direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C22×C5⋊Q16, C20.32C24, Dic10.28C23, (C2×C10)⋊9Q16, C10⋊3(C2×Q16), C5⋊3(C22×Q16), (C2×C20).212D4, C20.256(C2×D4), C4.32(C23×D5), (C22×Q8).7D5, C5⋊2C8.29C23, (C2×Q8).185D10, Q8.21(C22×D5), (C5×Q8).21C23, (C2×C20).549C23, (C22×C4).382D10, C10.151(C22×D4), (C22×C10).211D4, C23.107(C5⋊D4), (Q8×C10).230C22, (C22×C20).281C22, (C22×Dic10).19C2, (C2×Dic10).314C22, (Q8×C2×C10).6C2, C4.26(C2×C5⋊D4), (C2×C10).586(C2×D4), C2.24(C22×C5⋊D4), (C2×C4).155(C5⋊D4), (C22×C5⋊2C8).14C2, (C2×C4).630(C22×D5), C22.114(C2×C5⋊D4), (C2×C5⋊2C8).295C22, SmallGroup(320,1481)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 702 in 258 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C4 [×8], C22 [×7], C5, C8 [×4], C2×C4 [×6], C2×C4 [×12], Q8 [×4], Q8 [×16], C23, C10, C10 [×6], C2×C8 [×6], Q16 [×16], C22×C4, C22×C4 [×2], C2×Q8 [×6], C2×Q8 [×12], Dic5 [×4], C20, C20 [×3], C20 [×4], C2×C10 [×7], C22×C8, C2×Q16 [×12], C22×Q8, C22×Q8, C5⋊2C8 [×4], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×6], C2×C20 [×6], C5×Q8 [×4], C5×Q8 [×6], C22×C10, C22×Q16, C2×C5⋊2C8 [×6], C5⋊Q16 [×16], C2×Dic10 [×6], C2×Dic10 [×3], C22×Dic5, C22×C20, C22×C20, Q8×C10 [×6], Q8×C10 [×3], C22×C5⋊2C8, C2×C5⋊Q16 [×12], C22×Dic10, Q8×C2×C10, C22×C5⋊Q16
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, Q16 [×4], C2×D4 [×6], C24, D10 [×7], C2×Q16 [×6], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C22×Q16, C5⋊Q16 [×4], C2×C5⋊D4 [×6], C23×D5, C2×C5⋊Q16 [×6], C22×C5⋊D4, C22×C5⋊Q16
Generators and relations
G = < a,b,c,d,e | a2=b2=c5=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=d-1 >
►Regular action on 320 points
Generators in S320
(1 204)(2 205)(3 206)(4 207)(5 208)(6 201)(7 202)(8 203)(9 118)(10 119)(11 120)(12 113)(13 114)(14 115)(15 116)(16 117)(17 183)(18 184)(19 177)(20 178)(21 179)(22 180)(23 181)(24 182)(25 69)(26 70)(27 71)(28 72)(29 65)(30 66)(31 67)(32 68)(33 77)(34 78)(35 79)(36 80)(37 73)(38 74)(39 75)(40 76)(41 246)(42 247)(43 248)(44 241)(45 242)(46 243)(47 244)(48 245)(49 149)(50 150)(51 151)(52 152)(53 145)(54 146)(55 147)(56 148)(57 249)(58 250)(59 251)(60 252)(61 253)(62 254)(63 255)(64 256)(81 125)(82 126)(83 127)(84 128)(85 121)(86 122)(87 123)(88 124)(89 133)(90 134)(91 135)(92 136)(93 129)(94 130)(95 131)(96 132)(97 141)(98 142)(99 143)(100 144)(101 137)(102 138)(103 139)(104 140)(105 234)(106 235)(107 236)(108 237)(109 238)(110 239)(111 240)(112 233)(153 186)(154 187)(155 188)(156 189)(157 190)(158 191)(159 192)(160 185)(161 194)(162 195)(163 196)(164 197)(165 198)(166 199)(167 200)(168 193)(169 213)(170 214)(171 215)(172 216)(173 209)(174 210)(175 211)(176 212)(217 286)(218 287)(219 288)(220 281)(221 282)(222 283)(223 284)(224 285)(225 294)(226 295)(227 296)(228 289)(229 290)(230 291)(231 292)(232 293)(257 301)(258 302)(259 303)(260 304)(261 297)(262 298)(263 299)(264 300)(265 309)(266 310)(267 311)(268 312)(269 305)(270 306)(271 307)(272 308)(273 317)(274 318)(275 319)(276 320)(277 313)(278 314)(279 315)(280 316)
