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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 [×3], C4 [×2], C4 [×6], C4 [×3], C22, C5, C8 [×5], C2×C4, C2×C4 [×6], Q8 [×4], C10 [×3], C42 [×3], C4⋊C4 [×3], C2×C8 [×4], C2×Q8, C20 [×2], C20 [×6], C20 [×3], C2×C10, C4×C8 [×3], C4⋊C8 [×3], C4×Q8, C52C8 [×2], C52C8 [×3], C2×C20, C2×C20 [×6], C5×Q8 [×4], C8×Q8, C2×C52C8, C2×C52C8 [×3], C4×C20 [×3], C5×C4⋊C4 [×3], Q8×C10, C4×C52C8 [×3], C203C8 [×3], Q8×C20, Q8×C52C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], Q8 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×Q8, C4○D4, Dic5 [×4], D10 [×3], C4×Q8, C22×C8, C8○D4, C52C8 [×4], C2×Dic5 [×6], C22×D5, C8×Q8, C2×C52C8 [×6], 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 97 52)(2 282 98 53)(3 283 99 54)(4 284 100 55)(5 285 101 56)(6 286 102 49)(7 287 103 50)(8 288 104 51)(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)(17 270 199 291)(18 271 200 292)(19 272 193 293)(20 265 194 294)(21 266 195 295)(22 267 196 296)(23 268 197 289)(24 269 198 290)(25 61 245 159)(26 62 246 160)(27 63 247 153)(28 64 248 154)(29 57 241 155)(30 58 242 156)(31 59 243 157)(32 60 244 158)(33 129 274 69)(34 130 275 70)(35 131 276 71)(36 132 277 72)(37 133 278 65)(38 134 279 66)(39 135 280 67)(40 136 273 68)(41 145 261 212)(42 146 262 213)(43 147 263 214)(44 148 264 215)(45 149 257 216)(46 150 258 209)(47 151 259 210)(48 152 260 211)(73 225 123 254)(74 226 124 255)(75 227 125 256)(76 228 126 249)(77 229 127 250)(78 230 128 251)(79 231 121 252)(80 232 122 253)(81 172 301 167)(82 173 302 168)(83 174 303 161)(84 175 304 162)(85 176 297 163)(86 169 298 164)(87 170 299 165)(88 171 300 166)(89 142 183 234)(90 143 184 235)(91 144 177 236)(92 137 178 237)(93 138 179 238)(94 139 180 239)(95 140 181 240)(96 141 182 233)(105 318 192 221)(106 319 185 222)(107 320 186 223)(108 313 187 224)(109 314 188 217)(110 315 189 218)(111 316 190 219)(112 317 191 220)
(1 141 97 233)(2 142 98 234)(3 143 99 235)(4 144 100 236)(5 137 101 237)(6 138 102 238)(7 139 103 239)(8 140 104 240)(9 156 120 58)(10 157 113 59)(11 158 114 60)(12 159 115 61)(13 160 116 62)(14 153 117 63)(15 154 118 64)(16 155 119 57)(17 124 199 74)(18 125 200 75)(19 126 193 76)(20 127 194 77)(21 128 195 78)(22 121 196 79)(23 122 197 80)(24 123 198 73)(25 307 245 201)(26 308 246 202)(27 309 247 203)(28 310 248 204)(29 311 241 205)(30 312 242 206)(31 305 243 207)(32 306 244 208)(33 318 274 221)(34 319 275 222)(35 320 276 223)(36 313 277 224)(37 314 278 217)(38 315 279 218)(39 316 280 219)(40 317 273 220)(41 297 261 85)(42 298 262 86)(43 299 263 87)(44 300 264 88)(45 301 257 81)(46 302 258 82)(47 303 259 83)(48 304 260 84)(49 179 286 93)(50 180 287 94)(51 181 288 95)(52 182 281 96)(53 183 282 89)(54 184 283 90)(55 177 284 91)(56 178 285 92)(65 188 133 109)(66 189 134 110)(67 190 135 111)(68 191 136 112)(69 192 129 105)(70 185 130 106)(71 186 131 107)(72 187 132 108)(145 176 212 163)(146 169 213 164)(147 170 214 165)(148 171 215 166)(149 172 216 167)(150 173 209 168)(151 174 210 161)(152 175 211 162)(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 12 252 263 38)(2 39 264 