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## G = Q8×C5⋊2C8order 320 = 26·5

### Direct product of Q8 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — Q8×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C4×C5⋊2C8 — Q8×C5⋊2C8
 Lower central C5 — C10 — Q8×C5⋊2C8
 Upper central C1 — C2×C4 — C4×Q8

Generators and relations for Q8×C52C8
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=c-1 >

Subgroups: 190 in 102 conjugacy classes, 77 normal (22 characteristic)
C1, C2, C4, C4, C4, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C42, C4⋊C4, C2×C8, C2×Q8, C20, C20, C20, C2×C10, C4×C8, C4⋊C8, C4×Q8, C52C8, C52C8, C2×C20, C2×C20, C5×Q8, C8×Q8, C2×C52C8, C2×C52C8, C4×C20, C5×C4⋊C4, Q8×C10, C4×C52C8, C203C8, Q8×C20, Q8×C52C8
Quotients: C1, C2, C4, C22, C8, C2×C4, Q8, C23, D5, C2×C8, C22×C4, C2×Q8, C4○D4, Dic5, D10, C4×Q8, C22×C8, C8○D4, C52C8, C2×Dic5, C22×D5, C8×Q8, C2×C52C8, Q8×D5, Q82D5, C22×Dic5, C22×C52C8, Q8×Dic5, D4.Dic5, Q8×C52C8

