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## 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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C42, C2×C8, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C4×C8, C2×C42, C22×C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C22×C10, C2×C4×C8, C4×Dic5, C2×C5⋊C8, C22×Dic5, C22×C20, C4×C5⋊C8, C2×C4×Dic5, C22×C5⋊C8, C2×C4×C5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C42, C2×C8, C22×C4, F5, C4×C8, C2×C42, C22×C8, C5⋊C8, C2×F5, C2×C4×C8, D5⋊C8, C4×F5, C2×C5⋊C8, C22×F5, C4×C5⋊C8, 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)(17 147)(18 148)(19 149)(20 150)(21 151)(22 152)(23 145)(24 146)(25 194)(26 195)(27 196)(28 197)(29 198)(30 199)(31 200)(32 193)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 106)(42 107)(43 108)(44 109)(45 110)(46 111)(47 112)(48 105)(49 116)(50 117)(51 118)(52 119)(53 120)(54 113)(55 114)(56 115)(57 136)(58 129)(59 130)(60 131)(61 132)(62 133)(63 134)(64 135)(65 101)(66 102)(67 103)(68 104)(69 97)(70 98)(71 99)(72 100)(73 121)(74 122)(75 123)(76 124)(77 125)(78 126)(79 127)(80 128)(89 253)(90 254)(91 255)(92 256)(93 249)(94 250)(95 251)(96 252)(137 299)(138 300)(139 301)(140 302)(141 303)(142 304)(143 297)(144 298)(153 265)(154 266)(155 267)(156 268)(157 269)(158 270)(159 271)(160 272)(161 234)(162 235)(163 236)(164 237)(165 238)(166 239)(167 240)(168 233)(169 283)(170 284)(171 285)(172 286)(173 287)(174 288)(175 281)(176 282)(177 305)(178 306)(179 307)(180 308)(181 309)(182 310)(183 311)(184 312)(185 279)(186 280)(187 273)(188 274)(189 275)(190 276)(191 277)(192 278)(201 289)(202 290)(203 291)(204 292)(205 293)(206 294)(207 295)(208 296)(209 260)(210 261)(211 262)(212 263)(213 264)(214 257)(215 258)(216 259)(217 247)(218 248)(219 241)(220 242)(221 243)(222 244)(223 245)(224 246)(225 319)(226 320)(227 313)(228 314)(229 315)(230 316)(231 317)(232 318)
(1 110 297 22)(2 111 298 23)(3 112 299 24)(4 105 300 17)(5 106 301 18)(6 107 302 19)(7 108 303 20)(8 109 304 21)(9 42 140 149)(10 43 141 150)(11 44 142 151)(12 45 143 152)(13 46 144 145)(14 47 137 146)(15 48 138 147)(16 41 139 148)(25 94 99 63)(26 95 100 64)(27 96 101 57)(28 89 102 58)(29 90 103 59)(30 91 104 60)(31 92 97 61)(32 93 98 62)(33 309 121 221)(34 310 122 222)(35 311 123 223)(36 312 124 224)(37 305 125 217)(38 306 126 218)(39 307 127 219)(40 308 128 220)(49 232 156 278)(50 225 157 279)(51 226 158 280)(52 227 159 273)(53 228 160 274)(54 229 153 275)(55 230 154 276)(56 231 155 277)(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)(73 243 81 181)(74 244 82 182)(75 245 83 183)(76 246 84 184)(77 247 85 177)(78 248 86 178)(79 241 87 179)(80 242 88 180)(113 315 265 189)(114 316 266 190)(115 317 267 191)(116 318 268 192)(117 319 269 185)(118 320 270 186)(119 313 271 187)(120 314 272 188)(161 176 289 259)(162 169 290 260)(163 170 291 261)(164 171 292 262)(165 172 293 263)(166 173 294 264)(167 174 295 257)(168 175 296 258)(201 216 234 282)(202 209 235 283)(203 210 236 284)(204 211 237 285)(205 212 238 286)(206 213 239 287)(207 214 240 288)(208 215 233 281)
