direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C5×C22.4Q16, C20.58C42, C4⋊C4⋊4C20, (C2×C8)⋊4C20, (C2×C40)⋊18C4, C4.2(C4×C20), (C2×C10).48D8, C20.83(C4⋊C4), (C2×C20).70Q8, C22.7(C5×D8), (C2×C20).507D4, (C22×C40).5C2, (C22×C8).2C10, (C2×C10).19Q16, C23.52(C5×D4), C22.4(C5×Q16), (C2×C10).43SD16, C10.19(C2.D8), C10.14(C4.Q8), C22.9(C5×SD16), (C22×C10).213D4, C10.51(D4⋊C4), C20.150(C22⋊C4), C10.25(Q8⋊C4), (C22×C20).571C22, C10.43(C2.C42), C4.3(C5×C4⋊C4), (C5×C4⋊C4)⋊18C4, (C2×C4⋊C4).2C10, C2.2(C5×C2.D8), C2.2(C5×C4.Q8), (C10×C4⋊C4).29C2, (C2×C4).13(C5×Q8), (C2×C4).40(C2×C20), (C2×C4).112(C5×D4), C2.2(C5×D4⋊C4), C4.19(C5×C22⋊C4), C22.16(C5×C4⋊C4), C2.2(C5×Q8⋊C4), (C2×C10).87(C4⋊C4), (C2×C20).434(C2×C4), C22.29(C5×C22⋊C4), C2.5(C5×C2.C42), (C22×C4).104(C2×C10), (C2×C10).194(C22⋊C4), SmallGroup(320,145)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×C22.4Q16
G = < a,b,c,d,e | a5=b2=c2=d8=1, e2=cd4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=bd-1 >
Subgroups: 186 in 114 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C22×C4, C20, C20, C20, C2×C10, C2×C10, C2×C4⋊C4, C22×C8, C40, C2×C20, C2×C20, C2×C20, C22×C10, C22.4Q16, C5×C4⋊C4, C5×C4⋊C4, C2×C40, C2×C40, C22×C20, C22×C20, C10×C4⋊C4, C22×C40, C5×C22.4Q16
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, D8, SD16, Q16, C20, C2×C10, C2.C42, D4⋊C4, Q8⋊C4, C4.Q8, C2.D8, C2×C20, C5×D4, C5×Q8, C22.4Q16, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×D8, C5×SD16, C5×Q16, C5×C2.C42, C5×D4⋊C4, C5×Q8⋊C4, C5×C4.Q8, C5×C2.D8, C5×C22.4Q16
►Regular action on 320 points
Generators in S320
(1 89 22 117 109)(2 90 23 118 110)(3 91 24 119 111)(4 92 17 120 112)(5 93 18 113 105)(6 94 19 114 106)(7 95 20 115 107)(8 96 21 116 108)(9 43 35 27 97)(10 44 36 28 98)(11 45 37 29 99)(12 46 38 30 100)(13 47 39 31 101)(14 48 40 32 102)(15 41 33 25 103)(16 42 34 26 104)(49 176 73 65 57)(50 169 74 66 58)(51 170 75 67 59)(52 171 76 68 60)(53 172 77 69 61)(54 173 78 70 62)(55 174 79 71 63)(56 175 80 72 64)(81 229 236 213 220)(82 230 237 214 221)(83 231 238 215 222)(84 232 239 216 223)(85 225 240 209 224)(86 226 233 210 217)(87 227 234 211 218)(88 228 235 212 219)(121 153 145 137 129)(122 154 146 138 130)(123 155 147 139 131)(124 156 148 140 132)(125 157 149 141 133)(126 158 150 142 134)(127 159 151 143 135)(128 160 152 144 136)(161 280 272 264 256)(162 273 265 257 249)(163 274 266 258 250)(164 275 267 259 251)(165 276 268 260 252)(166 277 269 261 253)(167 278 270 262 254)(168 279 271 263 255)(177 192 248 193 208)(178 185 241 194 201)(179 186 242 195 202)(180 187 243 196 203)(181 188 244 197 204)(182 189 245 198 205)(183 190 246 199 206)(184 191 247 200 207)(281 313 305 297 289)(282 314 306 298 290)(283 315 307 299 291)(284 316 308 300 292)(285 317 309 301 293)(286 318 310 302 294)(287 319 311 303 295)(288 320 312 304 296)