(1 47)(2 48)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 311)(10 312)(11 305)(12 306)(13 307)(14 308)(15 309)(16 310)(17 197)(18 198)(19 199)(20 200)(21 193)(22 194)(23 195)(24 196)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(49 65)(50 66)(51 67)(52 68)(53 69)(54 70)(55 71)(56 72)(57 173)(58 174)(59 175)(60 176)(61 169)(62 170)(63 171)(64 172)(73 190)(74 191)(75 192)(76 185)(77 186)(78 187)(79 188)(80 189)(81 217)(82 218)(83 219)(84 220)(85 221)(86 222)(87 223)(88 224)(89 238)(90 239)(91 240)(92 233)(93 234)(94 235)(95 236)(96 237)(97 228)(98 229)(99 230)(100 231)(101 232)(102 225)(103 226)(104 227)(105 129)(106 130)(107 131)(108 132)(109 133)(110 134)(111 135)(112 136)(113 270)(114 271)(115 272)(116 265)(117 266)(118 267)(119 268)(120 269)(121 282)(122 283)(123 284)(124 285)(125 286)(126 287)(127 288)(128 281)(137 293)(138 294)(139 295)(140 296)(141 289)(142 290)(143 291)(144 292)(161 180)(162 181)(163 182)(164 183)(165 184)(166 177)(167 178)(168 179)(201 241)(202 242)(203 243)(204 244)(205 245)(206 246)(207 247)(208 248)(209 249)(210 250)(211 251)(212 252)(213 253)(214 254)(215 255)(216 256)(257 275)(258 276)(259 277)(260 278)(261 279)(262 280)(263 273)(264 274)(297 315)(298 316)(299 317)(300 318)(301 319)(302 320)(303 313)(304 314)
(1 110 167 117 291)(2 292 118 168 111)(3 112 161 119 293)(4 294 120 162 105)(5 106 163 113 295)(6 296 114 164 107)(7 108 165 115 289)(8 290 116 166 109)(9 193 240 205 231)(10 232 206 233 194)(11 195 234 207 225)(12 226 208 235 196)(13 197 236 201 227)(14 228 202 237 198)(15 199 238 203 229)(16 230 204 239 200)(17 95 241 104 307)(18 308 97 242 96)(19 89 243 98 309)(20 310 99 244 90)(21 91 245 100 311)(22 312 101 246 92)(23 93 247 102 305)(24 306 103 248 94)(25 313 36 214 224)(26 217 215 37 314)(27 315 38 216 218)(28 219 209 39 316)(29 317 40 210 220)(30 221 211 33 318)(31 319 34 212 222)(32 223 213 35 320)(41 136 180 268 137)(42 138 269 181 129)(43 130 182 270 139)(44 140 271 183 131)(45 132 184 272 141)(46 142 265 177 133)(47 134 178 266 143)(48 144 267 179 135)(49 263 185 58 128)(50 121 59 186 264)(51 257 187 60 122)(52 123 61 188 258)(53 259 189 62 124)(54 125 63 190 260)(55 261 191 64 126)(56 127 57 192 262)(65 273 76 174 281)(66 282 175 77 274)(67 275 78 176 283)(68 284 169 79 276)(69 277 80 170 285)(70 286 171 73 278)(71 279 74 172 287)(72 288 173 75 280)(81 255 157 304 146)(82 147 297 158 256)(83 249 159 298 148)(84 149 299 160 250)(85 251 153 300 150)(86 151 301 154 252)(87 253 155 302 152)(88 145 303 156 254)
(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 84 5 88)(2 83 6 87)(3 82 7 86)(4 81 8 85)(9 192 13 188)(10 191 14 187)(11 190 15 186)(12 189 16 185)(17 276 21 280)(18 275 22 279)(19 274 23 278)(20 273 24 277)(25 134 29 130)(26 133 30 129)(27 132 31 136)(28 131 32 135)(33 269 37 265)(34 268 38 272)(35 267 39 271)(36 266 40 270)(41 218 45 222)(42 217 46 221)(43 224 47 220)(44 223 48 219)(49 235 53 239)(50 234 54 238)(51 233 55 237)(52 240 56 236)(57 227 61 231)(58 226 62 230)(59 225 63 229)(60 232 64 228)(65 94 69 