253 13)(3 14 254 257 40)(4 33 258 255 15)(5 16 256 259 34)(6 35 260 249 9)(7 10 250 261 36)(8 37 262 251 11)(17 248 91 105 168)(18 161 106 92 241)(19 242 93 107 162)(20 163 108 94 243)(21 244 95 109 164)(22 165 110 96 245)(23 246 89 111 166)(24 167 112 90 247)(25 196 170 189 182)(26 183 190 171 197)(27 198 172 191 184)(28 177 192 173 199)(29 200 174 185 178)(30 179 186 175 193)(31 194 176 187 180)(32 181 188 169 195)(41 277 103 113 229)(42 230 114 104 278)(43 279 97 115 231)(44 232 116 98 280)(45 273 99 117 225)(46 226 118 100 274)(47 275 101 119 227)(48 228 120 102 276)(49 71 152 126 206)(50 207 127 145 72)(51 65 146 128 208)(52 201 121 147 66)(53 67 148 122 202)(54 203 123 149 68)(55 69 150 124 204)(56 205 125 151 70)(57 292 303 222 237)(58 238 223 304 293)(59 294 297 224 239)(60 240 217 298 295)(61 296 299 218 233)(62 234 219 300 289)(63 290 301 220 235)(64 236 221 302 291)(73 216 136 283 309)(74 310 284 129 209)(75 210 130 285 311)(76 312 286 131 211)(77 212 132 287 305)(78 306 288 133 213)(79 214 134 281 307)(80 308 282 135 215)(81 317 143 153 269)(82 270 154 144 318)(83 319 137 155 271)(84 272 156 138 320)(85 313 139 157 265)(86 266 158 140 314)(87 315 141 159 267)(88 268 160 142 316)
(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)| (1,281,97,52)(2,282,98,53)(3,283,99,54)(4,284,100,55)(5,285,101,56)(6,286,102,49)(7,287,103,50)(8,288,104,51)(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)(17,270,199,291)(18,271,200,292)(19,272,193,293)(20,265,194,294)(21,266,195,295)(22,267,196,296)(23,268,197,289)(24,269,198,290)(25,61,245,159)(26,62,246,160)(27,63,247,153)(28,64,248,154)(29,57,241,155)(30,58,242,156)(31,59,243,157)(32,60,244,158)(33,129,274,69)(34,130,275,70)(35,131,276,71)(36,132,277,72)(37,133,278,65)(38,134,279,66)(39,135,280,67)(40,136,273,68)(41,145,261,212)(42,146,262,213)(43,147,263,214)(44,148,264,215)(45,149,257,216)(46,150,258,209)(47,151,259,210)(48,152,260,211)(73,225,123,254)(74,226,124,255)(75,227,125,256)(76,228,126,249)(77,229,127,250)(78,230,128,251)(79,231,121,252)(80,232,122,253)(81,172,301,167)(82,173,302,168)(83,174,303,161)(84,175,304,162)(85,176,297,163)(86,169,298,164)(87,170,299,165)(88,171,300,166)(89,142,183,234)(90,143,184,235)(91,144,177,236)(92,137,178,237)(93,138,179,238)(94,139,180,239)(95,140,181,240)(96,141,182,233)(105,318,192,221)(106,319,185,222)(107,320,186,223)(108,313,187,224)(109,314,188,217)(110,315,189,218)(111,316,190,219)(112,317,191,220), (1,141,97,233)(2,142,98,234)(3,143,99,235)(4,144,100,236)(5,137,101,237)(6,138,102,238)(7,139,103,239)(8,140,104,240)(9,156,120,58)(10,157,113,59)(11,158,114,60)(12,159,115,61)(13,160,116,62)(14,153,117,63)(15,154,118,64)(16,155,119,57)(17,124,199,74)(18,125,200,75)(19,126,193,76)(20,127,194,77)(21,128,195,78)(22,121,196,79)(23,122,197,80)(24,123,198,73)(25,307,245,201)(26,308,246,202)(27,309,247,203)(28,310,248,204)(29,311,241,205)(30,312,242,206)(31,305,243,207)(32,306,244,208)(33,318,274,221)(34,319,275,222)(35,320,276,223)(36,313,277,224)(37,314,278,217)(38,315,279,218)(39,316,280,219)(40,317,273,220)(41,297,261,85)(42,298,262,86)(43,299,263,87)(44,300,264,88