direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×C8⋊C8, C8⋊3C40, C40⋊11C8, C20.59M4(2), (C2×C8).4C20, C2.1(C4×C40), C10.21(C4×C8), (C4×C40).31C2, (C4×C8).13C10, C20.85(C2×C8), C4.11(C2×C40), (C2×C40).54C4, C22.6(C4×C20), C4.9(C5×M4(2)), C42.91(C2×C10), (C2×C10).50C42, C10.15(C8⋊C4), (C4×C20).377C22, C2.1(C5×C8⋊C4), (C2×C4).80(C2×C20), (C2×C20).515(C2×C4), SmallGroup(320,127)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C8⋊C8
G = < a,b,c | a5=b8=c8=1, ab=ba, ac=ca, cbc-1=b5 >
Subgroups: 74 in 66 conjugacy classes, 58 normal (14 characteristic)
C1, C2, C2, C4, C22, C5, C8, C8, C2×C4, C2×C4, C10, C10, C42, C2×C8, C20, C2×C10, C4×C8, C4×C8, C40, C40, C2×C20, C2×C20, C8⋊C8, C4×C20, C2×C40, C4×C40, C4×C40, C5×C8⋊C8
Quotients: C1, C2, C4, C22, C5, C8, C2×C4, C10, C42, C2×C8, M4(2), C20, C2×C10, C4×C8, C8⋊C4, C40, C2×C20, C8⋊C8, C4×C20, C2×C40, C5×M4(2), C4×C40, C5×C8⋊C4, C5×C8⋊C8
►Regular action on 320 points
Generators in S320
(1 285 113 269 97)(2 286 114 270 98)(3 287 115 271 99)(4 288 116 272 100)(5 281 117 265 101)(6 282 118 266 102)(7 283 119 267 103)(8 284 120 268 104)(9 160 308 144 172)(10 153 309 137 173)(11 154 310 138 174)(12 155 311 139 175)(13 156 312 140 176)(14 157 305 141 169)(15 158 306 142 170)(16 159 307 143 171)(17 49 207 33 191)(18 50 208 34 192)(19 51 201 35 185)(20 52 202 36 186)(21 53 203 37 187)(22 54 204 38 188)(23 55 205 39 189)(24 56 206 40 190)(25 181 213 41 197)(26 182 214 42 198)(27 183 215 43 199)(28 184 216 44 200)(29 177 209 45 193)(30 178 210 46 194)(31 179 211 47 195)(32 180 212 48 196)(57 89 245 73 229)(58 90 246 74 230)(59 91 247 75 231)(60 92 248 76 232)(61 93 241 77 225)(62 94 242 78 226)(63 95 243 79 227)(64 96 244 80 228)(65 223 255 81 239)(66 224 256 82 240)(67 217 249 83 233)(68 218 250 84 234)(69 219 251 85 235)(70 220 252 86 236)(71 221 253 87 237)(72 222 254 88 238)(105 261 293 121 277)(106 262 294 122 278)(107 263 295 123 279)(108 264 296 124 280)(109 257 289 125 273)(110 258 290 126 274)(111 259 291 127 275)(112 260 292 128 276)(129 165 317 145 301)(130 166 318 146 302)(131 167 319 147 303)(132 168 320 148 304)(133 161 313 149 297)(134 162 314 150 298)(135 163 315 151 299)(136 164 316 152 300)
(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)
(1 221 23 138 262 57 181 129)(2 218 24 143 263 62 182 134)(3 223 17 140 264 59 183 131)(4 220 18 137 257 64 184 136)(5 217 19 142 258 61 177 133)(6 222 20 139 259 58 178 130)(7 219 21 144 260 63 179 135)(8 224 22 141 261 60 180 132)(9 128 243 47 315 119 85 203)(10 125 244 44 316 116 86 208)(11 122 245 41 317 113 87 205)(12 127 246 46 318 118 88 202)(13 124 247 43 319 115 81 207)(14 121 248 48 320 120 82 204)(15 126 241 45 313 117 83 201)(16 123 242 42 314 114 84 206)(25 301 97 71 189 310 106 229)(26 298 98 68 190 307 107 226)(27 303 99 65 191 312 108 231)(28 300 100 70 192 309 109 228)(29 297 101 67 185 306 110 225)(30 302 102 72 186 311 111 230)(31 299 103 69 187 308 112 227)(32 304 104 66 188 305 105 232)(33 156 280 75 199 147 271 239)(34 153 273 80 200 152 272 236)(35 158 274 77 193 149 265 233)(36 155 275 74 194 146 266 238)(37 160 276 79 195 151 267 235)(38 157 277 76 196 148 268 240)(39 154 278 73 197 145 269 237)(40 159 279 78 198 150 270 234)(49 176 296 91 215 167 287 255)(50 173 289 96 216 164 288 252)(51 170 290 93 209 161 281 249)(52 175 291 90 210 166 282 254)(53 172 292 95 211 163 283 251)(54 169 293 92 212 168 284 256)(55 174 294 89 213 165 285 253)(56 171 295 94 214 162 286 250)