Smallest permutation representation of Q8×C52C8
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)
(1 301 257 51)(2 302 258 52)(3 303 259 53)(4 304 260 54)(5 297 261 55)(6 298 262 56)(7 299 263 49)(8 300 264 50)(9 156 120 218)(10 157 113 219)(11 158 114 220)(12 159 115 221)(13 160 116 222)(14 153 117 223)(15 154 118 224)(16 155 119 217)(17 312 242 206)(18 305 243 207)(19 306 244 208)(20 307 245 201)(21 308 246 202)(22 309 247 203)(23 310 248 204)(24 311 241 205)(25 147 109 77)(26 148 110 78)(27 149 111 79)(28 150 112 80)(29 151 105 73)(30 152 106 74)(31 145 107 75)(32 146 108 76)(33 136 92 213)(34 129 93 214)(35 130 94 215)(36 131 95 216)(37 132 96 209)(38 133 89 210)(39 134 90 211)(40 135 91 212)(41 167 286 85)(42 168 287 86)(43 161 288 87)(44 162 281 88)(45 163 282 81)(46 164 283 82)(47 165 284 83)(48 166 285 84)(57 185 143 101)(58 186 144 102)(59 187 137 103)(60 188 138 104)(61 189 139 97)(62 190 140 98)(63 191 141 99)(64 192 142 100)(65 182 121 171)(66 183 122 172)(67 184 123 173)(68 177 124 174)(69 178 125 175)(70 179 126 176)(71 180 127 169)(72 181 128 170)(193 315 279 237)(194 316 280 238)(195 317 273 239)(196 318 274 240)(197 319 275 233)(198 320 276 234)(199 313 277 235)(200 314 278 236)(225 290 254 269)(226 291 255 270)(227 292 256 271)(228 293 249 272)(229 294 250 265)(230 295 251 266)(231 296 252 267)(232 289 253 268)
(1 12 92 250 193)(2 194 251 93 13)(3 14 94 252 195)(4 196 253 95 15)(5 16 96 254 197)(6 198 255 89 9)(7 10 90 256 199)(8 200 249 91 11)(17 167 186 106 176)(18 169 107 187 168)(19 161 188 108 170)(20 171 109 189 162)(21 163 190 110 172)(22 173 111 191 164)(23 165 192 112 174)(24 175 105 185 166)(25 97 88 245 182)(26 183 246 81 98)(27 99 82 247 184)(28 177 248 83 100)(29 101 84 241 178)(30 179 242 85 102)(31 103 86 243 180)(32 181 244 87 104)(33 229 279 257 115)(34 116 258 280 230)(35 231 273 259 117)(36 118 260 274 232)(37 225 275 261 119)(38 120 262 276 226)(39 227 277 263 113)(40 114 264 278 228)(41 58 152 126 206)(42 207 127 145 59)(43 60 146 128 208)(44 201 121 147 61)(45 62 148 122 202)(46 203 123 149 63)(47 64 150 124 204)(48 205 125 151 57)(49 219 134 292 235)(50 236 293 135 220)(51 221 136 294 237)(52 238 295 129 222)(53 223 130 296 239)(54 240 289 131 224)(55 217 132 290 233)(56 234 291 133 218)(65 77 139 281 307)(66 308 282 140 78)(67 79 141 283 309)(68 310 284 142 80)(69 73 143 285 311)(70 312 286 144 74)(71 75 137 287 305)(72 306 288 138 76)(153 215 267 317 303)(154 304 318 268 216)(155 209 269 319 297)(156 298 320 270 210)(157 211 271 313 299)(158 300 314 272 212)(159 213 265 315 301)(160 302 316 266 214)
(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), (1,301,257,51)(2,302,258,52)(3,303,259,53)(4,304,260,54)(5,297,261,55)(6,298,262,56)(7,299,263,49)(8,300,264,50)(9,156,120,218)(10,157,113,219)(11,158,114,220)(12,159,115,221)(13,160,116,222)(14,153,117,223)(15,154,118,224)(16,155,119,217)(17,312,242,206)(18,305,243,207)(19,306,244,208)(20,307,245,201)(21,308,246,202)(22,309,247,203)(23,310,248,204)(24,311,241,205)(25,147,109,77)(26,148,110,78)(27,149,111,79)(28,150,112,80)(29,151,105,73)(30,152,106,74)(31,145,107,75)(32,146,108,76)(33,136,92,213)(34,129,93,214)(35,130,94,215)(36,131,95,216)(37,132,96,209)(38,133,89,210)(39,134,90,211)(40,135,91,212)(41,167,286,85)(42,168,287,86)(43,161,288,87)(44,162,281,88)(45,163,282,81)(46,164,283,82)(47,165,284,83)(48,166,285,84)(57,185,143,101)(58,186,144,102)(59,187,137,103)(60,188,138,104)(61,189,139,97)(62,190,140,98)(63,191,141,99)(64,192,142,100)(65,182,121,171)(66,183,122,172)(67,184,123,173)(68,177,124,174)(69,178,125,175)(70,179,126,176)(71,180,127,169)(72,181,128,170)(193,315,279,237)(194,316,280,238)(195,317,273,239)(196,318,274,240)(197,319,275,233)(198,320,276,234)(199,313,277,235)(200,314,278,236)(225,290,254,269)(226,291,255,270)(227,292,256,271)(228,293,249,272)(229,294,250,265)(230,295,251,266)(231,296,252,267)(232,289,253,268), (1,12,92,250,193)(2,194,251,93,13)(3,14,94,252,195)(4,196,253,95,15)(5,16,96,254,197)(6,198,255,89,9)(7,10,90,256,199)(8,200,249,91,11)(17,167,186,106,176)(18,169,107,187,168)(19,161,188,108,170)(20,171,109,189,162)(21,163,190,110,172)(22,173,111,191,164)(23,165,192,112,174)(24,175,105,185,166)(25,97,88,245,182)(26,183,246,81,98)(27,99,82,247,184)(28,177,248,83,100)(29,101,84,241,178)(30,179,242,85,102)(31,103,86,243,180)(32,181,244,87,104)(33,229,279,257,115)(34,116,258,280,230)(35,231,273,259,117)(36,118,260,274,232)(37,225,275,261,119)(38,120,262,276,226)(39,227,277,263,113)(40,114,264,278,228)(41,58,152,126,206)(42,207,127,145,59)(43,60,146,128,208)(44,201,121,147,61)(45,62,148,122,202)(46,203,123,149,63)(47,64,150,124,204)(48,205,125,151,57)(49,219,134,292,235)(50,236,293,135,220)(51,221,136,294,237)(52,238,295,129,222)(53,223,130,296,239)(54,240,289,131,224)(55,217,132,290,233)(56,234,291,133,218)(65,77,139,281,307)(66,308,282,140,78)(67,79,141,283,309)(68,310,284,142,80)(69,73,143,285,311)(70,312,286,144,74)(71,75,137,287,305)(72,306,288,138,76)(153,215,267,317,303)(154,304,318,268,216)(155,209,269,319,297)(156,298,320,270,210)(157,211,271,313,299)(158,300,314,272,212)(159,213,265,315,301)(160,302,316,266,214), (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), (1,301,257,51)(2,302,258,52)(3,303,259,53)(4,304,260,54)(5,297,261,55)(6,298,262,56)(7,299,263,49)(8,300,264,50)(9,156,120,218)(10,157,113,219)(11,158,114,220)(12,159,115,221)(13,160,116,222)(14,153,117,223)(15,154,118,224)(16,155,119,217)(17,312,242,206)(18,305,243,207)(19,306,244,208)(20,307,245,201)(21,308,246,202)(22,309,247,203)(23,310,248,204)(24,311,241,205)(25,147,109,77)(26,148,110,78)(27,149,111,79)(28,150,112,80)(29,151,105,73)(30,152,106,74)(31,145,107,75)(32,146,108,76)(33,136,92,213)(34,129,93,214)(35,130,94,215)(36,131,95,216)(37,132,96,209)(38,133,89,210)(39,134,90,211)(40,135,91,212)(41,167,286,85)(42,168,287,86)(43,161,288,87)(44,162,281,88)(45,163,282,81)(46,164,283,82)(47,165,284,83)(48,166,285,84)(57,185,143,101)(58,186,144,102)(59,187,137,103)(60,188,138,104)(61,189,139,97)(62,190,140,98)(63,191,141,99)(64,192,142,100)(65,182,121,171)(66,183,122,172)(67,184,123,173)(68,177,124,174)(69,178,125,175)(70,179,126,176)(71,180,127,169)(72,181,128,170)(193,315,279,237)(194,316,280,238)(195,317,273,239)(196,318,274,240)(197,319,275,233)(198,320,276,234)(199,313,277,235)(200,314,278,236)(225,290,254,269)(226,291,255,270)(227,292,256,271)(228,293,249,272)(229,294,250,265)(230,295,251,266)(231,296,252,267)(232,289,253,268), (1,12,92,250,193)(2,194,251,93,13)(3,14,94,252,195)(4,196,253,95,15)(5,16,96,254,197)(6,198,255,89,9)(7,10,90,256,199)(8,200,249,91,11)(17,167,186,106,176)(18,169,107,187,168)(19,161,188,108,170)(20,171,109,189,162)(21,163,190,110,172)(22,173,111,191,164)(23,165,192,112,174)(24,175,105,185,166)(25,97,88,245,182)(26,183,246,81,98)(27,99,82,247,184)(28,177,248,83,100)(29,101,84,241,178)(30,179,242,85,102)(31,103,86,243,180)(32,181,244,87,104)(33,229,279,257,115)(34,116,258,280,230)(35,231,273,259,117)(36,118,260,274,232)(37,225,275,261,119)(38,120,262,276,226)(39,227,277,263,113)(40,114,264,278,228)(41,58,152,126,206)(42,207,127,145,59)(43,60,146,128,208)(44,201,121,147,61)(45,62,148,122,202)(46,203,123,149,63)(47,64,150,124,204)(48,205,125,151,57)(49,219,134,292,235)(50,236,293,135,220)(51,221,136,294,237)(52,238,295,129,222)(53,223,130,296,239)(54,240,289,131,224)(55,217,132,290,233)(56,234,291,133,218)(65,77,139,281,307)(66,308,282,140,78)(67,79,141,283,309)(68,310,284,142,80)(69,73,143,285,311)(70,312,286,144,74)(71,75,137,287,305)(72,306,288,138,76)(153,215,267,317,303)(154,304,318,268,216)(155,209,269,319,297)(156,298,320,270,210)(157,211,271,313,299)(158,300,314,272,212)(159,213,265,315,301)(160,302,316,266,214), (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)], [(1,301,257,51),(2,302,258,52),(3,303,259,53),(4,304,260,