Copied to
clipboard

## G = C2×C4×C5⋊C8order 320 = 26·5

### Direct product of C2×C4 and C5⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C2×C4×C5⋊C8
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C22×C5⋊C8 — C2×C4×C5⋊C8
 Lower central C5 — C2×C4×C5⋊C8
 Upper central C1 — C22×C4

Generators and relations for C2×C4×C5⋊C8
G = < a,b,c,d | a2=b4=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 378 in 162 conjugacy classes, 108 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C5, C8 [×8], C2×C4 [×6], C2×C4 [×12], C23, C10 [×3], C10 [×4], C42 [×4], C2×C8 [×12], C22×C4, C22×C4 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C4×C8 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×8], C2×Dic5 [×2], C2×Dic5 [×10], C2×C20 [×6], C22×C10, C2×C4×C8, C4×Dic5 [×4], C2×C5⋊C8 [×12], C22×Dic5 [×2], C22×C20, C4×C5⋊C8 [×4], C2×C4×Dic5, C22×C5⋊C8 [×2], C2×C4×C5⋊C8
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C8 [×8], C2×C4 [×18], C23, C42 [×4], C2×C8 [×12], C22×C4 [×3], F5, C4×C8 [×4], C2×C42, C22×C8 [×2], C5⋊C8 [×4], C2×F5 [×3], C2×C4×C8, D5⋊C8 [×2], C4×F5 [×2], C2×C5⋊C8 [×6], C22×F5, C4×C5⋊C8 [×4], C2×D5⋊C8, C2×C4×F5, C22×C5⋊C8, C2×C4×C5⋊C8

Smallest permutation representation of C2×C4×C5⋊C8
Regular action on 320 points
Generators in S320
(1 12)(2 13)(3 14)(4 15)(5 16)(6 9)(7 10)(8 11)(17 149)(18 150)(19 151)(20 152)(21 145)(22 146)(23 147)(24 148)(25 216)(26 209)(27 210)(28 211)(29 212)(30 213)(31 214)(32 215)(33 164)(34 165)(35 166)(36 167)(37 168)(38 161)(39 162)(40 163)(41 192)(42 185)(43 186)(44 187)(45 188)(46 189)(47 190)(48 191)(49 97)(50 98)(51 99)(52 100)(53 101)(54 102)(55 103)(56 104)(57 229)(58 230)(59 231)(60 232)(61 225)(62 226)(63 227)(64 228)(65 112)(66 105)(67 106)(68 107)(69 108)(70 109)(71 110)(72 111)(73 127)(74 128)(75 121)(76 122)(77 123)(78 124)(79 125)(80 126)(81 279)(82 280)(83 273)(84 274)(85 275)(86 276)(87 277)(88 278)(89 253)(90 254)(91 255)(92 256)(93 249)(94 250)(95 251)(96 252)(113 285)(114 286)(115 287)(116 288)(117 281)(118 282)(119 283)(120 284)(129 298)(130 299)(131 300)(132 301)(133 302)(134 303)(135 304)(136 297)(137 268)(138 269)(139 270)(140 271)(141 272)(142 265)(143 266)(144 267)(153 262)(154 263)(155 264)(156 257)(157 258)(158 259)(159 260)(160 261)(169 312)(170 305)(171 306)(172 307)(173 308)(174 309)(175 310)(176 311)(177 235)(178 236)(179 237)(180 238)(181 239)(182 240)(183 233)(184 234)(193 244)(194 245)(195 246)(196 247)(197 248)(198 241)(199 242)(200 243)(201 314)(202 315)(203 316)(204 317)(205 318)(206 319)(207 320)(208 313)(217 290)(218 291)(219 292)(220 293)(221 294)(222 295)(223 296)(224 289)
(1 113 57 262)(2 114 58 263)(3 115 59 264)(4 116 60 257)(5 117 61 258)(6 118 62 259)(7 119 63 260)(8 120 64 261)(9 282 226 158)(10 283 227 159)(11 284 228 160)(12 285 229 153)(13 286 230 154)(14 287 231 155)(15 288 232 156)(16 281 225 157)(17 182 191 295)(18 183 192 296)(19 184 185 289)(20 177 186 290)(21 178 187 291)(22 179 188 292)(23 180 189 293)(24 181 190 294)(25 166 176 75)(26 167 169 76)(27 168 170 77)(28 161 171 78)(29 162 172 79)(30 163 173 80)(31 164 174 73)(32 165 175 74)(33 309 127 214)(34 310 128 215)(35 311 121 216)(36 312 122 209)(37 305 123 210)(38 306 124 211)(39 307 125 212)(40 308 126 213)(41 223 150 233)(42 224 151 234)(43 217 152 235)(44 218 145 236)(45 219 146 237)(46 220 147 238)(47 221 148 239)(48 222 149 240)(49 205 137 278)(50 206 138 279)(51 207 139 280)(52 208 140 273)(53 201 141 274)(54 202 142 275)(55 203 143 276)(56 204 144 277)(65 136 196 252)(66 129 197 253)(67 130 198 254)(68 131 199 255)(69 132 200 256)(70 133 193 249)(71 134 194 250)(72 135 195 251)(81 98 319 269)(82 99 320 270)(83 100 313 271)(84 101 314 272)(85 102 315 265)(86 103 316 266)(87 104 317 267)(88 97 318 268)(89 105 298 248)(90 106 299 241)(91 107 300 242)(92 108 301 243)(93 109 302 244)(94 110 303 245)(95 111 304 246)(96 112 297 247)
(1 166 292 303 81)(2 304 167 82 293)(3 83 297 294 168)(4 295 84 161 298)(5 162 296 299 85)(6 300 163 86 289)(7 87 301 290 164)(8 291 88 165 302)(9 131 40 276 224)(10 277 132 217 33)(11 218 278 34 133)(12 35 219 134 279)(13 135 36 280 220)(14 273 136 221 37)(15 222 274 38 129)(16 39 223 130 275)(17 101 171 248 116)(18 241 102 117 172)(19 118 242 173 103)(20 174 119 104 243)(21 97 175 244 120)(22 245 98 113 176)(23 114 246 169 99)(24 170 115 100 247)(25 188 110 269 262)(26 270 189 263 111)(27 264 271 112 190)(28 105 257 191 272)(29 192 106 265 258)(30 266 185 259 107)(31 260 267 108 186)(32 109 261 187 268)(41 67 142 157 212)(42 158 68 213 143)(43 214 159 144 69)(44 137 215 70 160)(45 71 138 153 216)(46 154 72 209 139)(47 210 155 140 65)(48 141 211 66 156)(49 310 193 284 145)(50 285 311 146 194)(51 147 286 195 312)(52 196 148 305 287)(53 306 197 288 149)(54 281 307 150 198)(55 151 282 199 308)(56 200 152 309 283)(57 75 179 94 319)(58 95 76 320 180)(59 313 96 181 77)(60 182 314 78 89)(61 79 183 90 315)(62 91 80 316 184)(63 317 92 177 73)(64 178 318 74 93)(121 237 250 206 229)(122 207 238 230 251)(123 231 208 252 239)(124 253 232 240 201)(125 233 254 202 225)(126 203 234 226 255)(127 227 204 256 235)(128 249 228 236 205)
(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)

