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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), C4.11(C23×C10), (C23×C4).12C10, (C23×C20).27C2, C24.37(C2×C10), (C2×C10).385C24, (C2×C20).977C23, C22.14(C23×C10), C23.74(C22×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

Subgroups: 850, all normal (8 characteristic)
C1, C2, C2 [×14], C4 [×24], C22 [×35], C5, C2×C4 [×84], Q8 [×64], C23 [×15], C10, C10 [×14], C22×C4 [×42], C2×Q8 [×112], C24, C20 [×24], C2×C10 [×35], C23×C4 [×3], C22×Q8 [×28], C2×C20 [×84], C5×Q8 [×64], C22×C10 [×15], Q8×C23, C22×C20 [×42], Q8×C10 [×112], C23×C10, C23×C20 [×3], Q8×C2×C10 [×28], Q8×C22×C10

Quotients:
C1, C2 [×31], C22 [×155], C5, Q8 [×8], C23 [×155], C10 [×31], C2×Q8 [×28], C24 [×31], C2×C10 [×155], C22×Q8 [×14], C25, C5×Q8 [×8], C22×C10 [×155], Q8×C23, Q8×C10 [×28], C23×C10 [×31], Q8×C2×C10 [×14], C24×C10, Q8×C22×C10

Generators and relations
 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 >

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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
 

׿
×
⋊
ℤ
𝔽
○
≀