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 25)(8 26)(9 22)(10 23)(11 24)(12 17)(13 18)(14 19)(15 20)(16 21)(33 107)(34 108)(35 109)(36 110)(37 111)(38 112)(39 105)(40 106)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 113)(48 114)(49 123)(50 124)(51 125)(52 126)(53 127)(54 128)(55 121)(56 122)(57 131)(58 132)(59 133)(60 134)(61 135)(62 136)(63 129)(64 130)(65 139)(66 140)(67 141)(68 142)(69 143)(70 144)(71 137)(72 138)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 145)(80 146)(81 315)(82 316)(83 317)(84 318)(85 319)(86 320)(87 313)(88 314)(89 97)(90 98)(91 99)(92 100)(93 101)(94 102)(95 103)(96 104)(153 174)(154 175)(155 176)(156 169)(157 170)(158 171)(159 172)(160 173)(161 247)(162 248)(163 241)(164 242)(165 243)(166 244)(167 245)(168 246)(177 257)(178 258)(179 259)(180 260)(181 261)(182 262)(183 263)(184 264)(185 250)(186 251)(187 252)(188 253)(189 254)(190 255)(191 256)(192 249)(193 273)(194 274)(195 275)(196 276)(197 277)(198 278)(199 279)(200 280)(201 266)(202 267)(203 268)(204 269)(205 270)(206 271)(207 272)(208 265)(209 295)(210 296)(211 289)(212 290)(213 291)(214 292)(215 293)(216 294)(217 288)(218 281)(219 282)(220 283)(221 284)(222 285)(223 286)(224 287)(225 311)(226 312)(227 305)(228 306)(229 307)(230 308)(231 309)(232 310)(233 304)(234 297)(235 298)(236 299)(237 300)(238 301)(239 302)(240 303)
(1 123)(2 124)(3 125)(4 126)(5 127)(6 128)(7 121)(8 122)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 150)(18 151)(19 152)(20 145)(21 146)(22 147)(23 148)(24 149)(25 55)(26 56)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 63)(34 64)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 71)(42 72)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(81 248)(82 241)(83 242)(84 243)(85 244)(86 245)(87 246)(88 247)(89 155)(90 156)(91 157)(92 158)(93 159)(94 160)(95 153)(96 154)(97 176)(98 169)(99 170)(100 171)(101 172)(102 173)(103 174)(104 175)(105 135)(106 136)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 143)(114 144)(115 137)(116 138)(117 139)(118 140)(119 141)(120 142)(161 314)(162 315)(163 316)(164 317)(165 318)(166 319)(167 320)(168 313)(177 213)(178 214)(179 215)(180 216)(181 209)(182 210)(183 211)(184 212)(185 221)(186 222)(187 223)(188 224)(189 217)(190 218)(191 219)(192 220)(193 229)(194 230)(195 231)(196 232)(197 225)(198 226)(199 227)(200 228)(201 237)(202 238)(203 239)(204 240)(205 233)(206 234)(207 235)(208 236)(249 283)(250 284)(251 285)(252 286)(253 287)(254 288)(255 281)(256 282)(257 291)(258 292)(259 293)(260 294)(261 295)(262 296)(263 289)(264 290)(265 299)(266 300)(267 301)(268 302)(269 303)(270 304)(271 297)(272 298)(273 307)(274 308)(275 309)(276 310)(277 311)(278 312)(279 305)(280 306)
(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)
(1 218 127 186)(2 288 128 250)(3 224 121 192)(4 286 122 256)(5 222 123 190)(6 284 124 254)(7 220 125 188)(8 282 126 252)(9 305 77 275)(10 226 78 194)(11 311 79 273)(12 232 80 200)(13 309 73 279)(14 230 74 198)(15 307 75 277)(16 228 76 196)(17 310 146 280)(18 231 147 199)(19 308 148 278)(20 229 149 197)(21 306 150 276)(22 227 151 195)(23 312 152 274)(24 225 145 193)(25 283 51 253)(26 219 52 187)(27 281 53 251)(28 217 54 185)(29 287 55 249)(30 223 56 191)(31 285 49 255)(32 221 50 189)(33 291 59 261)(34 