90)(66 93 70 89)(67 92 71 96)(68 91 72 95)(73 309 77 305)(74 308 78 312)(75 307 79 311)(76 306 80 310)(97 176 101 172)(98 175 102 171)(99 174 103 170)(100 173 104 169)(105 146 109 150)(106 145 110 149)(107 152 111 148)(108 151 112 147)(113 156 117 160)(114 155 118 159)(115 154 119 158)(116 153 120 157)(121 207 125 203)(122 206 126 202)(123 205 127 201)(124 204 128 208)(137 216 141 212)(138 215 142 211)(139 214 143 210)(140 213 144 209)(161 297 165 301)(162 304 166 300)(163 303 167 299)(164 302 168 298)(177 318 181 314)(178 317 182 313)(179 316 183 320)(180 315 184 319)(193 262 197 258)(194 261 198 257)(195 260 199 264)(196 259 200 263)(241 284 245 288)(242 283 246 287)(243 282 247 286)(244 281 248 285)(249 296 253 292)(250 295 254 291)(251 294 255 290)(252 293 256 289)
G:=sub<Sym(320)| (1,204)(2,205)(3,206)(4,207)(5,208)(6,201)(7,202)(8,203)(9,118)(10,119)(11,120)(12,113)(13,114)(14,115)(15,116)(16,117)(17,183)(18,184)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,77)(34,78)(35,79)(36,80)(37,73)(38,74)(39,75)(40,76)(41,246)(42,247)(43,248)(44,241)(45,242)(46,243)(47,244)(48,245)(49,149)(50,150)(51,151)(52,152)(53,145)(54,146)(55,147)(56,148)(57,249)(58,250)(59,251)(60,252)(61,253)(62,254)(63,255)(64,256)(81,125)(82,126)(83,127)(84,128)(85,121)(86,122)(87,123)(88,124)(89,133)(90,134)(91,135)(92,136)(93,129)(94,130)(95,131)(96,132)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,233)(153,186)(154,187)(155,188)(156,189)(157,190)(158,191)(159,192)(160,185)(161,194)(162,195)(163,196)(164,197)(165,198)(166,199)(167,200)(168,193)(169,213)(170,214)(171,215)(172,216)(173,209)(174,210)(175,211)(176,212)(217,286)(218,287)(219,288)(220,281)(221,282)(222,283)(223,284)(224,285)(225,294)(226,295)(227,296)(228,289)(229,290)(230,291)(231,292)(232,293)(257,301)(258,302)(259,303)(260,304)(261,297)(262,298)(263,299)(264,300)(265,309)(266,310)(267,311)(268,312)(269,305)(270,306)(271,307)(272,308)(273,317)(274,318)(275,319)(276,320)(277,313)(278,314)(279,315)(280,316), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,311)(10,312)(11,305)(12,306)(13,307)(14,308)(15,309)(16,310)(17,197)(18,198)(19,199)(20,200)(21,193)(22,194)(23,195)(24,196)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,173)(58,174)(59,175)(60,176)(61,169)(62,170)(63,171)(64,172)(73,190)(74,191)(75,192)(76,185)(77,186)(78,187)(79,188)(80,189)(81,217)(82,218)(83,219)(84,220)(85,221)(86,222)(87,223)(88,224)(89,238)(90,239)(91,240)(92,233)(93,234)(94,235)(95,236)(96,237)(97,228)(98,229)(99,230)(100,231)(101,232)(102,225)(103,226)(104,227)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,270)(114,271)(115,272)(116,265)(117,266)(118,267)(119,268)(120,269)(121,282)(122,283)(123,284)(124,285)(125,286)(126,287)(127,288)(128,281)(137,293)(138,294)(139,295)(140,296)(141,289)(142,290)(143,291)(144,292)(161,180)(162,181)(163,182)(164,183)(165,184)(166,177)(167,178)(168,179)(201,241)(202,242)(203,243)(204,244)(205,245)(206,246)(207,247)(208,248)(209,249)(210,250)(211,251)(212,252)(213,253)(214,254)(215,255)(216,256