G:=sub<Sym(320)| (1,285,113,269,97)(2,286,114,270,98)(3,287,115,271,99)(4,288,116,272,100)(5,281,117,265,101)(6,282,118,266,102)(7,283,119,267,103)(8,284,120,268,104)(9,160,308,144,172)(10,153,309,137,173)(11,154,310,138,174)(12,155,311,139,175)(13,156,312,140,176)(14,157,305,141,169)(15,158,306,142,170)(16,159,307,143,171)(17,49,207,33,191)(18,50,208,34,192)(19,51,201,35,185)(20,52,202,36,186)(21,53,203,37,187)(22,54,204,38,188)(23,55,205,39,189)(24,56,206,40,190)(25,181,213,41,197)(26,182,214,42,198)(27,183,215,43,199)(28,184,216,44,200)(29,177,209,45,193)(30,178,210,46,194)(31,179,211,47,195)(32,180,212,48,196)(57,89,245,73,229)(58,90,246,74,230)(59,91,247,75,231)(60,92,248,76,232)(61,93,241,77,225)(62,94,242,78,226)(63,95,243,79,227)(64,96,244,80,228)(65,223,255,81,239)(66,224,256,82,240)(67,217,249,83,233)(68,218,250,84,234)(69,219,251,85,235)(70,220,252,86,236)(71,221,253,87,237)(72,222,254,88,238)(105,261,293,121,277)(106,262,294,122,278)(107,263,295,123,279)(108,264,296,124,280)(109,257,289,125,273)(110,258,290,126,274)(111,259,291,127,275)(112,260,292,128,276)(129,165,317,145,301)(130,166,318,146,302)(131,167,319,147,303)(132,168,320,148,304)(133,161,313,149,297)(134,162,314,150,298)(135,163,315,151,299)(136,164,316,152,300), (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), (1,221,23,138,262,57,181,129)(2,218,24,143,263,62,182,134)(3,223,17,140,264,59,183,131)(4,220,18,137,257,64,184,136)(5,217,19,142,258,61,177,133)(6,222,20,139,259,58,178,130)(7,219,21,144,260,63,179,135)(8,224,22,141,261,60,180,132)(9,128,243,47,315,119,85,203)(10,125,244,44,316,116,86,208)(11,122,245,41,317,113,87,205)(12,127,246,46,318,118,88,202)(13,124,247,43,319,115,81,207)(14,121,248,48,320,120,82,204)(15,126,241,45,313,117,83,201)(16,123,242,42,314,114,84,206)(25,301,97,71,189,310,106,229)(26,298,98,68,190,307,107,226)(27,303,99,65,191,312,108,231)(28,300,100,70,192,309,109,228)(29,297,101,67,185,306,110,225)(30,302,102,72,186,311,111,230)(31,299,103,69,187,308,112,227)(32,304,104,66,188,305,105,232)(33,156,280,75,199,147,271,239)(34,153,273,80,200,152,272,236)(35,158,274,77,193,149,265,233)(36,155,275,74,194,146,266,238)(37,160,276,79,195,151,267,235)(38,157,277,76,196,148,268,240)(39,154,278,73,197,145,269,237)(40,159,279,78,198,150,270,234)(49,176,296,91,215,167,287,255)(50,173,289,96,216,164,288,252)(51,170,290,93,209,161,281,249)(52,175,291,90,210,166,282,254)(53,172,292,95,211,163,283,251)(54,169,293,92,212,168,284,256)(55,174,294,89,213,165,285,253)(56,171,295,94,214,162,286,250)>;