212 60 180)(35 289 61 259)(36 210 62 178)(37 295 63 257)(38 216 64 184)(39 293 57 263)(40 214 58 182)(41 299 67 269)(42 235 68 203)(43 297 69 267)(44 233 70 201)(45 303 71 265)(46 239 72 207)(47 301 65 271)(48 237 66 205)(81 157 244 95)(82 169 245 102)(83 155 246 93)(84 175 247 100)(85 153 248 91)(86 173 241 98)(87 159 242 89)(88 171 243 104)(90 320 160 163)(92 318 154 161)(94 316 156 167)(96 314 158 165)(97 313 172 164)(99 319 174 162)(101 317 176 168)(103 315 170 166)(105 215 131 183)(106 292 132 262)(107 213 133 181)(108 290 134 260)(109 211 135 179)(110 296 136 258)(111 209 129 177)(112 294 130 264)(113 238 139 206)(114 300 140 270)(115 236 141 204)(116 298 142 268)(117 234 143 202)(118 304 144 266)(119 240 137 208)(120 302 138 272)
G:=sub<Sym(320)| (1,89,22,117,109)(2,90,23,118,110)(3,91,24,119,111)(4,92,17,120,112)(5,93,18,113,105)(6,94,19,114,106)(7,95,20,115,107)(8,96,21,116,108)(9,43,35,27,97)(10,44,36,28,98)(11,45,37,29,99)(12,46,38,30,100)(13,47,39,31,101)(14,48,40,32,102)(15,41,33,25,103)(16,42,34,26,104)(49,176,73,65,57)(50,169,74,66,58)(51,170,75,67,59)(52,171,76,68,60)(53,172,77,69,61)(54,173,78,70,62)(55,174,79,71,63)(56,175,80,72,64)(81,229,236,213,220)(82,230,237,214,221)(83,231,238,215,222)(84,232,239,216,223)(85,225,240,209,224)(86,226,233,210,217)(87,227,234,211,218)(88,228,235,212,219)(121,153,145,137,129)(122,154,146,138,130)(123,155,147,139,131)(124,156,148,140,132)(125,157,149,141,133)(126,158,150,142,134)(127,159,151,143,135)(128,160,152,144,136)(161,280,272,264,256)(162,273,265,257,249)(163,274,266,258,250)(164,275,267,259,251)(165,276,268,260,252)(166,277,269,261,253)(167,278,270,262,254)(168,279,271,263,255)(177,192,248,193,208)(178,185,241,194,201)(179,186,242,195,202)(180,187,243,196,203)(181,188,244,197,204)(182,189,245,198,205)(183,190,246,199,206)(184,191,247,200,207)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,105)(40,106)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,113)(48,114)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,121)(56,122)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,129)(64,130)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,137)(72,138)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,145)(80,146)(81,315)(82,316)(83,317)(84,318)(85,319)(86,320)(87,313)(88,314)(89,97)(90,98)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(153,174)(154,175)(155,176)(156,169)(157,170)(158,171)(159,172)(160,173)(161,247)(162,248)(163,241)(164,242)(165,243)(166,244)(167,245)(168,246)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)(185,250)(186,251)(187,252)(188,253)(189,254)(190,255)(191,256)(192,249)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)(201,266)(202,267)(203,268)(204,269)(205,270)(206,271)(207,272)(208,265)(209,295)(210,296)(211,289)(212,290)(213,291)(214,292)(215,293)(216,294)(217,288)(218,281)(219,282)(220,283)(221,284)(222,285)(223,286)(224,287)(225,311)(226,312)(227,305)(228,306)(229,307)(230,308)(231,309)(232,310)(233,304)(234,297)(235,298)(236,299)(237,300)(238,301)(239,302)(240