)(257,275)(258,276)(259,277)(260,278)(261,279)(262,280)(263,273)(264,274)(297,315)(298,316)(299,317)(300,318)(301,319)(302,320)(303,313)(304,314), (1,110,167,117,291)(2,292,118,168,111)(3,112,161,119,293)(4,294,120,162,105)(5,106,163,113,295)(6,296,114,164,107)(7,108,165,115,289)(8,290,116,166,109)(9,193,240,205,231)(10,232,206,233,194)(11,195,234,207,225)(12,226,208,235,196)(13,197,236,201,227)(14,228,202,237,198)(15,199,238,203,229)(16,230,204,239,200)(17,95,241,104,307)(18,308,97,242,96)(19,89,243,98,309)(20,310,99,244,90)(21,91,245,100,311)(22,312,101,246,92)(23,93,247,102,305)(24,306,103,248,94)(25,313,36,214,224)(26,217,215,37,314)(27,315,38,216,218)(28,219,209,39,316)(29,317,40,210,220)(30,221,211,33,318)(31,319,34,212,222)(32,223,213,35,320)(41,136,180,268,137)(42,138,269,181,129)(43,130,182,270,139)(44,140,271,183,131)(45,132,184,272,141)(46,142,265,177,133)(47,134,178,266,143)(48,144,267,179,135)(49,263,185,58,128)(50,121,59,186,264)(51,257,187,60,122)(52,123,61,188,258)(53,259,189,62,124)(54,125,63,190,260)(55,261,191,64,126)(56,127,57,192,262)(65,273,76,174,281)(66,282,175,77,274)(67,275,78,176,283)(68,284,169,79,276)(69,277,80,170,285)(70,286,171,73,278)(71,279,74,172,287)(72,288,173,75,280)(81,255,157,304,146)(82,147,297,158,256)(83,249,159,298,148)(84,149,299,160,250)(85,251,153,300,150)(86,151,301,154,252)(87,253,155,302,152)(88,145,303,156,254), (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), (1,84,5,88)(2,83,6,87)(3,82,7,86)(4,81,8,85)(9,192,13,188)(10,191,14,187)(11,190,15,186)(12,189,16,185)(17,276,21,280)(18,275,22,279)(19,274,23,278)(20,273,24,277)(25,134,29,130)(26,133,30,129)(27,132,31,136)(28,131,32,135)(33,269,37,265)(34,268,38,272)(35,267,39,271)(36,266,40,270)(41,218,45,222)(42,217,46,221)(43,224,47,220)(44,223,48,219)(49,235,53,239)(50,234,54,238)(51,233,55,237)(52,240,56,236)(57,227,61,231)(58,226,62,230)(59,225,63,229)(60,232,64,228)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(73,309,77,305)(74,308,78,312)(75,307,79,311)(76,306,80,310)(97,176,101,172)(98,175,102,171)(99,174,103,170)(100,173,104,169)(105,146,109,150)(106,145,110,149)(107,152,111,148)(108,151,112,147)(113,156,117,160)(114,155,118,159)(115,154,119,158)(116,153,120,157)(121,207,125,203)(122,206,126,202)(123,205,127,201)(124,204,128,208)(137,216,141,212)(138,215,142,211)(139,214,143,210)(140,213,144,209)(161,297,165,301)(162,304,166,300)(163,303,167,299)(164,302,168,298)(177,318,181,314)(178,317,182,313)(179,316,183,320)(180,315,184,319)(193,262,197,258)(194,261,198,257)(195,260,199,264)(196,259,200,263)(241,284,245,288)(242,283,246,287)(243,282,247,286)(244,281,248,285)(249,296,253,292)(250,295,254,291)(251,294,255,290)(252,293,256,289)>;
