direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Q8×C5⋊C8, C5⋊2(C8×Q8), (C5×Q8)⋊2C8, C2.3(Q8×F5), C20.9(C2×C8), (C2×Q8).9F5, C10.7(C4×Q8), (Q8×C10).5C4, C20⋊C8.5C2, C4⋊Dic5.15C4, C2.3(Q8.F5), C10.22(C8○D4), C10.22(C22×C8), Dic5.31(C2×Q8), (Q8×Dic5).15C2, Dic5.70(C4○D4), C22.52(C22×F5), (C4×Dic5).196C22, (C2×Dic5).356C23, C4.4(C2×C5⋊C8), (C4×C5⋊C8).4C2, C2.7(C22×C5⋊C8), (C2×C4).86(C2×F5), (C2×C20).64(C2×C4), (C2×C5⋊C8).41C22, (C2×C10).85(C22×C4), (C2×Dic5).76(C2×C4), SmallGroup(320,1124)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8×C5⋊C8
G = < a,b,c,d | a4=c5=d8=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >
Subgroups: 266 in 102 conjugacy classes, 64 normal (18 characteristic)
C1, C2, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×C8, C2×Q8, Dic5, Dic5, C20, C2×C10, C4×C8, C4⋊C8, C4×Q8, C5⋊C8, C5⋊C8, C2×Dic5, C2×Dic5, C2×C20, C5×Q8, C8×Q8, C4×Dic5, C4⋊Dic5, C2×C5⋊C8, C2×C5⋊C8, Q8×C10, C4×C5⋊C8, C20⋊C8, Q8×Dic5, Q8×C5⋊C8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, C2×C8, C22×C4, C2×Q8, C4○D4, F5, C4×Q8, C22×C8, C8○D4, C5⋊C8, C2×F5, C8×Q8, C2×C5⋊C8, C22×F5, Q8.F5, Q8×F5, C22×C5⋊C8, Q8×C5⋊C8
►Regular action on 320 points
Generators in S320
(1 281 257 44)(2 282 258 45)(3 283 259 46)(4 284 260 47)(5 285 261 48)(6 286 262 41)(7 287 263 42)(8 288 264 43)(9 312 120 206)(10 305 113 207)(11 306 114 208)(12 307 115 201)(13 308 116 202)(14 309 117 203)(15 310 118 204)(16 311 119 205)(17 218 242 156)(18 219 243 157)(19 220 244 158)(20 221 245 159)(21 222 246 160)(22 223 247 153)(23 224 248 154)(24 217 241 155)(25 265 109 294)(26 266 110 295)(27 267 111 296)(28 268 112 289)(29 269 105 290)(30 270 106 291)(31 271 107 292)(32 272 108 293)(33 121 92 65)(34 122 93 66)(35 123 94 67)(36 124 95 68)(37 125 96 69)(38 126 89 70)(39 127 90 71)(40 128 91 72)(49 86 299 168)(50 87 300 161)(51 88 301 162)(52 81 302 163)(53 82 303 164)(54 83 304 165)(55 84 297 166)(56 85 298 167)(57 197 143 275)(58 198 144 276)(59 199 137 277)(60 200 138 278)(61 193 139 279)(62 194 140 280)(63 195 141 273)(64 196 142 274)(73 225 151 254)(74 226 152 255)(75 227 145 256)(76 228 146 249)(77 229 147 250)(78 230 148 251)(79 231 149 252)(80 232 150 253)(97 315 189 237)(98 316 190 238)(99 317 191 239)(100 318 192 240)(101 319 185 233)(102 320 186 234)(103 313 187 235)(104 314 188 236)(129 183 214 172)(130 184 215 173)(131 177 216 174)(132 178 209 175)(133 179 210 176)(134 180 211 169)(135 181 212 170)(136 182 213 171)
(1 297 257 55)(2 298 258 56)(3 299 259 49)(4 300 260 50)(5 301 261 51)(6 302 262 52)(7 303 263 53)(8 304 264 54)(9 160 120 222)(10 153 113 223)(11 154 114 224)(12 155 115 217)(13 156 116 218)(14 157 117 219)(15 158 118 220)(16 159 119 221)(17 308 242 202)(18 309 243 203)(19 310 244 204)(20 311 245 205)(21 312 246 206)(22 305 247 207)(23 306 248 208)(24 307 241 201)(25 151 109 73)(26 152 110 74)(27 145 111 75)(28 146 112 76)(29 147 105 77)(30 148 106 78)(31 149 107 79)(32 150 108 80)(33 132 92 209)(34 133 93 210)(35 134 94 211)(36 135 95 212)(37 136 96 213)(38 129 89 214)(39 130 90 215)(40 131 91 216)(41 163 286 81)(42 164 287 82)(43 165 