,303), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,121)(8,122)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,150)(18,151)(19,152)(20,145)(21,146)(22,147)(23,148)(24,149)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,71)(42,72)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(81,248)(82,241)(83,242)(84,243)(85,244)(86,245)(87,246)(88,247)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,153)(96,154)(97,176)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,143)(114,144)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(161,314)(162,315)(163,316)(164,317)(165,318)(166,319)(167,320)(168,313)(177,213)(178,214)(179,215)(180,216)(181,209)(182,210)(183,211)(184,212)(185,221)(186,222)(187,223)(188,224)(189,217)(190,218)(191,219)(192,220)(193,229)(194,230)(195,231)(196,232)(197,225)(198,226)(199,227)(200,228)(201,237)(202,238)(203,239)(204,240)(205,233)(206,234)(207,235)(208,236)(249,283)(250,284)(251,285)(252,286)(253,287)(254,288)(255,281)(256,282)(257,291)(258,292)(259,293)(260,294)(261,295)(262,296)(263,289)(264,290)(265,299)(266,300)(267,301)(268,302)(269,303)(270,304)(271,297)(272,298)(273,307)(274,308)(275,309)(276,310)(277,311)(278,312)(279,305)(280,306), (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), (1,218,127,186)(2,288,128,250)(3,224,121,192)(4,286,122,256)(5,222,123,190)(6,284,124,254)(7,220,125,188)(8,282,126,252)(9,305,77,275)(10,226,78,194)(11,311,79,273)(12,232,80,200)(13,309,73,279)(14,230,74,198)(15,307,75,277)(16,228,76,196)(17,310,146,280)(18,231,147,199)(19,308,148,278)(20,229,149,197)(21,306,150,276)(22,227,151,195)(23,312,152,274)(24,225,145,193)(25,283,51,253)(26,219,52,187)(27,281,53,251)(28,217,54,185)(29,287,55,249)(30,223,56,191)(31,285,49,255)(32,221,50,189)(33,291,59,261)(34,212,60,180)(35,289,61,259)(36,210,62,178)(37,295,63,257)(38,216,64,184)(39,293,57,263)(40,214,58,182)(41,299,67,269)(42,235,68,203)(43,297,69,267)(44,233,70,201)(45,303,71,265)(46,239,72,207)(47,301,65,271)(48,237,66,205)(81,157,244,95)(82,169,245,102)(83,155,246,93)(84,175,247,100)(85,153,248,91)(86,173,241,98)(87,159,242,89)(88,171,243,104)(90,320,160,163)(92,318,154,161)(94,316,156,167)(96,314,158,165)(97,313,172,164)(99,319,174,162)(101,317,176,168)(103,315,170,166)(105,215,131,183)(106,292,132,262)(107,213,133,181)(108,290,134,260)(109,211,135,179)(110,296,136,258)(111,209,129,177)(112,294,130,264)(113,238,139,206)(114,300,140,270)(115,236,141,204)(116,298,142,268)(117,234,143,202)(118,304,144,266)(119,240,137,208)(120,302,138,272)>;
G:=Group( (1,89,22,117,109)(2,90,23,118,110)(3,91,24,119,111)(4,92,17,120,112)(5,93,18,113,105)(6,94,19,114,106)(7,95,20,115,107)(8,96,21,116,108)(9,43,35,27,97)(10,44,36,28,98)(11,45,37,29,99)(12,46,38,30,100)(13,47,39,31,101)(14,48,40,32,102)(15,41,33,25,103)(16,42,34,26,104)(49,176,73,65,57)(50,169,74,66,58)(51,170,75,67,59)(52,171,76,68,60)(53,172,77,69,61)(54,173,78,70,62)(55,174,79,71,63)(56,175,80,72,64)(81,229,236,213,220)(82,230,237,214,221)(83,231,238,215,222)(84,232,239,216,223)(85,225,240,209,224)(86