G:=Group( (1,285,113,269,97)(2,286,114,270,98)(3,287,115,271,99)(4,288,116,272,100)(5,281,117,265,101)(6,282,118,266,102)(7,283,119,267,103)(8,284,120,268,104)(9,160,308,144,172)(10,153,309,137,173)(11,154,310,138,174)(12,155,311,139,175)(13,156,312,140,176)(14,157,305,141,169)(15,158,306,142,170)(16,159,307,143,171)(17,49,207,33,191)(18,50,208,34,192)(19,51,201,35,185)(20,52,202,36,186)(21,53,203,37,187)(22,54,204,38,188)(23,55,205,39,189)(24,56,206,40,190)(25,181,213,41,197)(26,182,214,42,198)(27,183,215,43,199)(28,184,216,44,200)(29,177,209,45,193)(30,178,210,46,194)(31,179,211,47,195)(32,180,212,48,196)(57,89,245,73,229)(58,90,246,74,230)(59,91,247,75,231)(60,92,248,76,232)(61,93,241,77,225)(62,94,242,78,226)(63,95,243,79,227)(64,96,244,80,228)(65,223,255,81,239)(66,224,256,82,240)(67,217,249,83,233)(68,218,250,84,234)(69,219,251,85,235)(70,220,252,86,236)(71,221,253,87,237)(72,222,254,88,238)(105,261,293,121,277)(106,262,294,122,278)(107,263,295,123,279)(108,264,296,124,280)(109,257,289,125,273)(110,258,290,126,274)(111,259,291,127,275)(112,260,292,128,276)(129,165,317,145,301)(130,166,318,146,302)(131,167,319,147,303)(132,168,320,148,304)(133,161,313,149,297)(134,162,314,150,298)(135,163,315,151,299)(136,164,316,152,300), (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), (1,221,23,138,262,57,181,129)(2,218,24,143,263,62,182,134)(3,223,17,140,264,59,183,131)(4,220,18,137,257,64,184,136)(5,217,19,142,258,61,177,133)(6,222,20,139,259,58,178,130)(7,219,21,144,260,63,179,135)(8,224,22,141,261,60,180,132)(9,128,243,47,315,119,85,203)(10,125,244,44,316,116,86,208)(11,122,245,41,317,113,87,205)(12,127,246,46,318,118,88,202)(13,124,247,43,319,115,81,207)(14,121,248,48,320,120,82,204)(15,126,241,45,313,117,83,201)(16,123,242,42,314,114,84,206)(25,301,97,71,189,310,106,229)(26,298,98,68,190,307,107,226)(27,303,99,65,191,312,108,231)(28,300,100,70,192,309,109,228)(29,297,101,67,185,306,110,225)(30,302,102,72,186,311,111,230)(31,299,103,69,187,308,112,227)(32,304,104,66,188,305,105,232)(33,156,280,75,199,147,271,239)(34,153,273,80,200,152,272,236)(35,158,274,77,193,149,265,233)(36,155,275,74,194,146,266,238)(37,160,276,79,195,151,267,235)(38,157,277,76,196,148,268,240)(39,154,278,73,197,145,269,237)(40,159,279,78,198,150,270,234)(49,176,296,91,215,167,287,255)(50,173,289,96,216,164,288,252)(51,170,290,93,209,161,281,249)(52,175,291,90,210,166,282,254)(53,172,292,95,211,163,283,251)(54,169,293,92,212,168,284,256)(55,174,294,89,213,165,285,253)(56,171,295,94,214,162,286,250) );
G=PermutationGroup([[(1,285,113,269,97),(2,286,114,270,98),(3,287,115,271,99),(4,288,116,272,100),(5,281,117,265,101),(6,282,118,266,102),(7,283,119,267,103),(8,284,120,268,104),(9,160,308,144,172),(10,153,309,137,173),(11,154,310,138,174),(12,155,311,139,175),(13,156,312,140,176),(14,157,305,141,169),(15,158,306,142,170),(16,159,307,143,171),(17,49,207,33,191),(18,50,208,34,192),(19,51,201,35,185),(20,52,202,36,186),(21,53,203,37,187),(22,54,204,38,188),(23,55,205,39,189),(24,56,206,40,190),(25,181,213,41,197),(26,182,214,42,198),(27,183,215,43,199),(28,184,216,44,200),(29,177,209,45,193),(30,178,210,46,194),(31,179,211,47,195),(32,180,212,48