288 83)(44 166 281 84)(45 167 282 85)(46 168 283 86)(47 161 284 87)(48 162 285 88)(57 189 143 97)(58 190 144 98)(59 191 137 99)(60 192 138 100)(61 185 139 101)(62 186 140 102)(63 187 141 103)(64 188 142 104)(65 178 121 175)(66 179 122 176)(67 180 123 169)(68 181 124 170)(69 182 125 171)(70 183 126 172)(71 184 127 173)(72 177 128 174)(193 319 279 233)(194 320 280 234)(195 313 273 235)(196 314 274 236)(197 315 275 237)(198 316 276 238)(199 317 277 239)(200 318 278 240)(225 294 254 265)(226 295 255 266)(227 296 256 267)(228 289 249 268)(229 290 250 269)(230 291 251 270)(231 292 252 271)(232 293 253 272)
(1 12 92 250 193)(2 251 13 194 93)(3 195 252 94 14)(4 95 196 15 253)(5 16 96 254 197)(6 255 9 198 89)(7 199 256 90 10)(8 91 200 11 249)(17 186 176 167 106)(18 168 187 107 169)(19 108 161 170 188)(20 171 109 189 162)(21 190 172 163 110)(22 164 191 111 173)(23 112 165 174 192)(24 175 105 185 166)(25 97 88 245 182)(26 246 98 183 81)(27 184 247 82 99)(28 83 177 100 248)(29 101 84 241 178)(30 242 102 179 85)(31 180 243 86 103)(32 87 181 104 244)(33 229 279 257 115)(34 258 230 116 280)(35 117 259 273 231)(36 274 118 232 260)(37 225 275 261 119)(38 262 226 120 276)(39 113 263 277 227)(40 278 114 228 264)(41 152 206 58 126)(42 59 145 127 207)(43 128 60 208 146)(44 201 121 147 61)(45 148 202 62 122)(46 63 149 123 203)(47 124 64 204 150)(48 205 125 151 57)(49 235 292 134 219)(50 135 236 220 293)(51 221 136 294 237)(52 295 222 238 129)(53 239 296 130 223)(54 131 240 224 289)(55 217 132 290 233)(56 291 218 234 133)(65 77 139 281 307)(66 282 78 308 140)(67 309 283 141 79)(68 142 310 80 284)(69 73 143 285 311)(70 286 74 312 144)(71 305 287 137 75)(72 138 306 76 288)(153 303 317 267 215)(154 268 304 216 318)(155 209 269 319 297)(156 320 210 298 270)(157 299 313 271 211)(158 272 300 212 314)(159 213 265 315 301)(160 316 214 302 266)
(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)| (1,281,257,44)(2,282,258,45)(3,283,259,46)(4,284,260,47)(5,285,261,48)(6,286,262,41)(7,287,263,42)(8,288,264,43)(9,312,120,206)(10,305,113,207)(11,306,114,208)(12,307,115,201)(13,308,116,202)(14,309,117,203)(15,310,118,204)(16,311,119,205)(17,218,242,156)(18,219,243,157)(19,220,244,158)(20,221,245,159)(21,222,246,160)(22,223,247,153)(23,224,248,154)(24,217,241,155)(25,265,109,294)(26,266,110,295)(27,267,111,296)(28,268,112,289)(29,269,105,290)(30,270,106,291)(31,271,107,292)(32,272,108,293)(33,121,92,65)(34,122,93,66)(35,123,94,67)(36,124,95,68)(37,125,96,69)(38,126,89,70)(39,127,90,71)(40,128,91,72)(49,86,299,168)(50,87,300,161)(51,88,301,162)(52,81,302,163)(53,82,303,164)(54,83,304,165)(55,84,297,166)(56,85,298,167)(57,197,143,275)(58,198,144,276)(59,199,137,277)(60,200,138,278)(61,193,139,279)(62,194,140,280)(63,195,141,273)(64,196,142,274)(73,225,151,254)(74,226,152,255)(75,227,145,256)(76,228,146,249)(77,229,147,250)(78,230,148,251)(79,231,149,252)(80,232,150,253)(97,315,189,237)(98,316,190,238)(99,317,191,239)(100,318,192,240)(101,319,185,233)(102,320,186,234)(103,313,187,235)(104,314,188,236)(129,183,214,172)(130,184,215,173)(131,177,216,174)(132,178,209,175)(133,179,210