,226,233,210,217)(87,227,234,211,218)(88,228,235,212,219)(121,153,145,137,129)(122,154,146,138,130)(123,155,147,139,131)(124,156,148,140,132)(125,157,149,141,133)(126,158,150,142,134)(127,159,151,143,135)(128,160,152,144,136)(161,280,272,264,256)(162,273,265,257,249)(163,274,266,258,250)(164,275,267,259,251)(165,276,268,260,252)(166,277,269,261,253)(167,278,270,262,254)(168,279,271,263,255)(177,192,248,193,208)(178,185,241,194,201)(179,186,242,195,202)(180,187,243,196,203)(181,188,244,197,204)(182,189,245,198,205)(183,190,246,199,206)(184,191,247,200,207)(281,313,305,297,289)(282,314,306,298,290)(283,315,307,299,291)(284,316,308,300,292)(285,317,309,301,293)(286,318,310,302,294)(287,319,311,303,295)(288,320,312,304,296), (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,25)(8,26)(9,22)(10,23)(11,24)(12,17)(13,18)(14,19)(15,20)(16,21)(33,107)(34,108)(35,109)(36,110)(37,111)(38,112)(39,105)(40,106)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,113)(48,114)(49,123)(50,124)(51,125)(52,126)(53,127)(54,128)(55,121)(56,122)(57,131)(58,132)(59,133)(60,134)(61,135)(62,136)(63,129)(64,130)(65,139)(66,140)(67,141)(68,142)(69,143)(70,144)(71,137)(72,138)(73,147)(74,148)(75,149)(76,150)(77,151)(78,152)(79,145)(80,146)(81,315)(82,316)(83,317)(84,318)(85,319)(86,320)(87,313)(88,314)(89,97)(90,98)(91,99)(92,100)(93,101)(94,102)(95,103)(96,104)(153,174)(154,175)(155,176)(156,169)(157,170)(158,171)(159,172)(160,173)(161,247)(162,248)(163,241)(164,242)(165,243)(166,244)(167,245)(168,246)(177,257)(178,258)(179,259)(180,260)(181,261)(182,262)(183,263)(184,264)(185,250)(186,251)(187,252)(188,253)(189,254)(190,255)(191,256)(192,249)(193,273)(194,274)(195,275)(196,276)(197,277)(198,278)(199,279)(200,280)(201,266)(202,267)(203,268)(204,269)(205,270)(206,271)(207,272)(208,265)(209,295)(210,296)(211,289)(212,290)(213,291)(214,292)(215,293)(216,294)(217,288)(218,281)(219,282)(220,283)(221,284)(222,285)(223,286)(224,287)(225,311)(226,312)(227,305)(228,306)(229,307)(230,308)(231,309)(232,310)(233,304)(234,297)(235,298)(236,299)(237,300)(238,301)(239,302)(240,303), (1,123)(2,124)(3,125)(4,126)(5,127)(6,128)(7,121)(8,122)(9,73)(10,74)(11,75)(12,76)(13,77)(14,78)(15,79)(16,80)(17,150)(18,151)(19,152)(20,145)(21,146)(22,147)(23,148)(24,149)(25,55)(26,56)(27,49)(28,50)(29,51)(30,52)(31,53)(32,54)(33,63)(34,64)(35,57)(36,58)(37,59)(38,60)(39,61)(40,62)(41,71)(42,72)(43,65)(44,66)(45,67)(46,68)(47,69)(48,70)(81,248)(82,241)(83,242)(84,243)(85,244)(86,245)(87,246)(88,247)(89,155)(90,156)(91,157)(92,158)(93,159)(94,160)(95,153)(96,154)(97,176)(98,169)(99,170)(100,171)(101,172)(102,173)(103,174)(104,175)(105,135)(106,136)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,143)(114,144)(115,137)(116,138)(117,139)(118,140)(119,141)(120,142)(161,314)(162,315)(163,316)(164,317)(165,318)(166,319)(167,320)(168,313)(177,213)(178,214)(179,215)(180,216)(181,209)(182,210)(183,211)(184,212)(185,221)(186,222)(187,223)(188,224)(189,217)(190,218)(191,219)(192,220)(193,229)(194,230)(195,231)(196,232)(197,225)(198,226)(199,227