G:=Group( (1,204)(2,205)(3,206)(4,207)(5,208)(6,201)(7,202)(8,203)(9,118)(10,119)(11,120)(12,113)(13,114)(14,115)(15,116)(16,117)(17,183)(18,184)(19,177)(20,178)(21,179)(22,180)(23,181)(24,182)(25,69)(26,70)(27,71)(28,72)(29,65)(30,66)(31,67)(32,68)(33,77)(34,78)(35,79)(36,80)(37,73)(38,74)(39,75)(40,76)(41,246)(42,247)(43,248)(44,241)(45,242)(46,243)(47,244)(48,245)(49,149)(50,150)(51,151)(52,152)(53,145)(54,146)(55,147)(56,148)(57,249)(58,250)(59,251)(60,252)(61,253)(62,254)(63,255)(64,256)(81,125)(82,126)(83,127)(84,128)(85,121)(86,122)(87,123)(88,124)(89,133)(90,134)(91,135)(92,136)(93,129)(94,130)(95,131)(96,132)(97,141)(98,142)(99,143)(100,144)(101,137)(102,138)(103,139)(104,140)(105,234)(106,235)(107,236)(108,237)(109,238)(110,239)(111,240)(112,233)(153,186)(154,187)(155,188)(156,189)(157,190)(158,191)(159,192)(160,185)(161,194)(162,195)(163,196)(164,197)(165,198)(166,199)(167,200)(168,193)(169,213)(170,214)(171,215)(172,216)(173,209)(174,210)(175,211)(176,212)(217,286)(218,287)(219,288)(220,281)(221,282)(222,283)(223,284)(224,285)(225,294)(226,295)(227,296)(228,289)(229,290)(230,291)(231,292)(232,293)(257,301)(258,302)(259,303)(260,304)(261,297)(262,298)(263,299)(264,300)(265,309)(266,310)(267,311)(268,312)(269,305)(270,306)(271,307)(272,308)(273,317)(274,318)(275,319)(276,320)(277,313)(278,314)(279,315)(280,316), (1,47)(2,48)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,311)(10,312)(11,305)(12,306)(13,307)(14,308)(15,309)(16,310)(17,197)(18,198)(19,199)(20,200)(21,193)(22,194)(23,195)(24,196)(25,145)(26,146)(27,147)(28,148)(29,149)(30,150)(31,151)(32,152)(33,153)(34,154)(35,155)(36,156)(37,157)(38,158)(39,159)(40,160)(49,65)(50,66)(51,67)(52,68)(53,69)(54,70)(55,71)(56,72)(57,173)(58,174)(59,175)(60,176)(61,169)(62,170)(63,171)(64,172)(73,190)(74,191)(75,192)(76,185)(77,186)(78,187)(79,188)(80,189)(81,217)(82,218)(83,219)(84,220)(85,221)(86,222)(87,223)(88,224)(89,238)(90,239)(91,240)(92,233)(93,234)(94,235)(95,236)(96,237)(97,228)(98,229)(99,230)(100,231)(101,232)(102,225)(103,226)(104,227)(105,129)(106,130)(107,131)(108,132)(109,133)(110,134)(111,135)(112,136)(113,270)(114,271)(115,272)(116,265)(117,266)(118,267)(119,268)(120,269)(121,282)(122,283)(123,284)(124,285)(125,286)(126,287)(127,288)(128,281)(137,293)(138,294)(139,295)(140,296)(141,289)(142,290)(143,291)(144,292)(161,180)(162,181)(163,182)(164,183)(165,184)(166,177)(167,178)(168,179)(201,241)(202,242)(203,243)(204,244)(205,245)(206,246)(207,247)(208,248)(209,249)(210,250)(211,251)(212,252)(213,253)(214,254)(215,255)(216,256)(257,275)(258,276)(259,277)(260,278)(261,279)(262,280)(263,273)(264,274)(297,315)(298,316)(299,317)(300,318)(301,319)(302,320)(303,313)(304,314), (1,110,167,117,291)(2,292,118,168,111)(3,112,161,119,293)(4,294,120,162,105)(5,106,163,113,295)(6,296,114,164,107)(7,108,165,115,289)(8,290,116,166,109)(9,193,240,205,231)(10,232,206,233,194)(11,195,234,207,225)(12,226,208,235,196)(13,197,236,201,227)(14,228,202,237,198)(15,199,238,203,229)(16,230,204,239,200)(17,95,241,104,307)(18,308,97,242,96)(19,89,243,98,309)(20,310,99,244,90)(21,91,245,100,311)(22,312,101,246,92)(23,93,247,102,305)(24,306,103,248,94)(25,313,36,214,224)(26,217,215,37,314)(27,315,38,216,218)(28,219,209,39,316)(29,317,40,210,220)(30,221,211,33,318)(31,319,34,212,222)(32,223,213,35,320)(41,136,180,268,137)(42,138,269,181,129)(43