,196),(57,89,245,73,229),(58,90,246,74,230),(59,91,247,75,231),(60,92,248,76,232),(61,93,241,77,225),(62,94,242,78,226),(63,95,243,79,227),(64,96,244,80,228),(65,223,255,81,239),(66,224,256,82,240),(67,217,249,83,233),(68,218,250,84,234),(69,219,251,85,235),(70,220,252,86,236),(71,221,253,87,237),(72,222,254,88,238),(105,261,293,121,277),(106,262,294,122,278),(107,263,295,123,279),(108,264,296,124,280),(109,257,289,125,273),(110,258,290,126,274),(111,259,291,127,275),(112,260,292,128,276),(129,165,317,145,301),(130,166,318,146,302),(131,167,319,147,303),(132,168,320,148,304),(133,161,313,149,297),(134,162,314,150,298),(135,163,315,151,299),(136,164,316,152,300)], [(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)], [(1,221,23,138,262,57,181,129),(2,218,24,143,263,62,182,134),(3,223,17,140,264,59,183,131),(4,220,18,137,257,64,184,136),(5,217,19,142,258,61,177,133),(6,222,20,139,259,58,178,130),(7,219,21,144,260,63,179,135),(8,224,22,141,261,60,180,132),(9,128,243,47,315,119,85,203),(10,125,244,44,316,116,86,208),(11,122,245,41,317,113,87,205),(12,127,246,46,318,118,88,202),(13,124,247,43,319,115,81,207),(14,121,248,48,320,120,82,204),(15,126,241,45,313,117,83,201),(16,123,242,42,314,114,84,206),(25,301,97,71,189,310,106,229),(26,298,98,68,190,307,107,226),(27,303,99,65,191,312,108,231),(28,300,100,70,192,309,109,228),(29,297,101,67,185,306,110,225),(30,302,102,72,186,311,111,230),(31,299,103,69,187,308,112,227),(32,304,104,66,188,305,105,232),(33,156,280,75,199,147,271,239),(34,153,273,80,200,152,272,236),(35,158,274,77,193,149,265,233),(36,155,275,74,194,146,266,238),(37,160,276,79,195,151,267,235),(38,157,277,76,196,148,268,240),(39,154,278,73,197,145,269,237),(40,159,279,78,198,150,270,234),(49,176,296,91,215,167,287,255),(50,173,289,96,216,164,288,252),(51,170,290,93,209,161,281,249),(52,175,291,90,210,166,282,254),(53,172,292,95,211,163,283,251),(54,169,293,92,212,168,284,256),(55,174,294,89,213,165,285,253),(56,171,295,94,214,162,286,250)]])
200 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8X | 10A | ··· | 10L | 20A | ··· | 20AV | 40A | ··· | 40CR |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
200 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | ||||||||
| image | C1 | C2 | C4 | C5 | C8 | C10 | C20 | C40 | M4(2) | C5×M4(2) |
| kernel | C5×C8⋊C8 | C4×C40 | C2×C40 | C8⋊C8 | C40 | C4×C8 | C2×C8 | C8 | C20 | C4 |
| # reps | 1 | 3 | 12 | 4 | 16 | 12 | 48 | 64 | 8 | 32 |
Matrix representation of C5×C8⋊C8 ►in GL3(𝔽41) generated by
| 1 | 0 | 0 |
| 0 | 18 | 0 |
| 0 | 0 | 18 |
| 9 | 0 | 0 |
| 0 | 0 | 32 |
| 0 | 1 | 0 |
| 38 | 0 | 0 |
| 0 | 38 | 0 |
| 0 | 0 | 3 |
G:=sub<GL(3,GF(41))| [1,0,0,0,18,0,0,0,18],[9,0,0,0,0,1,0,32,0],[38,0,0,0,38,0,0,0,3] >;
C5×C8⋊C8 in GAP, Magma, Sage, TeX
C_5\times C_8\rtimes C_8
% in TeX
G:=Group("C5xC8:C8"); // GroupNames label
G:=SmallGroup(320,127);
// by ID
G=gap.SmallGroup(320,127);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,140,1149,288,136,172]);
// 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^5>;
// generators/relations