)(200,228)(201,237)(202,238)(203,239)(204,240)(205,233)(206,234)(207,235)(208,236)(249,283)(250,284)(251,285)(252,286)(253,287)(254,288)(255,281)(256,282)(257,291)(258,292)(259,293)(260,294)(261,295)(262,296)(263,289)(264,290)(265,299)(266,300)(267,301)(268,302)(269,303)(270,304)(271,297)(272,298)(273,307)(274,308)(275,309)(276,310)(277,311)(278,312)(279,305)(280,306), (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), (1,218,127,186)(2,288,128,250)(3,224,121,192)(4,286,122,256)(5,222,123,190)(6,284,124,254)(7,220,125,188)(8,282,126,252)(9,305,77,275)(10,226,78,194)(11,311,79,273)(12,232,80,200)(13,309,73,279)(14,230,74,198)(15,307,75,277)(16,228,76,196)(17,310,146,280)(18,231,147,199)(19,308,148,278)(20,229,149,197)(21,306,150,276)(22,227,151,195)(23,312,152,274)(24,225,145,193)(25,283,51,253)(26,219,52,187)(27,281,53,251)(28,217,54,185)(29,287,55,249)(30,223,56,191)(31,285,49,255)(32,221,50,189)(33,291,59,261)(34,212,60,180)(35,289,61,259)(36,210,62,178)(37,295,63,257)(38,216,64,184)(39,293,57,263)(40,214,58,182)(41,299,67,269)(42,235,68,203)(43,297,69,267)(44,233,70,201)(45,303,71,265)(46,239,72,207)(47,301,65,271)(48,237,66,205)(81,157,244,95)(82,169,245,102)(83,155,246,93)(84,175,247,100)(85,153,248,91)(86,173,241,98)(87,159,242,89)(88,171,243,104)(90,320,160,163)(92,318,154,161)(94,316,156,167)(96,314,158,165)(97,313,172,164)(99,319,174,162)(101,317,176,168)(103,315,170,166)(105,215,131,183)(106,292,132,262)(107,213,133,181)(108,290,134,260)(109,211,135,179)(110,296,136,258)(111,209,129,177)(112,294,130,264)(113,238,139,206)(114,300,140,270)(115,236,141,204)(116,298,142,268)(117,234,143,202)(118,304,144,266)(119,240,137,208)(120,302,138,272) );
G=PermutationGroup([[(1,89,22,117,109),(2,90,23,118,110),(3,91,24,119,111),(4,92,17,120,112),(5,93,18,113,105),(6,94,19,114,106),(7,95,20,115,107),(8,96,21,116,108),(9,43,35,27,97),(10,44,36,28,98),(11,45,37,29,99),(12,46,38,30,100),(13,47,39,31,101),(14,48,40,32,102),(15,41,33,25,103),(16,42,34,26,104),(49,176,73,65,57),(50,169,74,66,58),(51,170,75,67,59),(52,171,76,68,60),(53,172,77,69,61),(54,173,78,70,62),(55,174,79,71,63),(56,175,80,72,64),(81,229,236,213,220),(82,230,237,214,221),(83,231,238,215,222),(84,232,239,216,223),(85,225,240,209,224),(86,226,233,210,217),(87,227,234,211,218),(88,228,235,212,219),(121,153,145,137,129),(122,154,146,138,130),(123,155,147,139,131),(124,156,148,140,132),(125,157,149,141,133),(126,158,150,142,134),(127,159,151,143,135),(128,160,152,144,136),(161,280,272,264,256),(162,273,265,257,249),(163,274,266,258,250),(164,275,267,259,251),(165,276,268,260,252),(166,277,269,261,253),(167,278,270,262,254),(168,279,271,263,255),(177,192,248,193,208),(178,185,241,194,201),(179,186,242,195,202),(180,187,243,196,203),(181,188,244,197,204),(182,189,245,198,205),(183,190,246,199,206),(184,191,247,200,207),(281,313,305,297,289),(282,314,306,298,290),(283,315,307,299,291),(284,316,308,300,292),(285,317,309,301,293),(286,318,310,302,294),(287,319,311,303,295),(288,320,312,304,296)], [(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