,130,182,270,139)(44,140,271,183,131)(45,132,184,272,141)(46,142,265,177,133)(47,134,178,266,143)(48,144,267,179,135)(49,263,185,58,128)(50,121,59,186,264)(51,257,187,60,122)(52,123,61,188,258)(53,259,189,62,124)(54,125,63,190,260)(55,261,191,64,126)(56,127,57,192,262)(65,273,76,174,281)(66,282,175,77,274)(67,275,78,176,283)(68,284,169,79,276)(69,277,80,170,285)(70,286,171,73,278)(71,279,74,172,287)(72,288,173,75,280)(81,255,157,304,146)(82,147,297,158,256)(83,249,159,298,148)(84,149,299,160,250)(85,251,153,300,150)(86,151,301,154,252)(87,253,155,302,152)(88,145,303,156,254), (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), (1,84,5,88)(2,83,6,87)(3,82,7,86)(4,81,8,85)(9,192,13,188)(10,191,14,187)(11,190,15,186)(12,189,16,185)(17,276,21,280)(18,275,22,279)(19,274,23,278)(20,273,24,277)(25,134,29,130)(26,133,30,129)(27,132,31,136)(28,131,32,135)(33,269,37,265)(34,268,38,272)(35,267,39,271)(36,266,40,270)(41,218,45,222)(42,217,46,221)(43,224,47,220)(44,223,48,219)(49,235,53,239)(50,234,54,238)(51,233,55,237)(52,240,56,236)(57,227,61,231)(58,226,62,230)(59,225,63,229)(60,232,64,228)(65,94,69,90)(66,93,70,89)(67,92,71,96)(68,91,72,95)(73,309,77,305)(74,308,78,312)(75,307,79,311)(76,306,80,310)(97,176,101,172)(98,175,102,171)(99,174,103,170)(100,173,104,169)(105,146,109,150)(106,145,110,149)(107,152,111,148)(108,151,112,147)(113,156,117,160)(114,155,118,159)(115,154,119,158)(116,153,120,157)(121,207,125,203)(122,206,126,202)(123,205,127,201)(124,204,128,208)(137,216,141,212)(138,215,142,211)(139,214,143,210)(140,213,144,209)(161,297,165,301)(162,304,166,300)(163,303,167,299)(164,302,168,298)(177,318,181,314)(178,317,182,313)(179,316,183,320)(180,315,184,319)(193,262,197,258)(194,261,198,257)(195,260,199,264)(196,259,200,263)(241,284,245,288)(242,283,246,287)(243,282,247,286)(244,281,248,285)(249,296,253,292)(250,295,254,291)(251,294,255,290)(252,293,256,289) );
G=PermutationGroup([(1,204),(2,205),(3,206),(4,207),(5,208),(6,201),(7,202),(8,203),(9,118),(10,119),(11,120),(12,113),(13,114),(14,115),(15,116),(16,117),(17,183),(18,184),(19,177),(20,178),(21,179),(22,180),(23,181),(24,182),(25,69),(26,70),(27,71),(28,72),(29,65),(30,66),(31,67),(32,68),(33,77),(34,78),(35,79),(36,80),(37,73),(38,74),(39,75),(40,76),(41,246),(42,247),(43,248),(44,241),(45,242),(46,243),(47,244),(48,245),(49,149),(50,150),(51,151),(52,152),(53,145),(54,146),(55,147),(56,148),(57,249),(58,250),(59,251),(60,252),(61,253),(62,254),(63,255),(64,256),(81,125),(82,126),(83,127),(84,128),(85,121),(86,122),(87,123),(88,124),(89,133),(90,134),(91,135),(92,136),(93,129),(94,130),(95,131),(96,132),(97,141),(98,142),(99,143),(100,144),(101,137),(102,138),(103,139),(104,140),(105,234),(106,235),(107,236),(108,237),(109,238),(110,239),(111,240),(112,233),(153,186),(154,187),(155,188),(156,189),(157,190),(158,191),(159,192),(160,185),(161,194),(162,195),(163,196),(164,197),(165,198),(166,199),(167,200),(168,193),(169,213),(170,214),(171,215),(172,216),(173,209),(174,210),(175,211),(176,212),(217,286),(218,