,176)(134,180,211,169)(135,181,212,170)(136,182,213,171), (1,297,257,55)(2,298,258,56)(3,299,259,49)(4,300,260,50)(5,301,261,51)(6,302,262,52)(7,303,263,53)(8,304,264,54)(9,160,120,222)(10,153,113,223)(11,154,114,224)(12,155,115,217)(13,156,116,218)(14,157,117,219)(15,158,118,220)(16,159,119,221)(17,308,242,202)(18,309,243,203)(19,310,244,204)(20,311,245,205)(21,312,246,206)(22,305,247,207)(23,306,248,208)(24,307,241,201)(25,151,109,73)(26,152,110,74)(27,145,111,75)(28,146,112,76)(29,147,105,77)(30,148,106,78)(31,149,107,79)(32,150,108,80)(33,132,92,209)(34,133,93,210)(35,134,94,211)(36,135,95,212)(37,136,96,213)(38,129,89,214)(39,130,90,215)(40,131,91,216)(41,163,286,81)(42,164,287,82)(43,165,288,83)(44,166,281,84)(45,167,282,85)(46,168,283,86)(47,161,284,87)(48,162,285,88)(57,189,143,97)(58,190,144,98)(59,191,137,99)(60,192,138,100)(61,185,139,101)(62,186,140,102)(63,187,141,103)(64,188,142,104)(65,178,121,175)(66,179,122,176)(67,180,123,169)(68,181,124,170)(69,182,125,171)(70,183,126,172)(71,184,127,173)(72,177,128,174)(193,319,279,233)(194,320,280,234)(195,313,273,235)(196,314,274,236)(197,315,275,237)(198,316,276,238)(199,317,277,239)(200,318,278,240)(225,294,254,265)(226,295,255,266)(227,296,256,267)(228,289,249,268)(229,290,250,269)(230,291,251,270)(231,292,252,271)(232,293,253,272), (1,12,92,250,193)(2,251,13,194,93)(3,195,252,94,14)(4,95,196,15,253)(5,16,96,254,197)(6,255,9,198,89)(7,199,256,90,10)(8,91,200,11,249)(17,186,176,167,106)(18,168,187,107,169)(19,108,161,170,188)(20,171,109,189,162)(21,190,172,163,110)(22,164,191,111,173)(23,112,165,174,192)(24,175,105,185,166)(25,97,88,245,182)(26,246,98,183,81)(27,184,247,82,99)(28,83,177,100,248)(29,101,84,241,178)(30,242,102,179,85)(31,180,243,86,103)(32,87,181,104,244)(33,229,279,257,115)(34,258,230,116,280)(35,117,259,273,231)(36,274,118,232,260)(37,225,275,261,119)(38,262,226,120,276)(39,113,263,277,227)(40,278,114,228,264)(41,152,206,58,126)(42,59,145,127,207)(43,128,60,208,146)(44,201,121,147,61)(45,148,202,62,122)(46,63,149,123,203)(47,124,64,204,150)(48,205,125,151,57)(49,235,292,134,219)(50,135,236,220,293)(51,221,136,294,237)(52,295,222,238,129)(53,239,296,130,223)(54,131,240,224,289)(55,217,132,290,233)(56,291,218,234,133)(65,77,139,281,307)(66,282,78,308,140)(67,309,283,141,79)(68,142,310,80,284)(69,73,143,285,311)(70,286,74,312,144)(71,305,287,137,75)(72,138,306,76,288)(153,303,317,267,215)(154,268,304,216,318)(155,209,269,319,297)(156,320,210,298,270)(157,299,313,271,211)(158,272,300,212,314)(159,213,265,315,301)(160,316,214,302,266), (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( (1,281,257,44)(2,282,258,45)(3,283,259,46)(4,284,260,47)(5,285,261,48)(6,286,262,41)(7,287,263,42)(8,288,264,43)(9,312,120,206)(10,305,113,207)(11,306,114,208)(12,307,115,201)(13,308,116,202)(14,309,117,203)(15,310,118,204)(16,311,119,205)(17,218,242,156)(18,219,243,157)(19,220,244,158)(20,221,245,159)(21,222,246,160)(22,223,247,153)(23,224,248,154)(24,217,241,155)(25,265,109,294)(26,266,110,295)(27,267,111,296)(28,268,112,289)(29,269,105,290)(30,270,106,291)(31,271,107,292)(32,272,108