,25),(8,26),(9,22),(10,23),(11,24),(12,17),(13,18),(14,19),(15,20),(16,21),(33,107),(34,108),(35,109),(36,110),(37,111),(38,112),(39,105),(40,106),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,113),(48,114),(49,123),(50,124),(51,125),(52,126),(53,127),(54,128),(55,121),(56,122),(57,131),(58,132),(59,133),(60,134),(61,135),(62,136),(63,129),(64,130),(65,139),(66,140),(67,141),(68,142),(69,143),(70,144),(71,137),(72,138),(73,147),(74,148),(75,149),(76,150),(77,151),(78,152),(79,145),(80,146),(81,315),(82,316),(83,317),(84,318),(85,319),(86,320),(87,313),(88,314),(89,97),(90,98),(91,99),(92,100),(93,101),(94,102),(95,103),(96,104),(153,174),(154,175),(155,176),(156,169),(157,170),(158,171),(159,172),(160,173),(161,247),(162,248),(163,241),(164,242),(165,243),(166,244),(167,245),(168,246),(177,257),(178,258),(179,259),(180,260),(181,261),(182,262),(183,263),(184,264),(185,250),(186,251),(187,252),(188,253),(189,254),(190,255),(191,256),(192,249),(193,273),(194,274),(195,275),(196,276),(197,277),(198,278),(199,279),(200,280),(201,266),(202,267),(203,268),(204,269),(205,270),(206,271),(207,272),(208,265),(209,295),(210,296),(211,289),(212,290),(213,291),(214,292),(215,293),(216,294),(217,288),(218,281),(219,282),(220,283),(221,284),(222,285),(223,286),(224,287),(225,311),(226,312),(227,305),(228,306),(229,307),(230,308),(231,309),(232,310),(233,304),(234,297),(235,298),(236,299),(237,300),(238,301),(239,302),(240,303)], [(1,123),(2,124),(3,125),(4,126),(5,127),(6,128),(7,121),(8,122),(9,73),(10,74),(11,75),(12,76),(13,77),(14,78),(15,79),(16,80),(17,150),(18,151),(19,152),(20,145),(21,146),(22,147),(23,148),(24,149),(25,55),(26,56),(27,49),(28,50),(29,51),(30,52),(31,53),(32,54),(33,63),(34,64),(35,57),(36,58),(37,59),(38,60),(39,61),(40,62),(41,71),(42,72),(43,65),(44,66),(45,67),(46,68),(47,69),(48,70),(81,248),(82,241),(83,242),(84,243),(85,244),(86,245),(87,246),(88,247),(89,155),(90,156),(91,157),(92,158),(93,159),(94,160),(95,153),(96,154),(97,176),(98,169),(99,170),(100,171),(101,172),(102,173),(103,174),(104,175),(105,135),(106,136),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(113,143),(114,144),(115,137),(116,138),(117,139),(118,140),(119,141),(120,142),(161,314),(162,315),(163,316),(164,317),(165,318),(166,319),(167,320),(168,313),(177,213),(178,214),(179,215),(180,216),(181,209),(182,210),(183,211),(184,212),(185,221),(186,222),(187,223),(188,224),(189,217),(190,218),(191,219),(192,220),(193,229),(194,230),(195,231),(196,232),(197,225),(198,226),(199,227),(200,228),(201,237),(202,238),(203,239),(204,240),(205,233),(206,234),(207,235),(208,236),(249,283),(250,284),(251,285),(252,286),(253,287),(254,288),(255,281),(256,282),(257,291),(258,292),(259,293),(260,294),(261,295),(262,296),(263,289),(264,290),(265,299),(266,300),(267,301),(268,302),(269,303),(270,304),(271,297),(272,298),(273,307),(274,308),(275,309),(276,310),(277,311),(278,312),(279,305),(280,306)], [(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)], [(1,218,127,186),(2,288