35,317,43,264,159),(13,244,282,34,318,42,257,158),(14,243,283,33,319,41,258,157),(15,242,284,40,320,48,259,156),(16,241,285,39,313,47,260,155),(17,303,25,293,141,309,145,267),(18,302,26,292,142,308,146,266),(19,301,27,291,143,307,147,265),(20,300,28,290,144,306,148,272),(21,299,29,289,137,305,149,271),(22,298,30,296,138,312,150,270),(23,297,31,295,139,311,151,269),(24,304,32,294,140,310,152,268),(65,229,73,201,107,237,119,250),(66,228,74,208,108,236,120,249),(67,227,75,207,109,235,113,256),(68,226,76,206,110,234,114,255),(69,225,77,205,111,233,115,254),(70,232,78,204,112,240,116,253),(71,231,79,203,105,239,117,252),(72,230,80,202,106,238,118,251),(81,130,184,172,93,164,276,126),(82,129,177,171,94,163,277,125),(83,136,178,170,95,162,278,124),(84,135,179,169,96,161,279,123),(85,134,180,176,89,168,280,122),(86,133,181,175,90,167,273,121),(87,132,182,174,91,166,274,128),(88,131,183,173,92,165,275,127)]])
140 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 8I | ··· | 8P | 10A | ··· | 10L | 20A | ··· | 20P | 20Q | ··· | 20AF | 40A | ··· | 40AF | 40AG | ··· | 40BL |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C4 | C5 | C8 | C10 | C10 | C20 | C40 | D4 | Q8 | M4(2) | D8 | Q16 | C8.C4 | C5×D4 | C5×Q8 | C5×M4(2) | C5×D8 | C5×Q16 | C5×C8.C4 |
| kernel | C5×C8⋊1C8 | C4×C40 | C5×C4⋊C8 | C2×C40 | C8⋊1C8 | C40 | C4×C8 | C4⋊C8 | C2×C8 | C8 | C2×C20 | C2×C20 | C20 | C20 | C20 | C10 | C2×C4 | C2×C4 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 4 | 8 | 16 | 32 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C5×C8⋊1C8 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 12 | 29 |
| 0 | 0 | 12 | 12 |
| 27 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 28 | 2 |
| 0 | 0 | 2 | 13 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,12,12,0,0,29,12],[27,0,0,0,0,1,0,0,0,0,28,2,0,0,2,13] >;
C5×C8⋊1C8 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes_1C_8
% in TeX
G:=Group("C5xC8:1C8"); // GroupNames label
G:=SmallGroup(320,140);
// by ID
G=gap.SmallGroup(320,140);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,708,2803,136,172]);
// Polycyclic
G:=Group<a,b,c|a^5=b^8=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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