,293)(33,121,92,65)(34,122,93,66)(35,123,94,67)(36,124,95,68)(37,125,96,69)(38,126,89,70)(39,127,90,71)(40,128,91,72)(49,86,299,168)(50,87,300,161)(51,88,301,162)(52,81,302,163)(53,82,303,164)(54,83,304,165)(55,84,297,166)(56,85,298,167)(57,197,143,275)(58,198,144,276)(59,199,137,277)(60,200,138,278)(61,193,139,279)(62,194,140,280)(63,195,141,273)(64,196,142,274)(73,225,151,254)(74,226,152,255)(75,227,145,256)(76,228,146,249)(77,229,147,250)(78,230,148,251)(79,231,149,252)(80,232,150,253)(97,315,189,237)(98,316,190,238)(99,317,191,239)(100,318,192,240)(101,319,185,233)(102,320,186,234)(103,313,187,235)(104,314,188,236)(129,183,214,172)(130,184,215,173)(131,177,216,174)(132,178,209,175)(133,179,210,176)(134,180,211,169)(135,181,212,170)(136,182,213,171), (1,297,257,55)(2,298,258,56)(3,299,259,49)(4,300,260,50)(5,301,261,51)(6,302,262,52)(7,303,263,53)(8,304,264,54)(9,160,120,222)(10,153,113,223)(11,154,114,224)(12,155,115,217)(13,156,116,218)(14,157,117,219)(15,158,118,220)(16,159,119,221)(17,308,242,202)(18,309,243,203)(19,310,244,204)(20,311,245,205)(21,312,246,206)(22,305,247,207)(23,306,248,208)(24,307,241,201)(25,151,109,73)(26,152,110,74)(27,145,111,75)(28,146,112,76)(29,147,105,77)(30,148,106,78)(31,149,107,79)(32,150,108,80)(33,132,92,209)(34,133,93,210)(35,134,94,211)(36,135,95,212)(37,136,96,213)(38,129,89,214)(39,130,90,215)(40,131,91,216)(41,163,286,81)(42,164,287,82)(43,165,288,83)(44,166,281,84)(45,167,282,85)(46,168,283,86)(47,161,284,87)(48,162,285,88)(57,189,143,97)(58,190,144,98)(59,191,137,99)(60,192,138,100)(61,185,139,101)(62,186,140,102)(63,187,141,103)(64,188,142,104)(65,178,121,175)(66,179,122,176)(67,180,123,169)(68,181,124,170)(69,182,125,171)(70,183,126,172)(71,184,127,173)(72,177,128,174)(193,319,279,233)(194,320,280,234)(195,313,273,235)(196,314,274,236)(197,315,275,237)(198,316,276,238)(199,317,277,239)(200,318,278,240)(225,294,254,265)(226,295,255,266)(227,296,256,267)(228,289,249,268)(229,290,250,269)(230,291,251,270)(231,292,252,271)(232,293,253,272), (1,12,92,250,193)(2,251,13,194,93)(3,195,252,94,14)(4,95,196,15,253)(5,16,96,254,197)(6,255,9,198,89)(7,199,256,90,10)(8,91,200,11,249)(17,186,176,167,106)(18,168,187,107,169)(19,108,161,170,188)(20,171,109,189,162)(21,190,172,163,110)(22,164,191,111,173)(23,112,165,174,192)(24,175,105,185,166)(25,97,88,245,182)(26,246,98,183,81)(27,184,247,82,99)(28,83,177,100,248)(29,101,84,241,178)(30,242,102,179,85)(31,180,243,86,103)(32,87,181,104,244)(33,229,279,257,115)(34,258,230,116,280)(35,117,259,273,231)(36,274,118,232,260)(37,225,275,261,119)(38,262,226,120,276)(39,113,263,277,227)(40,278,114,228,264)(41,152,206,58,126)(42,59,145,127,207)(43,128,60,208,146)(44,201,121,147,61)(45,148,202,62,122)(46,63,149,123,203)(47,124,64,204,150)(48,205,125,151,57)(49,235,292,134,219)(50,135,236,220,293)(51,221,136,294,237)(52,295,222,238,129)(53,239,296,130,223)(54,131,240,224,289)(55,217,132,290,233)(56,291,218,234,133)(65,77,139,281,307)(66,282,78,308,140)(67,309,283,141,79)(68,142,310,80,284)(69,73,143,285,311)(70,286,74,312,144)(71,305,287,137,75)(72,138,306,76,288)(153,303,