54),(5,297,261,55),(6,298,262,56),(7,299,263,49),(8,300,264,50),(9,156,120,218),(10,157,113,219),(11,158,114,220),(12,159,115,221),(13,160,116,222),(14,153,117,223),(15,154,118,224),(16,155,119,217),(17,312,242,206),(18,305,243,207),(19,306,244,208),(20,307,245,201),(21,308,246,202),(22,309,247,203),(23,310,248,204),(24,311,241,205),(25,147,109,77),(26,148,110,78),(27,149,111,79),(28,150,112,80),(29,151,105,73),(30,152,106,74),(31,145,107,75),(32,146,108,76),(33,136,92,213),(34,129,93,214),(35,130,94,215),(36,131,95,216),(37,132,96,209),(38,133,89,210),(39,134,90,211),(40,135,91,212),(41,167,286,85),(42,168,287,86),(43,161,288,87),(44,162,281,88),(45,163,282,81),(46,164,283,82),(47,165,284,83),(48,166,285,84),(57,185,143,101),(58,186,144,102),(59,187,137,103),(60,188,138,104),(61,189,139,97),(62,190,140,98),(63,191,141,99),(64,192,142,100),(65,182,121,171),(66,183,122,172),(67,184,123,173),(68,177,124,174),(69,178,125,175),(70,179,126,176),(71,180,127,169),(72,181,128,170),(193,315,279,237),(194,316,280,238),(195,317,273,239),(196,318,274,240),(197,319,275,233),(198,320,276,234),(199,313,277,235),(200,314,278,236),(225,290,254,269),(226,291,255,270),(227,292,256,271),(228,293,249,272),(229,294,250,265),(230,295,251,266),(231,296,252,267),(232,289,253,268)], [(1,12,92,250,193),(2,194,251,93,13),(3,14,94,252,195),(4,196,253,95,15),(5,16,96,254,197),(6,198,255,89,9),(7,10,90,256,199),(8,200,249,91,11),(17,167,186,106,176),(18,169,107,187,168),(19,161,188,108,170),(20,171,109,189,162),(21,163,190,110,172),(22,173,111,191,164),(23,165,192,112,174),(24,175,105,185,166),(25,97,88,245,182),(26,183,246,81,98),(27,99,82,247,184),(28,177,248,83,100),(29,101,84,241,178),(30,179,242,85,102),(31,103,86,243,180),(32,181,244,87,104),(33,229,279,257,115),(34,116,258,280,230),(35,231,273,259,117),(36,118,260,274,232),(37,225,275,261,119),(38,120,262,276,226),(39,227,277,263,113),(40,114,264,278,228),(41,58,152,126,206),(42,207,127,145,59),(43,60,146,128,208),(44,201,121,147,61),(45,62,148,122,202),(46,203,123,149,63),(47,64,150,124,204),(48,205,125,151,57),(49,219,134,292,235),(50,236,293,135,220),(51,221,136,294,237),(52,238,295,129,222),(53,223,130,296,239),(54,240,289,131,224),(55,217,132,290,233),(56,234,291,133,218),(65,77,139,281,307),(66,308,282,140,78),(67,79,141,283,309),(68,310,284,142,80),(69,73,143,285,311),(70,312,286,144,74),(71,75,137,287,305),(72,306,288,138,76),(153,215,267,317,303),(154,304,318,268,216),(155,209,269,319,297),(156,298,320,270,210),(157,211,271,313,299),(158,300,314,272,212),(159,213,265,315,301),(160,302,316,266,214)], [(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 2B 2C 4A 4B 4C 4D 4E ··· 4P 5A 5B 8A ··· 8H 8I ··· 8T 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 1 1 1 1 2 ··· 2 2 2 5 ··· 5 10 ··· 10 2 ··· 2 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + - + + - - - + image C1 C2 C2 C2 C4 C4 C8 Q8 D5 C4○D4 D10 Dic5 Dic5 C8○D4 C5⋊2C8 Q8×D5 Q8⋊2D5 D4.Dic5 kernel Q8×C5⋊2C8 C4×C5⋊2C8 C20⋊3C8 Q8×C20 C5×C4⋊C4 Q8×C10 C5×Q8 C5⋊2C8 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C10 Q8 C4 C4 C2 # reps 1 3 3 1 6 2 16 2 2 2 6 6 2 4 16 2 2 4

Matrix representation of Q8×C52C8 in GL4(𝔽41) generated by

 0 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 38 20 0 0 20 3 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 3 0 0 0 0 3 0 0 0 0 35 37 0 0 1 6
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[38,20,0,0,20,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[3,0,0,0,0,3,0,0,0,0,35,1,0,0,37,6] >;

Q8×C52C8 in GAP, Magma, Sage, TeX

Q_8\times C_5\rtimes_2C_8
% in TeX

G:=Group("Q8xC5:2C8");
// GroupNames label

G:=SmallGroup(320,650);
// by ID

G=gap.SmallGroup(320,650);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,219,100,136,12550]);
// 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^-1>;
// generators/relations

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