287),(219,288),(220,281),(221,282),(222,283),(223,284),(224,285),(225,294),(226,295),(227,296),(228,289),(229,290),(230,291),(231,292),(232,293),(257,301),(258,302),(259,303),(260,304),(261,297),(262,298),(263,299),(264,300),(265,309),(266,310),(267,311),(268,312),(269,305),(270,306),(271,307),(272,308),(273,317),(274,318),(275,319),(276,320),(277,313),(278,314),(279,315),(280,316)], [(1,47),(2,48),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,311),(10,312),(11,305),(12,306),(13,307),(14,308),(15,309),(16,310),(17,197),(18,198),(19,199),(20,200),(21,193),(22,194),(23,195),(24,196),(25,145),(26,146),(27,147),(28,148),(29,149),(30,150),(31,151),(32,152),(33,153),(34,154),(35,155),(36,156),(37,157),(38,158),(39,159),(40,160),(49,65),(50,66),(51,67),(52,68),(53,69),(54,70),(55,71),(56,72),(57,173),(58,174),(59,175),(60,176),(61,169),(62,170),(63,171),(64,172),(73,190),(74,191),(75,192),(76,185),(77,186),(78,187),(79,188),(80,189),(81,217),(82,218),(83,219),(84,220),(85,221),(86,222),(87,223),(88,224),(89,238),(90,239),(91,240),(92,233),(93,234),(94,235),(95,236),(96,237),(97,228),(98,229),(99,230),(100,231),(101,232),(102,225),(103,226),(104,227),(105,129),(106,130),(107,131),(108,132),(109,133),(110,134),(111,135),(112,136),(113,270),(114,271),(115,272),(116,265),(117,266),(118,267),(119,268),(120,269),(121,282),(122,283),(123,284),(124,285),(125,286),(126,287),(127,288),(128,281),(137,293),(138,294),(139,295),(140,296),(141,289),(142,290),(143,291),(144,292),(161,180),(162,181),(163,182),(164,183),(165,184),(166,177),(167,178),(168,179),(201,241),(202,242),(203,243),(204,244),(205,245),(206,246),(207,247),(208,248),(209,249),(210,250),(211,251),(212,252),(213,253),(214,254),(215,255),(216,256),(257,275),(258,276),(259,277),(260,278),(261,279),(262,280),(263,273),(264,274),(297,315),(298,316),(299,317),(300,318),(301,319),(302,320),(303,313),(304,314)], [(1,110,167,117,291),(2,292,118,168,111),(3,112,161,119,293),(4,294,120,162,105),(5,106,163,113,295),(6,296,114,164,107),(7,108,165,115,289),(8,290,116,166,109),(9,193,240,205,231),(10,232,206,233,194),(11,195,234,207,225),(12,226,208,235,196),(13,197,236,201,227),(14,228,202,237,198),(15,199,238,203,229),(16,230,204,239,200),(17,95,241,104,307),(18,308,97,242,96),(19,89,243,98,309),(20,310,99,244,90),(21,91,245,100,311),(22,312,101,246,92),(23,93,247,102,305),(24,306,103,248,94),(25,313,36,214,224),(26,217,215,37,314),(27,315,38,216,218),(28,219,209,39,316),(29,317,40,210,220),(30,221,211,33,318),(31,319,34,212,222),(32,223,213,35,320),(41,136,180,268,137),(42,138,269,181,129),(43,130,182,270,139),(44,140,271,183,131),(45,132,184,272,141),(46,142,265,177,133),(47,134,178,266,143),(48,144,267,179,135),(49,263,185,58,128),(50,121,59,186,264),(51,257,187,60,122),(52,123,61,188,258),(53,259,189,62,124),(54,125,63,190,260),(55,261,191,64,126),(56,127,57,192,262),(65,273,76,174,281),(66,282,175,77,274),(67,275,78,176,283),(68,284,169,79,276),(69,277,80,170,285),(70,286,171,73,278),(71,279,74,172,287),(72,288,173,75,280),