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

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

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

80 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4X 5 8A ··· 8AF 10A ··· 10G 20A ··· 20H order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 ··· 1 1 ··· 1 5 ··· 5 4 5 ··· 5 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + + - + + image C1 C2 C2 C2 C4 C4 C4 C4 C8 C8 F5 C5⋊C8 C2×F5 C2×F5 D5⋊C8 C4×F5 kernel C2×C4×C5⋊C8 C4×C5⋊C8 C2×C4×Dic5 C22×C5⋊C8 C4×Dic5 C2×C5⋊C8 C22×Dic5 C22×C20 C2×Dic5 C2×C20 C22×C4 C2×C4 C2×C4 C23 C22 C22 # reps 1 4 1 2 4 16 2 2 16 16 1 4 2 1 4 4

Matrix representation of C2×C4×C5⋊C8 in GL7(𝔽41)

 1 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 1 0 0 40 0 0 0 0 1 0 40 0 0 0 0 0 1 40
,
 27 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 21 40 28 21 0 0 0 8 20 0 1 0 0 0 21 0 40 29 0 0 0 20 28 20 1

G:=sub<GL(7,GF(41))| [1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,40,40,40,40],[27,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,21,8,21,20,0,0,0,40,20,0,28,0,0,0,28,0,40,20,0,0,0,21,1,29,1] >;

C2×C4×C5⋊C8 in GAP, Magma, Sage, TeX

C_2\times C_4\times C_5\rtimes C_8
% in TeX

G:=Group("C2xC4xC5:C8");
// GroupNames label

G:=SmallGroup(320,1084);
// by ID

G=gap.SmallGroup(320,1084);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