)(45,301,257,81)(46,302,258,82)(47,303,259,83)(48,304,260,84)(49,179,286,93)(50,180,287,94)(51,181,288,95)(52,182,281,96)(53,183,282,89)(54,184,283,90)(55,177,284,91)(56,178,285,92)(65,188,133,109)(66,189,134,110)(67,190,135,111)(68,191,136,112)(69,192,129,105)(70,185,130,106)(71,186,131,107)(72,187,132,108)(145,176,212,163)(146,169,213,164)(147,170,214,165)(148,171,215,166)(149,172,216,167)(150,173,209,168)(151,174,210,161)(152,175,211,162)(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), (1,12,252,263,38)(2,39,264,253,13)(3,14,254,257,40)(4,33,258,255,15)(5,16,256,259,34)(6,35,260,249,9)(7,10,250,261,36)(8,37,262,251,11)(17,248,91,105,168)(18,161,106,92,241)(19,242,93,107,162)(20,163,108,94,243)(21,244,95,109,164)(22,165,110,96,245)(23,246,89,111,166)(24,167,112,90,247)(25,196,170,189,182)(26,183,190,171,197)(27,198,172,191,184)(28,177,192,173,199)(29,200,174,185,178)(30,179,186,175,193)(31,194,176,187,180)(32,181,188,169,195)(41,277,103,113,229)(42,230,114,104,278)(43,279,97,115,231)(44,232,116,98,280)(45,273,99,117,225)(46,226,118,100,274)(47,275,101,119,227)(48,228,120,102,276)(49,71,152,126,206)(50,207,127,145,72)(51,65,146,128,208)(52,201,121,147,66)(53,67,148,122,202)(54,203,123,149,68)(55,69,150,124,204)(56,205,125,151,70)(57,292,303,222,237)(58,238,223,304,293)(59,294,297,224,239)(60,240,217,298,295)(61,296,299,218,233)(62,234,219,300,289)(63,290,301,220,235)(64,236,221,302,291)(73,216,136,283,309)(74,310,284,129,209)(75,210,130,285,311)(76,312,286,131,211)(77,212,132,287,305)(78,306,288,133,213)(79,214,134,281,307)(80,308,282,135,215)(81,317,143,153,269)(82,270,154,144,318)(83,319,137,155,271)(84,272,156,138,320)(85,313,139,157,265)(86,266,158,140,314)(87,315,141,159,267)(88,268,160,142,316), (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( (1,281,97,52)(2,282,98,53)(3,283,99,54)(4,284,100,55)(5,285,101,56)(6,286,102,49)(7,287,103,50)(8,288,104,51)(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)(17,270,199,291)(18,271,200,292)(19,272,193,293)(20,265,194,294)(21,266,195,295)(22,267,196,296)(23,268,197,289)(24,269,198,290)(25,61,245,159)(26,62,246,160)(27,63,247,153)(28,64,248,154)(29,57,241,155)(30,58,242,156)(31,59,243,157)(32,60,244,158)(33,129,274,69)(34,130,275,70)(35,131,276,71)(36,132,277,72)(37,133,278,65)(38,134,279,66)(39,135,280,67)(40,136,273,68)(41,145,261,212)(42,146,262,213)(43,147,263,214)(44,148,264,215)(45,149,257,216)(46,150,258,209)(47,151,259,210)(48,152,260,211)(73,225,123,254)(74,226,124,255)(75,227,125,256)(76,228,126,249)(77,229,127,250)(78,230,128,251)(79,231,121,252)(80,232,122,253)(81,172,301,167)(82,173,302,168)(83,174,303,161)(84,175,304,162)(85,176,297,163)(86,169,298,164)(87,170,299,165)(88,171,300,166)(89,142,183,234)(90,143,184,235)(91,144,177,236)(92,137,178,237)(93,138,179,238)(94,139,180,239)(95,140,181,240)(96,141,182,233)(105,318,192,221)(106,319,185,222)(107,320,186,223)(108,313,187,224)(109,314,188,217)(110,315,189,218)(111,316,190,219)(112,317,191,220), (1,141,97,233)(2,142,98,234)(3,143,99,235)(4,144,100,236)(5,137,101,237)(6,138,102,238)(7,139,103,239)(8,140,104,240)(9,156,120,58)(10,157,113