(1 83 292 63 185)(2 64 84 186 293)(3 187 57 294 85)(4 295 188 86 58)(5 87 296 59 189)(6 60 88 190 289)(7 191 61 290 81)(8 291 192 82 62)(9 131 40 276 201)(10 277 132 202 33)(11 203 278 34 133)(12 35 204 134 279)(13 135 36 280 205)(14 273 136 206 37)(15 207 274 38 129)(16 39 208 130 275)(17 174 272 248 102)(18 241 175 103 265)(19 104 242 266 176)(20 267 97 169 243)(21 170 268 244 98)(22 245 171 99 269)(23 100 246 270 172)(24 271 101 173 247)(25 117 110 183 262)(26 184 118 263 111)(27 264 177 112 119)(28 105 257 120 178)(29 113 106 179 258)(30 180 114 259 107)(31 260 181 108 115)(32 109 261 116 182)(41 307 215 198 54)(42 199 308 55 216)(43 56 200 209 309)(44 210 49 310 193)(45 311 211 194 50)(46 195 312 51 212)(47 52 196 213 305)(48 214 53 306 197)(65 287 217 146 159)(66 147 288 160 218)(67 153 148 219 281)(68 220 154 282 149)(69 283 221 150 155)(70 151 284 156 222)(71 157 152 223 285)(72 224 158 286 145)(73 303 317 92 162)(74 93 304 163 318)(75 164 94 319 297)(76 320 165 298 95)(77 299 313 96 166)(78 89 300 167 314)(79 168 90 315 301)(80 316 161 302 91)(121 141 231 256 235)(122 249 142 236 232)(123 237 250 225 143)(124 226 238 144 251)(125 137 227 252 239)(126 253 138 240 228)(127 233 254 229 139)(128 230 234 140 255)
(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)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,145)(24,146)(25,194)(26,195)(27,196)(28,197)(29,198)(30,199)(31,200)(32,193)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,105)(49,116)(50,117)(51,118)(52,119)(53,120)(54,113)(55,114)(56,115)(57,136)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,101)(66,102)(67,103)(68,104)(69,97)(70,98)(71,99)(72,100)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(89,253)(90,254)(91,255)(92,256)(93,249)(94,250)(95,251)(96,252)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,297)(144,298)(153,265)(154,266)(155,267)(156,268)(157,269)(158,270)(159,271)(160,272)(161,234)(162,235)(163,236)(164,237)(165,238)(166,239)(167,240)(168,233)(169,283)(170,284)(171,285)(172,286)(173,287)(174,288)(175,281)(176,282)(177,305)(178,306)(179,307)(180,308)(181,309)(182,310)(183,311)(184,312)(185,279)(186,280)(187,273)(188,274)(189,275)(190,276)(191,277)(192,278)(201,289)(202,290)(203,291)(204,292)(205,293)(206,294)(207,295)(208,296)(209,260)(210,261)(211,262)(212,263)(213,264)(214,257)(215,258)(216,259)(217,247)(218,248)(219,241)(220,242)(221,243)(222,244)(223,245)(224,246)(225,319)(226,320)(227,313)(228,314)(229,315)(230,316)(231,317)(232,318), (1,110,297,22)(2,111,298,23)(3,112,299,24)(4,105,300,17)(5,106,301,18)(6,107,302,19)(7,108,303,20)(8,109,304,21)(9,42,140,149)(10,43,141,150)(11,44,142,151)(12,45,143,152)(13,46,144,145)(14,47,137,146)(15,48,138,147)(16,41,139,148)(25,94,99,63)(26,95,100,64)(27,96,101,57)(28,89,102,58)(29,90,103,59)(30,91,104,60)(31,92,97,61)(32,93,98,62)(33,309,121,221)(34,310,122,222)(35,311,123,223)(36,312,124,224)(37,305,125,217)(38,306,126,218)(39,307,127,219)(40,308,128,220)(49,232,156,278)(50,225,157,279)(51,226,158,280)(52,227,159,273)(53,228,160,274)(54,229,153,275)(55,230,154,276)(56,231,155,277)(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)(73,243,81,181)(74,244,82,182)(75,245,83,183)(76,246,84,184)(77,247,85,177)(78,248,86,178)(79,241,87,179)(80,242,88,180)(113,315,265,189)(114,316,266,190)(115,317,267,191)(116,318,268,192)(117,319,269,185)(118,320,270,186)(119,313,271,187)(120,314,272,188)(161,176,289,259)(162,169,290,260)(163,170,291,261)(164,171,292,262)(165,172,293,263)(166,173,294,264)(167,174,295,257)(168,175,296,258)(201,216,234,282)(202,209,235,283)(203,210,236,284)(204,211,237,285)(205,212,238,286)(206,213,239,287)(207,214,240,288)(208,215,233,281), (1,83,292,63,185)(2,64,84,186,293)(3,187,57,294,85)(4,295,188,86,58)(5,87,296,59,189)(6,60,88,190,289)(7,191,61,290,81)(8,291,192,82,62)(9,131,40,276,201)(10,277,132,202,33)(11,203,278,34,133)(12,35,204,134,279)(13,135,36,280,205)(14,273,136,206,37)(15,207,274,38,129)(16,39,208,130,275)(17,174,272,248,102)(18,241,175,103,265)(19,104,242,266,176)(20,267,97,169,243)(21,170,268,244,98)(22,245,171,99,269)(23,100,246,270,172)(24,271,101,173,247)(25,117,110,183,262)(26,184,118,263,111)(27,264,177,112,119)(28,105,257,120,178)(29,113,106,179,258)(30,180,114,259,107)(31,260,181,108,115)(32,109,261,116,182)(41,307,215,198,54)(42,199,308,55,216)(43,56,200,209,309)(44,210,49,310,193)(45,311,211,194,50)(46,195,312,51,212)(47,52,196,213,305)(48,214,53,306,197)(65,287,217,146,159)(66,147,288,160,218)(67,153,148,219,281)(68,220,154,282,149)(69,283,221,150,155)(70,151,284,156,222)(71,157,152,223,285)(72,224,158,286,145)(73,303,317,92,162)(74,93,304,163,318)(75,164,94,319,297)(76,320,165,298,95)(77,299,313,96,166)(78,89,300,167,314)(79,168,90,315,301)(80,316,161,302,91)(121,141,231,256,235)(122,249,142,236,232)(123,237,250,225,143)(124,226,238,144,251)(125,137,227,252,239)(126,253,138,240,228)(127,233,254,229,139)(128,230,234,140,255), (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)(17,147)(18,148)(19,149)(20,150)(21,151)(22,152)(23,145)(24,146)(25,194)(26,195)(27,196)(28,197)(29,198)(30,199)(31,200)(32,193)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,106)(42,107)(43,108)(44,109)(45,110)(46,111)(47,112)(48,105)(49,116)(50,117)(51,118)(52,119)(53,120)(54,113)(55,114)(56,115)(57,136)(58,129)(59,130)(60,131)(61,132)(62,133)(63,134)(64,135)(65,101)(66,102)(67,103)(68,104)(69,97)(70,98)(71,99)(72,100)(73,121)(74,122)(75,123)(76,124)(77,125)(78,126)(79,127)(80,128)(89,253)(90,254)(91,255)(92,256)(93,249)(94,250)(95,251)(96,252)(137,299)(138,300)(139,301)(140,302)(141,303)(142,304)(143,297)(144,298)(153,265)(154,266)(155,267)(156,268)(157,269)(158,270)(159,271)(160,272)(161,234)(162,235)(163,236)(164,237)(165,238)(166,239)(167,240)(168,233)(169,283)(170,284)(171,285)(172,286)(173,287)(174,288)(175,281)(176,282)(177,305)(178,306)(179,307)(180,308)(181,309)(182,310)(183,311)(184,312)(185,279)(186,280)(187,273)(188,274)(189,275)(190,276)(191,277)(192,278)(201,289)(202,290)(203,291)(204,292)(205,293)(206,294)(207,295)(208,296)(209,260)(210,261)(211,262)(212,263)(213,264)(214,257)(215,258)(216,259)(217,247)(218,248)(219,241)(220,242)(221,243)(222,244)(223,245)(224,246)(225,319)(226,320)(227,313)(228,314)(229,315)(230,316)(231,317)(232,318), (1,110,297,22)(2,111,298,23)(3,112,299,24)(4,105,300,17)(5,106,301,18)(6,107,302,19)(7,108,303,20)(8,109,304,21)(9,42,140,149)(10,43,141,150)(11,44,142,151)(12,45,143,152)(13,46,144,145)(14,47,137,146)(15,48,138,147)(16,41,139,148)(25,94,99,63)(26,95,100,64)(27,96,101,57)(28,89,102,58)(29,90,103,59)(30,91,104,60)(31,92,97,61)(32,93,98,62)(33,309,121,221)(34,310,122,222)(35,311,123,223)(36,312,124,224)(37,305,125,217)(38,306,126,218)(39,307,127,219)(40,308,128,220)(49,232,156,278)(50,225,157,279)(51,226,158,280)(52,227,159,273)(53,228,160,274)(54,229,153,275)(55,230,154,276)(56,231,155,277)(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)(73,243,81,181)(74,244,82,182)(75,245,83,183)(76,246,84,184)(77,247,85,177)(78,248,86,178)(79,241,87,179)(80,242,88,180)(113,315,265,189)(114,316,266,190)(115,317,267,191)(116,318,268,192)(117,319,269,185)(118,320,270,186)(119,313,271,187)(120,314,272,188)(161,176,289,259)(162,169,290,260)(163,170,291,261)(164,171,292,262)(165,172,293,263)(166,173,294,264)(167,174,295,257)(168,175,296,258)(201,216,234,282)(202,209,235,283)(203,210,236,284)(204,211,237,285)(205,212,238,286)(206,213,239,287)(207,214,240,288)(208,215,233,281), (1,83,292,63,185)(2,64,84,186,293)(3,187,57,294,85)(4,295,188,86,58)(5,87,296,59,189)(6,60,88,190,289)(7,191,61,290,81)(8,291,192,82,62)(9,131,40,276,201)(10,277,132,202,33)(11,203,278,34,133)(12,35,204,134,279)(13,135,36,280,205)(14,273,136,206,37)(15,207,274,38,129)(16,39,208,130,275)(17,174,272,248,102)(18,241,175,103,265)(19,104,242,266,176)(20,267,97,169,243)(21,170,268,244,98)(22,245,171,99,269)(23,100,246,270,172)(24,271,101,173,247)(25,117,110,183,262)(26,184,118,263,111)(27,264,177,112,119)(28,105,257,120,178)(29,113,106,179,258)(30,180,114,259,107)(31,260,181,108,115)(32,109,261,116,182)(41,307,215,198,54)(42,199,308,55,216)(43,56,200,209,309)(44,210,49,310,193)(45,311,211,194,50)(46,195,312,51,212)(47,52,196,213,305)(48,214,53,306,197)(65,287,217,146,159)(66,147,288,160,218)(67,153,148,219,281)(68,220,154,282,149)(69,283,221,150,155)(70,151,284,156,222)(71,157,152,223,285)(72,224,158,286,145)(73,303,317,92,162)(74,93,304,163,318)(75,164,94,319,297)(76,320,165,298,95)(77,299,313,96,166)(78,89,300,167,314)(79,168,90,315,301)(80,316,161,302,91)(121,141,231,256,235)(122,249,142,236,232)(123,237,250,225,143)(124,226,238,144,251)(125,137,227,252,239)(126,253,138,240,228)(127,233,254,229,139)(128,230,234,140,255), (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),(17,147),(18,148),(19,149),(20,150),(21,151),(22,152),(23,145),(24,146),(25,194),(26,195),(27,196),(28,197),(29,198),(30,199),(31,200),(32,193),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,106),(42,107),(43,108),(44,109),(45,110),(46,111),(47,112),(48,105),(49,116),(50,117),(51,118),(52,119),(53,120),(54,113),(55,114),(56,115),(57,136),(58,129),(59,130),(60,131),(61,132),(62,133),(63,134),(64,135),(65,101),(66,102),(67,103),(68,104),(69,97),(70,98),(71,99),(72,100),(73,121),(74,122),(75,123),(76,124),(77,125),(78,126),(79,127),(80,128),(89,253),(90,254),(91,255),(92,256),(93,249),(94,250),(95,251),(96,252),(137,299),(138,300),(139,301),(140,302),(141,303),(142,304),(143,297),(144,298),(153,265),(154,266),(155,267),(156,268),(157,269),(158,270),(159,271),(160,272),(161,234),(162,235),(163,236),(164,237),(165,238),(166,239),(167,240),(168,233),(169,283),(170,284),(171,285),(172,286),(173,287),(