(81,255,157,304,146),(82,147,297,158,256),(83,249,159,298,148),(84,149,299,160,250),(85,251,153,300,150),(86,151,301,154,252),(87,253,155,302,152),(88,145,303,156,254)], [(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)], [(1,84,5,88),(2,83,6,87),(3,82,7,86),(4,81,8,85),(9,192,13,188),(10,191,14,187),(11,190,15,186),(12,189,16,185),(17,276,21,280),(18,275,22,279),(19,274,23,278),(20,273,24,277),(25,134,29,130),(26,133,30,129),(27,132,31,136),(28,131,32,135),(33,269,37,265),(34,268,38,272),(35,267,39,271),(36,266,40,270),(41,218,45,222),(42,217,46,221),(43,224,47,220),(44,223,48,219),(49,235,53,239),(50,234,54,238),(51,233,55,237),(52,240,56,236),(57,227,61,231),(58,226,62,230),(59,225,63,229),(60,232,64,228),(65,94,69,90),(66,93,70,89),(67,92,71,96),(68,91,72,95),(73,309,77,305),(74,308,78,312),(75,307,79,311),(76,306,80,310),(97,176,101,172),(98,175,102,171),(99,174,103,170),(100,173,104,169),(105,146,109,150),(106,145,110,149),(107,152,111,148),(108,151,112,147),(113,156,117,160),(114,155,118,159),(115,154,119,158),(116,153,120,157),(121,207,125,203),(122,206,126,202),(123,205,127,201),(124,204,128,208),(137,216,141,212),(138,215,142,211),(139,214,143,210),(140,213,144,209),(161,297,165,301),(162,304,166,300),(163,303,167,299),(164,302,168,298),(177,318,181,314),(178,317,182,313),(179,316,183,320),(180,315,184,319),(193,262,197,258),(194,261,198,257),(195,260,199,264),(196,259,200,263),(241,284,245,288),(242,283,246,287),(243,282,247,286),(244,281,248,285),(249,296,253,292),(250,295,254,291),(251,294,255,290),(252,293,256,289)])
Matrix representation ►G ⊆ GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 1 | 6 | 0 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 29 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 40 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 40 |
| 0 | 0 | 0 | 0 | 40 | 30 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,6,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,0,6,1,0,0,0,0,0,0,12,12,0,0,0,0,29,12],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,11,40,0,0,0,0,40,30] >;
68 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 8A | ··· | 8H | 10A | ··· | 10N | 20A | ··· | 20X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | ··· | 4 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | D4 | D4 | D5 | Q16 | D10 | D10 | C5⋊D4 | C5⋊D4 | C5⋊Q16 |
| kernel | C22×C5⋊Q16 | C22×C5⋊2C8 | C2×C5⋊Q16 | C22×Dic10 | Q8×C2×C10 | C2×C20 | C22×C10 | C22×Q8 | C2×C10 | C22×C4 | C2×Q8 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 12 | 1 | 1 | 3 | 1 | 2 | 8 | 2 | 12 | 12 | 4 | 8 |
In GAP, Magma, Sage, TeX
C_2^2\times C_5\rtimes Q_{16} % in TeX
G:=Group("C2^2xC5:Q16"); // GroupNames label
G:=SmallGroup(320,1481);
// by ID
G=gap.SmallGroup(320,1481);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,675,136,1684,235,102,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^5=d^8=1,e^2=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,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
×
⋊
ℤ
𝔽
○
≀