,128,250),(3,224,121,192),(4,286,122,256),(5,222,123,190),(6,284,124,254),(7,220,125,188),(8,282,126,252),(9,305,77,275),(10,226,78,194),(11,311,79,273),(12,232,80,200),(13,309,73,279),(14,230,74,198),(15,307,75,277),(16,228,76,196),(17,310,146,280),(18,231,147,199),(19,308,148,278),(20,229,149,197),(21,306,150,276),(22,227,151,195),(23,312,152,274),(24,225,145,193),(25,283,51,253),(26,219,52,187),(27,281,53,251),(28,217,54,185),(29,287,55,249),(30,223,56,191),(31,285,49,255),(32,221,50,189),(33,291,59,261),(34,212,60,180),(35,289,61,259),(36,210,62,178),(37,295,63,257),(38,216,64,184),(39,293,57,263),(40,214,58,182),(41,299,67,269),(42,235,68,203),(43,297,69,267),(44,233,70,201),(45,303,71,265),(46,239,72,207),(47,301,65,271),(48,237,66,205),(81,157,244,95),(82,169,245,102),(83,155,246,93),(84,175,247,100),(85,153,248,91),(86,173,241,98),(87,159,242,89),(88,171,243,104),(90,320,160,163),(92,318,154,161),(94,316,156,167),(96,314,158,165),(97,313,172,164),(99,319,174,162),(101,317,176,168),(103,315,170,166),(105,215,131,183),(106,292,132,262),(107,213,133,181),(108,290,134,260),(109,211,135,179),(110,296,136,258),(111,209,129,177),(112,294,130,264),(113,238,139,206),(114,300,140,270),(115,236,141,204),(116,298,142,268),(117,234,143,202),(118,304,144,266),(119,240,137,208),(120,302,138,272)]])
140 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 5A | 5B | 5C | 5D | 8A | ··· | 8H | 10A | ··· | 10AB | 20A | ··· | 20P | 20Q | ··· | 20AV | 40A | ··· | 40AF |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 | 40 | ··· | 40 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | ··· | 2 |
140 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | + | - | ||||||||||||||
| image | C1 | C2 | C2 | C4 | C4 | C5 | C10 | C10 | C20 | C20 | D4 | Q8 | D4 | D8 | SD16 | Q16 | C5×D4 | C5×Q8 | C5×D4 | C5×D8 | C5×SD16 | C5×Q16 |
| kernel | C5×C22.4Q16 | C10×C4⋊C4 | C22×C40 | C5×C4⋊C4 | C2×C40 | C22.4Q16 | C2×C4⋊C4 | C22×C8 | C4⋊C4 | C2×C8 | C2×C20 | C2×C20 | C22×C10 | C2×C10 | C2×C10 | C2×C10 | C2×C4 | C2×C4 | C23 | C22 | C22 | C22 |
| # reps | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 32 | 16 | 2 | 1 | 1 | 2 | 4 | 2 | 8 | 4 | 4 | 8 | 16 | 8 |
Matrix representation of C5×C22.4Q16 ►in GL4(𝔽41) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 1 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 9 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 15 | 26 |
| 0 | 0 | 15 | 15 |
| 1 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 19 | 3 |
| 0 | 0 | 3 | 22 |
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,32,0,0,0,0,15,15,0,0,26,15],[1,0,0,0,0,32,0,0,0,0,19,3,0,0,3,22] >;
C5×C22.4Q16 in GAP, Magma, Sage, TeX
C_5\times C_2^2._4Q_{16} % in TeX
G:=Group("C5xC2^2.4Q16"); // GroupNames label
G:=SmallGroup(320,145);
// by ID
G=gap.SmallGroup(320,145);
# by ID
G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,248,10085,124]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^8=1,e^2=c*d^4,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=b*d^-1>;
// generators/relations