317,267,215)(154,268,304,216,318)(155,209,269,319,297)(156,320,210,298,270)(157,299,313,271,211)(158,272,300,212,314)(159,213,265,315,301)(160,316,214,302,266), (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([[(1,281,257,44),(2,282,258,45),(3,283,259,46),(4,284,260,47),(5,285,261,48),(6,286,262,41),(7,287,263,42),(8,288,264,43),(9,312,120,206),(10,305,113,207),(11,306,114,208),(12,307,115,201),(13,308,116,202),(14,309,117,203),(15,310,118,204),(16,311,119,205),(17,218,242,156),(18,219,243,157),(19,220,244,158),(20,221,245,159),(21,222,246,160),(22,223,247,153),(23,224,248,154),(24,217,241,155),(25,265,109,294),(26,266,110,295),(27,267,111,296),(28,268,112,289),(29,269,105,290),(30,270,106,291),(31,271,107,292),(32,272,108,293),(33,121,92,65),(34,122,93,66),(35,123,94,67),(36,124,95,68),(37,125,96,69),(38,126,89,70),(39,127,90,71),(40,128,91,72),(49,86,299,168),(50,87,300,161),(51,88,301,162),(52,81,302,163),(53,82,303,164),(54,83,304,165),(55,84,297,166),(56,85,298,167),(57,197,143,275),(58,198,144,276),(59,199,137,277),(60,200,138,278),(61,193,139,279),(62,194,140,280),(63,195,141,273),(64,196,142,274),(73,225,151,254),(74,226,152,255),(75,227,145,256),(76,228,146,249),(77,229,147,250),(78,230,148,251),(79,231,149,252),(80,232,150,253),(97,315,189,237),(98,316,190,238),(99,317,191,239),(100,318,192,240),(101,319,185,233),(102,320,186,234),(103,313,187,235),(104,314,188,236),(129,183,214,172),(130,184,215,173),(131,177,216,174),(132,178,209,175),(133,179,210,176),(134,180,211,169),(135,181,212,170),(136,182,213,171)], [(1,297,257,55),(2,298,258,56),(3,299,259,49),(4,300,260,50),(5,301,261,51),(6,302,262,52),(7,303,263,53),(8,304,264,54),(9,160,120,222),(10,153,113,223),(11,154,114,224),(12,155,115,217),(13,156,116,218),(14,157,117,219),(15,158,118,220),(16,159,119,221),(17,308,242,202),(18,309,243,203),(19,310,244,204),(20,311,245,205),(21,312,246,206),(22,305,247,207),(23,306,248,208),(24,307,241,201),(25,151,109,73),(26,152,110,74),(27,145,111,75),(28,146,112,76),(29,147,105,77),(30,148,106,78),(31,149,107,79),(32,150,108,80),(33,132,92,209),(34,133,93,210),(35,134,94,211),(36,135,95,212),(37,136,96,213),(38,129,89,214),(39,130,90,215),(40,131,91,216),(41,163,286,81),(42,164,287,82),(43,165,288,83),(44,166,281,84),(45,167,282,85),(46,168,283,86),(47,161,284,87),(48,162,285,88),(57,189,143,97),(58,190,144,98),(59,191,137,99),(60,192,138,100),(61,185,139,101),(62,186,140,102),(63,187,141,103),(64,188,142,104),(65,178,121,175),(66,179,122,176),(67,180,123,169),(68,181,124,170),(69,182,125,171),(70,183,126,172),(71,184,127,173),(72,177,128,174),(193,319,279,233),(194,320,280,234),(195,313,273,235),(196,314,274,236),(197,315,275,237),(198,316,276,238),(199,317,277,239),(200,318,278,240),(225,294,254,265),(226,295,255,266),(227,296,256,267),(228,289,249,268),(229,290,250,269),(230,291,251,270),(231,292,252,271),(232,293,253,272)], [(1,12,92,250,193),(2,251,13,194,93),(3,195,252,94,14),(4,95,196,15,253),(5,16,96,254,197),(6,255,9,198,89),(7,199,256,90,10),(8,91,200,11,249),(17,186,176,167,106),(18,168,187,107,169),(19,108,161,170,188),(20