,59)(11,158,114,60)(12,159,115,61)(13,160,116,62)(14,153,117,63)(15,154,118,64)(16,155,119,57)(17,124,199,74)(18,125,200,75)(19,126,193,76)(20,127,194,77)(21,128,195,78)(22,121,196,79)(23,122,197,80)(24,123,198,73)(25,307,245,201)(26,308,246,202)(27,309,247,203)(28,310,248,204)(29,311,241,205)(30,312,242,206)(31,305,243,207)(32,306,244,208)(33,318,274,221)(34,319,275,222)(35,320,276,223)(36,313,277,224)(37,314,278,217)(38,315,279,218)(39,316,280,219)(40,317,273,220)(41,297,261,85)(42,298,262,86)(43,299,263,87)(44,300,264,88)(45,301,257,81)(46,302,258,82)(47,303,259,83)(48,304,260,84)(49,179,286,93)(50,180,287,94)(51,181,288,95)(52,182,281,96)(53,183,282,89)(54,184,283,90)(55,177,284,91)(56,178,285,92)(65,188,133,109)(66,189,134,110)(67,190,135,111)(68,191,136,112)(69,192,129,105)(70,185,130,106)(71,186,131,107)(72,187,132,108)(145,176,212,163)(146,169,213,164)(147,170,214,165)(148,171,215,166)(149,172,216,167)(150,173,209,168)(151,174,210,161)(152,175,211,162)(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), (1,12,252,263,38)(2,39,264,253,13)(3,14,254,257,40)(4,33,258,255,15)(5,16,256,259,34)(6,35,260,249,9)(7,10,250,261,36)(8,37,262,251,11)(17,248,91,105,168)(18,161,106,92,241)(19,242,93,107,162)(20,163,108,94,243)(21,244,95,109,164)(22,165,110,96,245)(23,246,89,111,166)(24,167,112,90,247)(25,196,170,189,182)(26,183,190,171,197)(27,198,172,191,184)(28,177,192,173,199)(29,200,174,185,178)(30,179,186,175,193)(31,194,176,187,180)(32,181,188,169,195)(41,277,103,113,229)(42,230,114,104,278)(43,279,97,115,231)(44,232,116,98,280)(45,273,99,117,225)(46,226,118,100,274)(47,275,101,119,227)(48,228,120,102,276)(49,71,152,126,206)(50,207,127,145,72)(51,65,146,128,208)(52,201,121,147,66)(53,67,148,122,202)(54,203,123,149,68)(55,69,150,124,204)(56,205,125,151,70)(57,292,303,222,237)(58,238,223,304,293)(59,294,297,224,239)(60,240,217,298,295)(61,296,299,218,233)(62,234,219,300,289)(63,290,301,220,235)(64,236,221,302,291)(73,216,136,283,309)(74,310,284,129,209)(75,210,130,285,311)(76,312,286,131,211)(77,212,132,287,305)(78,306,288,133,213)(79,214,134,281,307)(80,308,282,135,215)(81,317,143,153,269)(82,270,154,144,318)(83,319,137,155,271)(84,272,156,138,320)(85,313,139,157,265)(86,266,158,140,314)(87,315,141,159,267)(88,268,160,142,316), (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([(1,281,97,52),(2,282,98,53),(3,283,99,54),(4,284,100,55),(5,285,101,56),(6,286,102,49),(7,287,103,50),(8,288,104,51),(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),(17,270,199,291),(18,271,200,292),(19,272,193,293),(20,265,194,294),(21,266,195,295),(22,267,196,296),(23,268,197,289),(24,269,198,290),(25,61,245,159),(26,62,246,160),(27,63,247,153),(28,64,248,154),(29,57,241,155),(30,58,242,156),(31,59,243,157),(32,60,244,158),(33,129,274,69),(34,130,275,70),(35,131,276,71),(36,132,277,72),(37,133,278,65),(38,134,279,66),(39,135,280,67),(40,136,273,68),(41,145,261,212),(42,146,262,213),(43,147,263,214),(44,148,264,215),(45,149,257,216),(46,150,258,209),(47,151,259,210),(48,152,260,211),(73,225,123,254),(74,226,124,255),(75,227,125,256),(76,228,126,249),(77,229,127,250),(78,230