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## G = C10×C4⋊C8order 320 = 26·5

### Direct product of C10 and C4⋊C8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C10×C4⋊C8
 Chief series C1 — C2 — C4 — C2×C4 — C2×C20 — C2×C40 — C5×C4⋊C8 — C10×C4⋊C8
 Lower central C1 — C2 — C10×C4⋊C8
 Upper central C1 — C22×C20 — C10×C4⋊C8

Generators and relations for C10×C4⋊C8
G = < a,b,c | a10=b4=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 162 in 138 conjugacy classes, 114 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C2×C8, C22×C4, C20, C20, C20, C2×C10, C2×C10, C4⋊C8, C2×C42, C22×C8, C40, C2×C20, C2×C20, C2×C20, C22×C10, C2×C4⋊C8, C4×C20, C2×C40, C2×C40, C22×C20, C5×C4⋊C8, C2×C4×C20, C22×C40, C10×C4⋊C8
Quotients:

Smallest permutation representation of 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)
(1 183 90 165)(2 184 81 166)(3 185 82 167)(4 186 83 168)(5 187 84 169)(6 188 85 170)(7 189 86 161)(8 190 87 162)(9 181 88 163)(10 182 89 164)(11 295 23 307)(12 296 24 308)(13 297 25 309)(14 298 26 310)(15 299 27 301)(16 300 28 302)(17 291 29 303)(18 292 30 304)(19 293 21 305)(20 294 22 306)(31 281 64 258)(32 282 65 259)(33 283 66 260)(34 284 67 251)(35 285 68 252)(36 286 69 253)(37 287 70 254)(38 288 61 255)(39 289 62 256)(40 290 63 257)(41 275 311 236)(42 276 312 237)(43 277 313 238)(44 278 314 239)(45 279 315 240)(46 280 316 231)(47 271 317 232)(48 272 318 233)(49 273 319 234)(50 274 320 235)(51 245 79 263)(52 246 80 264)(53 247 71 265)(54 248 72 266)(55 249 73 267)(56 250 74 268)(57 241 75 269)(58 242 76 270)(59 243 77 261)(60 244 78 262)(91 176 124 209)(92 177 125 210)(93 178 126 201)(94 179 127 202)(95 180 128 203)(96 171 129 204)(97 172 130 205)(98 173 121 206)(99 174 122 207)(100 175 123 208)(101 218 145 230)(102 219 146 221)(103 220 147 222)(104 211 148 223)(105 212 149 224)(106 213 150 225)(107 214 141 226)(108 215 142 227)(109 216 143 228)(110 217 144 229)(111 196 139 157)(112 197 140 158)(113 198 131 159)(114 199 132 160)(115 200 133 151)(116 191 134 152)(117 192 135 153)(118 193 136 154)(119 194 137 155)(120 195 138 156)
(1 282 108 245 139 307 94 276)(2 283 109 246 140 308 95 277)(3 284 110 247 131 309 96 278)(4 285 101 248 132 310 97 279)(5 286 102 249 133 301 98 280)(6 287 103 250 134 302 99 271)(7 288 104 241 135 303 100 272)(8 289 105 242 136 304 91 273)(9 290 106 243 137 305 92 274)(10 281 107 244 138 306 93 275)(11 202 312 165 65 227 79 196)(12 203 313 166 66 228 80 197)(13 204 314 167 67 229 71 198)(14 205 315 168 68 230 72 199)(15 206 316 169 69 221 73 200)(16 207 317 170 70 222 74 191)(17 208 318 161 61 223 75 192)(18 209 319 162 62 224 76 193)(19 210 320 163 63 225 77 194)(20 201 311 164 64 226 78 195)(21 177 50 181 40 213 59 155)(22 178 41 182 31 214 60 156)(23 179 42 183 32 215 51 157)(24 180 43 184 33 216 52 158)(25 171 44 185 34 217 53 159)(26 172 45 186 35 218 54 160)(27 173 46 187 36 219 55 151)(28 174 47 188 37 220 56 152)(29 175 48 189 38 211 57 153)(30 176 49 190 39 212 58 154)(81 260 143 264 112 296 128 238)(82 251 144 265 113 297 129 239)(83 252 145 266 114 298 130 240)(84 253 146 267 115 299 121 231)(85 254 147 268 116 300 122 232)(86 255 148 269 117 291 123 233)(87 256 149 270 118 292 124 234)(88 257 150 261 119 293 125 235)(89 258 141 262 120 294 126 236)(90 259 142 263 111 295 127 237)

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

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

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

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 2 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 C5 C8 C10 C10 C10 C20 C20 C40 D4 Q8 M4(2) C5×D4 C5×Q8 C5×M4(2) kernel C10×C4⋊C8 C5×C4⋊C8 C2×C4×C20 C22×C40 C4×C20 C22×C20 C2×C4⋊C8 C2×C20 C4⋊C8 C2×C42 C22×C8 C42 C22×C4 C2×C4 C2×C20 C2×C20 C2×C10 C2×C4 C2×C4 C22 # reps 1 4 1 2 4 4 4 16 16 4 8 16 16 64 2 2 4 8 8 16

Matrix representation of C10×C4⋊C8 in GL4(𝔽41) generated by

 1 0 0 0 0 40 0 0 0 0 18 0 0 0 0 18
,
 40 0 0 0 0 1 0 0 0 0 40 23 0 0 32 1
,
 38 0 0 0 0 1 0 0 0 0 37 16 0 0 38 4
G:=sub<GL(4,GF(41))| [1,0,0,0,0,40,0,0,0,0,18,0,0,0,0,18],[40,0,0,0,0,1,0,0,0,0,40,32,0,0,23,1],[38,0,0,0,0,1,0,0,0,0,37,38,0,0,16,4] >;

C10×C4⋊C8 in GAP, Magma, Sage, TeX

C_{10}\times C_4\rtimes C_8
% in TeX

G:=Group("C10xC4:C8");
// GroupNames label

G:=SmallGroup(320,923);
// by ID

G=gap.SmallGroup(320,923);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,560,589,288,124]);
// Polycyclic

G:=Group<a,b,c|a^10=b^4=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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