,171,109,189,162),(21,190,172,163,110),(22,164,191,111,173),(23,112,165,174,192),(24,175,105,185,166),(25,97,88,245,182),(26,246,98,183,81),(27,184,247,82,99),(28,83,177,100,248),(29,101,84,241,178),(30,242,102,179,85),(31,180,243,86,103),(32,87,181,104,244),(33,229,279,257,115),(34,258,230,116,280),(35,117,259,273,231),(36,274,118,232,260),(37,225,275,261,119),(38,262,226,120,276),(39,113,263,277,227),(40,278,114,228,264),(41,152,206,58,126),(42,59,145,127,207),(43,128,60,208,146),(44,201,121,147,61),(45,148,202,62,122),(46,63,149,123,203),(47,124,64,204,150),(48,205,125,151,57),(49,235,292,134,219),(50,135,236,220,293),(51,221,136,294,237),(52,295,222,238,129),(53,239,296,130,223),(54,131,240,224,289),(55,217,132,290,233),(56,291,218,234,133),(65,77,139,281,307),(66,282,78,308,140),(67,309,283,141,79),(68,142,310,80,284),(69,73,143,285,311),(70,286,74,312,144),(71,305,287,137,75),(72,138,306,76,288),(153,303,317,267,215),(154,268,304,216,318),(155,209,269,319,297),(156,320,210,298,270),(157,299,313,271,211),(158,272,300,212,314),(159,213,265,315,301),(160,316,214,302,266)], [(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)]])
50 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | ··· | 4P | 5 | 8A | ··· | 8H | 8I | ··· | 8T | 10A | 10B | 10C | 20A | ··· | 20F |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 8 | ··· | 8 | 8 | ··· | 8 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 4 | 5 | ··· | 5 | 10 | ··· | 10 | 4 | 4 | 4 | 8 | ··· | 8 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 |
| type | + | + | + | + | - | + | + | - | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | Q8 | C4○D4 | C8○D4 | F5 | C2×F5 | C5⋊C8 | Q8.F5 | Q8×F5 |
| kernel | Q8×C5⋊C8 | C4×C5⋊C8 | C20⋊C8 | Q8×Dic5 | C4⋊Dic5 | Q8×C10 | C5×Q8 | C5⋊C8 | Dic5 | C10 | C2×Q8 | C2×C4 | Q8 | C2 | C2 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 16 | 2 | 2 | 4 | 1 | 3 | 4 | 1 | 1 |
Matrix representation of Q8×C5⋊C8 ►in GL8(𝔽41)
| 1 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 40 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 26 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 32 | 23 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 40 | 40 | 40 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 38 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 9 | 16 | 31 |
| 0 | 0 | 0 | 0 | 7 | 22 | 32 | 10 |
| 0 | 0 | 0 | 0 | 10 | 29 | 19 | 26 |
| 0 | 0 | 0 | 0 | 31 | 38 | 12 | 22 |
G:=sub<GL(8,GF(41))| [1,40,0,0,0,0,0,0,2,40,0,0,0,0,0,0,0,0,26,4,0,0,0,0,0,0,5,15,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,0,0,0,0,0,0,0,23,9,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,19,7,10,31,0,0,0,0,9,22,29,38,0,0,0,0,16,32,19,12,0,0,0,0,31,10,26,22] >;
Q8×C5⋊C8 in GAP, Magma, Sage, TeX
Q_8\times C_5\rtimes C_8
% in TeX
G:=Group("Q8xC5:C8"); // GroupNames label
G:=SmallGroup(320,1124);
// by ID
G=gap.SmallGroup(320,1124);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,219,100,136,6278,1595]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^5=d^8=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations