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G = C2×C4×C5⋊C8  order 320 = 26·5

Direct product of C2×C4 and C5⋊C8

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C2×C4×C5⋊C8, Dic5.12C42, C10⋊2(C4×C8), C20⋊9(C2×C8), (C2×C20)⋊5C8, (C2×Dic5)⋊7C8, Dic5⋊8(C2×C8), C23.57(C2×F5), (C22×C4).27F5, C22.20(C4×F5), (C22×C20).32C4, C10.18(C22×C8), (C2×C10).19C42, C10.16(C2×C42), (C4×Dic5).46C4, C22.14(D5⋊C8), C22.40(C22×F5), (C22×Dic5).26C4, Dic5.35(C22×C4), (C4×Dic5).355C22, (C2×Dic5).337C23, (C22×Dic5).265C22, C5⋊3(C2×C4×C8), C2.4(C2×C4×F5), C2.3(C2×D5⋊C8), C2.2(C22×C5⋊C8), (C22×C5⋊C8).7C2, C22.13(C2×C5⋊C8), (C2×C10).33(C2×C8), (C2×C5⋊C8).44C22, (C2×C4).169(C2×F5), (C2×C4×Dic5).50C2, (C2×C20).177(C2×C4), (C2×C10).53(C22×C4), (C22×C10).53(C2×C4), (C2×Dic5).177(C2×C4), SmallGroup(320,1084)

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

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

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

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

(1 105 297 22)(2 106 298 23)(3 107 299 24)(4 108 300 17)(5 109 301 18)(6 110 302 19)(7 111 303 20)(8 112 304 21)(9 70 138 193)(10 71 139 194)(11 72 140 195)(12 65 141 196)(13 66 142 197)(14 67 143 198)(15 68 144 199)(16 69 137 200)(25 99 94 50)(26 100 95 51)(27 101 96 52)(28 102 89 53)(29 103 90 54)(30 104 91 55)(31 97 92 56)(32 98 93 49)(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)(41 217 149 233)(42 218 150 234)(43 219 151 235)(44 220 152 236)(45 221 145 237)(46 222 146 238)(47 223 147 239)(48 224 148 240)(57 227 159 273)(58 228 160 274)(59 229 153 275)(60 230 154 276)(61 231 155 277)(62 232 156 278)(63 225 157 279)(64 226 158 280)(73 246 87 174)(74 247 88 175)(75 248 81 176)(76 241 82 169)(77 242 83 170)(78 243 84 171)(79 244 85 172)(80 245 86 173)(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)(129 204 251 284)(130 205 252 285)(131 206 253 286)(132 207 254 287)(133 208 255 288)(134 201 256 281)(135 202 249 282)(136 203 250 283)(161 289 259 178)(162 290 260 179)(163 291 261 180)(164 292 262 181)(165 293 263 182)(166 294 264 183)(167 295 257 184)(168 296 258 177)

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

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

Matrix representation ►G ⊆ GL7(𝔽41)

1000000
04000000
00400000
0001000
0000100
0000010
0000001
,
1000000
0100000
00320000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
00000040
00010040
00001040
00000140
,
27000000
0100000
0010000
00021402821
00082001
0002104029
0002028201

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] >;
 

80 conjugacy classes

class 1 2A···2G4A···4H4I···4X 5 8A···8AF10A···10G20A···20H
order12···24···44···458···810···1020···20
size11···11···15···545···54···44···4

80 irreducible representations

dim1111111111444444
type+++++-++
imageC1C2C2C2C4C4C4C4C8C8F5C5⋊C8C2×F5C2×F5D5⋊C8C4×F5
kernelC2×C4×C5⋊C8C4×C5⋊C8C2×C4×Dic5C22×C5⋊C8C4×Dic5C2×C5⋊C8C22×Dic5C22×C20C2×Dic5C2×C20C22×C4C2×C4C2×C4C23C22C22
# reps1412416221616142144

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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