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G = Q8×C22×C10  order 320 = 26·5

Direct product of C22×C10 and Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: Q8×C22×C10, C10.22C25, C20.88C24, C2.2(C24×C10), (C23×C4).12C10, C4.11(C23×C10), (C23×C20).27C2, C24.37(C2×C10), (C2×C20).977C23, (C2×C10).385C24, C23.74(C22×C10), C22.14(C23×C10), (C22×C10).471C23, (C23×C10).120C22, (C22×C20).605C22, (C2×C4).145(C22×C10), (C22×C4).132(C2×C10), SmallGroup(320,1630)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2 — Q8×C22×C10
Chief series
C1 — C2 — C10 — C20 — C5×Q8 — Q8×C10 — Q8×C2×C10 — Q8×C22×C10
Lower central
C1 — C2 — Q8×C22×C10
Upper central
C1 — C23×C10 — Q8×C22×C10

Generators and relations for Q8×C22×C10
 G = < a,b,c,d,e | a2=b2=c10=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 850, all normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C2×C4, Q8, C23, C10, C10, C22×C4, C2×Q8, C24, C20, C2×C10, C23×C4, C22×Q8, C2×C20, C5×Q8, C22×C10, Q8×C23, C22×C20, Q8×C10, C23×C10, C23×C20, Q8×C2×C10, Q8×C22×C10
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C24, C2×C10, C22×Q8, C25, C5×Q8, C22×C10, Q8×C23, Q8×C10, C23×C10, Q8×C2×C10, C24×C10, Q8×C22×C10

Smallest permutation representation of Q8×C22×C10
►Regular action on 320 points
Generators in S320
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 81)(11 258)(12 259)(13 260)(14 251)(15 252)(16 253)(17 254)(18 255)(19 256)(20 257)(21 79)(22 80)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 61)(40 62)(41 59)(42 60)(43 51)(44 52)(45 53)(46 54)(47 55)(48 56)(49 57)(50 58)(91 163)(92 164)(93 165)(94 166)(95 167)(96 168)(97 169)(98 170)(99 161)(100 162)(101 159)(102 160)(103 151)(104 152)(105 153)(106 154)(107 155)(108 156)(109 157)(110 158)(111 143)(112 144)(113 145)(114 146)(115 147)(116 148)(117 149)(118 150)(119 141)(120 142)(121 139)(122 140)(123 131)(124 132)(125 133)(126 134)(127 135)(128 136)(129 137)(130 138)(171 243)(172 244)(173 245)(174 246)(175 247)(176 248)(177 249)(178 250)(179 241)(180 242)(181 239)(182 240)(183 231)(184 232)(185 233)(186 234)(187 235)(188 236)(189 237)(190 238)(191 223)(192 224)(193 225)(194 226)(195 227)(196 228)(197 229)(198 230)(199 221)(200 222)(201 219)(202 220)(203 211)(204 212)(205 213)(206 214)(207 215)(208 216)(209 217)(210 218)(261 319)(262 320)(263 311)(264 312)(265 313)(266 314)(267 315)(268 316)(269 317)(270 318)(271 303)(272 304)(273 305)(274 306)(275 307)(276 308)(277 309)(278 310)(279 301)(280 302)(281 299)(282 300)(283 291)(284 292)(285 293)(286 294)(287 295)(288 296)(289 297)(290 298)

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

(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)

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

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

200 conjugacy classes

class 1 2A···2O4A···4X5A5B5C5D10A···10BH20A···20CR
order12···24···4555510···1020···20
size11···12···211111···12···2

200 irreducible representations

dim11111122
type+++-
imageC1C2C2C5C10C10Q8C5×Q8
kernelQ8×C22×C10C23×C20Q8×C2×C10Q8×C23C23×C4C22×Q8C22×C10C23
# reps1328412112832

Matrix representation of Q8×C22×C10 ►in GL5(𝔽41)

10000
040000
004000
000400
000040
,
10000
040000
00100
00010
00001
,
400000
040000
00100
000230
000023
,
10000
01000
004000
000320
000179
,
10000
01000
00100
000112
0002130

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,23,0,0,0,0,0,23],[1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,32,17,0,0,0,0,9],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,11,21,0,0,0,2,30] >;
 

Q8×C22×C10 in GAP, Magma, Sage, TeX 

Q_8\times C_2^2\times C_{10}
 
% in TeX
 
G:=Group("Q8xC2^2xC10");
 
// GroupNames label
 
G:=SmallGroup(320,1630);
 
// by ID
 
G=gap.SmallGroup(320,1630);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-5,-2,1120,2269,1128]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^10=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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×
⋊
ℤ
𝔽
○
≀
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