,128,251),(79,231,121,252),(80,232,122,253),(81,172,301,167),(82,173,302,168),(83,174,303,161),(84,175,304,162),(85,176,297,163),(86,169,298,164),(87,170,299,165),(88,171,300,166),(89,142,183,234),(90,143,184,235),(91,144,177,236),(92,137,178,237),(93,138,179,238),(94,139,180,239),(95,140,181,240),(96,141,182,233),(105,318,192,221),(106,319,185,222),(107,320,186,223),(108,313,187,224),(109,314,188,217),(110,315,189,218),(111,316,190,219),(112,317,191,220)], [(1,141,97,233),(2,142,98,234),(3,143,99,235),(4,144,100,236),(5,137,101,237),(6,138,102,238),(7,139,103,239),(8,140,104,240),(9,156,120,58),(10,157,113,59),(11,158,114,60),(12,159,115,61),(13,160,116,62),(14,153,117,63),(15,154,118,64),(16,155,119,57),(17,124,199,74),(18,125,200,75),(19,126,193,76),(20,127,194,77),(21,128,195,78),(22,121,196,79),(23,122,197,80),(24,123,198,73),(25,307,245,201),(26,308,246,202),(27,309,247,203),(28,310,248,204),(29,311,241,205),(30,312,242,206),(31,305,243,207),(32,306,244,208),(33,318,274,221),(34,319,275,222),(35,320,276,223),(36,313,277,224),(37,314,278,217),(38,315,279,218),(39,316,280,219),(40,317,273,220),(41,297,261,85),(42,298,262,86),(43,299,263,87),(44,300,264,88),(45,301,257,81),(46,302,258,82),(47,303,259,83),(48,304,260,84),(49,179,286,93),(50,180,287,94),(51,181,288,95),(52,182,281,96),(53,183,282,89),(54,184,283,90),(55,177,284,91),(56,178,285,92),(65,188,133,109),(66,189,134,110),(67,190,135,111),(68,191,136,112),(69,192,129,105),(70,185,130,106),(71,186,131,107),(72,187,132,108),(145,176,212,163),(146,169,213,164),(147,170,214,165),(148,171,215,166),(149,172,216,167),(150,173,209,168),(151,174,210,161),(152,175,211,162),(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)], [(1,12,252,263,38),(2,39,264,253,13),(3,14,254,257,40),(4,33,258,255,15),(5,16,256,259,34),(6,35,260,249,9),(7,10,250,261,36),(8,37,262,251,11),(17,248,91,105,168),(18,161,106,92,241),(19,242,93,107,162),(20,163,108,94,243),(21,244,95,109,164),(22,165,110,96,245),(23,246,89,111,166),(24,167,112,90,247),(25,196,170,189,182),(26,183,190,171,197),(27,198,172,191,184),(28,177,192,173,199),(29,200,174,185,178),(30,179,186,175,193),(31,194,176,187,180),(32,181,188,169,195),(41,277,103,113,229),(42,230,114,104,278),(43,279,97,115,231),(44,232,116,98,280),(45,273,99,117,225),(46,226,118,100,274),(47,275,101,119,227),(48,228,120,102,276),(49,71,152,126,206),(50,207,127,145,72),(51,65,146,128,208),(52,201,121,147,66),(53,67,148,122,202),(54,203,123,149,68),(55,69,150,124,204),(56,205,125,151,70),(57,292,303,222,237),(58,238,223,304,293),(59,294,297,224,239),(60,240,217,298,295),(61,296,299,218,233),(62,234,219,300,289),(63,290,301,220,235),(64,236,221,302,291),(73,216,136,283,309),(74,310,284,129,209),(75,210,130,285,311),(76,312,286,131,211),(77,212,132,287,305),(78,306,288,133,213),(79,214,134,281,307),(80,308,282,135,215),(81,317,143,153,269),(82,270,154,144,318),(83,319,137,155,271),(84,272,156,138,320),(85,313,139,157,265),(86,266,158,140,314),(87,315,141,159,267),(88,268,160,142,316)], [(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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