174,288),(175,281),(176,282),(177,305),(178,306),(179,307),(180,308),(181,309),(182,310),(183,311),(184,312),(185,279),(186,280),(187,273),(188,274),(189,275),(190,276),(191,277),(192,278),(201,289),(202,290),(203,291),(204,292),(205,293),(206,294),(207,295),(208,296),(209,260),(210,261),(211,262),(212,263),(213,264),(214,257),(215,258),(216,259),(217,247),(218,248),(219,241),(220,242),(221,243),(222,244),(223,245),(224,246),(225,319),(226,320),(227,313),(228,314),(229,315),(230,316),(231,317),(232,318)], [(1,110,297,22),(2,111,298,23),(3,112,299,24),(4,105,300,17),(5,106,301,18),(6,107,302,19),(7,108,303,20),(8,109,304,21),(9,42,140,149),(10,43,141,150),(11,44,142,151),(12,45,143,152),(13,46,144,145),(14,47,137,146),(15,48,138,147),(16,41,139,148),(25,94,99,63),(26,95,100,64),(27,96,101,57),(28,89,102,58),(29,90,103,59),(30,91,104,60),(31,92,97,61),(32,93,98,62),(33,309,121,221),(34,310,122,222),(35,311,123,223),(36,312,124,224),(37,305,125,217),(38,306,126,218),(39,307,127,219),(40,308,128,220),(49,232,156,278),(50,225,157,279),(51,226,158,280),(52,227,159,273),(53,228,160,274),(54,229,153,275),(55,230,154,276),(56,231,155,277),(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),(73,243,81,181),(74,244,82,182),(75,245,83,183),(76,246,84,184),(77,247,85,177),(78,248,86,178),(79,241,87,179),(80,242,88,180),(113,315,265,189),(114,316,266,190),(115,317,267,191),(116,318,268,192),(117,319,269,185),(118,320,270,186),(119,313,271,187),(120,314,272,188),(161,176,289,259),(162,169,290,260),(163,170,291,261),(164,171,292,262),(165,172,293,263),(166,173,294,264),(167,174,295,257),(168,175,296,258),(201,216,234,282),(202,209,235,283),(203,210,236,284),(204,211,237,285),(205,212,238,286),(206,213,239,287),(207,214,240,288),(208,215,233,281)], [(1,83,292,63,185),(2,64,84,186,293),(3,187,57,294,85),(4,295,188,86,58),(5,87,296,59,189),(6,60,88,190,289),(7,191,61,290,81),(8,291,192,82,62),(9,131,40,276,201),(10,277,132,202,33),(11,203,278,34,133),(12,35,204,134,279),(13,135,36,280,205),(14,273,136,206,37),(15,207,274,38,129),(16,39,208,130,275),(17,174,272,248,102),(18,241,175,103,265),(19,104,242,266,176),(20,267,97,169,243),(21,170,268,244,98),(22,245,171,99,269),(23,100,246,270,172),(24,271,101,173,247),(25,117,110,183,262),(26,184,118,263,111),(27,264,177,112,119),(28,105,257,120,178),(29,113,106,179,258),(30,180,114,259,107),(31,260,181,108,115),(32,109,261,116,182),(41,307,215,198,54),(42,199,308,55,216),(43,56,200,209,309),(44,210,49,310,193),(45,311,211,194,50),(46,195,312,51,212),(47,52,196,213,305),(48,214,53,306,197),(65,287,217,146,159),(66,147,288,160,218),(67,153,148,219,281),(68,220,154,282,149),(69,283,221,150,155),(70,151,284,156,222),(71,157,152,223,285),(72,224,158,286,145),(73,303,317,92,162),(74,93,304,163,318),(75,164,94,319,297),(76,320,165,298,95),(77,299,313,96,166),(78,89,300,167,314),(79,168,90,315,301),(80,316,161,302,91),(121,141,231,256,235),(122,249,142,236,232),(123,237,250,225,143),(124,226,238,144,251),(125,137,227,252,239),(126,253,138,240,228),(127,233,254,229,139),(128,230,234,140,255)], [(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

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