direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×Q82, C10.1672+ (1+4), C4⋊Q8.14C10, C4.20(Q8×C10), (C4×Q8).10C10, (Q8×C20).23C2, C20.126(C2×Q8), C42.52(C2×C10), C10.66(C22×Q8), (C4×C20).293C22, (C2×C20).966C23, (C2×C10).378C24, C22.52(C23×C10), (Q8×C10).186C22, C2.19(C5×2+ (1+4)), C2.12(Q8×C2×C10), (C5×C4⋊Q8).29C2, C4⋊C4.78(C2×C10), (C2×Q8).29(C2×C10), (C5×C4⋊C4).403C22, (C2×C4).39(C22×C10), SmallGroup(320,1560)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 266 in 212 conjugacy classes, 182 normal (8 characteristic)
C1, C2, C2 [×2], C4 [×12], C4 [×9], C22, C5, C2×C4 [×15], Q8 [×8], Q8 [×6], C10, C10 [×2], C42 [×9], C4⋊C4 [×18], C2×Q8 [×8], C20 [×12], C20 [×9], C2×C10, C4×Q8 [×6], C4⋊Q8 [×9], C2×C20 [×15], C5×Q8 [×8], C5×Q8 [×6], Q82, C4×C20 [×9], C5×C4⋊C4 [×18], Q8×C10 [×8], Q8×C20 [×6], C5×C4⋊Q8 [×9], C5×Q82
Quotients:
C1, C2 [×15], C22 [×35], C5, Q8 [×8], C23 [×15], C10 [×15], C2×Q8 [×12], C24, C2×C10 [×35], C22×Q8 [×2], 2+ (1+4), C5×Q8 [×8], C22×C10 [×15], Q82, Q8×C10 [×12], C23×C10, Q8×C2×C10 [×2], C5×2+ (1+4), C5×Q82
Generators and relations
G = < a,b,c,d,e | a5=b4=d4=1, c2=b2, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►Regular action on 320 points
Generators in S320
(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 67 21 75)(2 68 22 71)(3 69 23 72)(4 70 24 73)(5 66 25 74)(6 285 20 280)(7 281 16 276)(8 282 17 277)(9 283 18 278)(10 284 19 279)(11 291 317 288)(12 292 318 289)(13 293 319 290)(14 294 320 286)(15 295 316 287)(26 60 34 65)(27 56 35 61)(28 57 31 62)(29 58 32 63)(30 59 33 64)(36 87 41 95)(37 88 42 91)(38 89 43 92)(39 90 44 93)(40 86 45 94)(46 80 54 85)(47 76 55 81)(48 77 51 82)(49 78 52 83)(50 79 53 84)(96 147 101 155)(97 148 102 151)(98 149 103 152)(99 150 104 153)(100 146 105 154)(106 140 114 145)(107 136 115 141)(108 137 111 142)(109 138 112 143)(110 139 113 144)(116 167 121 175)(117 168 122 171)(118 169 123 172)(119 170 124 173)(120 166 125 174)(126 160 134 165)(127 156 135 161)(128 157 131 162)(129 158 132 163)(130 159 133 164)(176 235 181 227)(177 231 182 228)(178 232 183 229)(179 233 184 230)(180 234 185 226)(186 225 194 220)(187 221 195 216)(188 222 191 217)(189 223 192 218)(190 224 193 219)(196 255 201 247)(197 251 202 248)(198 252 203 249)(199 253 204 250)(200 254 205 246)(206 245 214 240)(207 241 215 236)(208 242 211 237)(209 243 212 238)(210 244 213 239)(256 315 261 307)(257 311 262 308)(258 312 263 309)(259 313 264 310)(260 314 265 306)(266 305 274 300)(267 301 275 296)(268 302 271 297)(269 303 272 298)(270 304 273 299)
(1 207 21 215)(2 208 22 211)(3 209 23 212)(4 210 24 213)(5 206 25 214)(6 154 20 146)(7 155 16 147)(8 151 17 148)(9 152 18 149)(10 153 19 150)(11 142 317 137)(12 143 318 138)(13 144 319 139)(14 145 320 140)(15 141 316 136)(26 200 34 205)(27 196 35 201)(28 197 31 202)(29 198 32 203)(30 199 33 204)(36 181 41 176)(37 182 42 177)(38 183 43 178)(39 184 44 179)(40 185 45 180)(46 194 54 186)(47 195 55 187)(48 191 51 188)(49 192 52 189)(50 193 53 190)(56 247 61 255)(57 248 62 251)(58 249 63 252)(59 250 64 253)(60 246 65 254)(66 240 74 245)(67 236 75 241)(68 237 71 242)(69 238 72 243)(70 239 73 244)(76 221 81 216)(77 222 82 217)(78 223 83 218)(79 224 84 219)(80 225 85 220)(86 234 94 226)(87 235 95 227)(88 231 91 228)(89 232 92 229)(90 233 93 230)(96 281 101 276)(97 282 102 277)(98 283 103 278)(99 284 104 279)(100 285 105 280)(106 294 114 286)(107 295 115 287)(108 291 111 288)(109 292 112 289)(110 293 113 290)(116 267 121 275)(117 268 122 271)(118 269 123 272)(119 270 124 273)(120 266 125 274)(126 260 134 265)(127 256 135 261)(128 257 131 262)(129 258 132 263)(130 259 133 264)(156 307 161 315)(157 308 162 311)(158 309 163 312)(159 310 164 313)(160 306 165 314)(166 300 174 305)(167 296 175 301)(168 297 171 302)(169 298 172 303)(170 299 173 304)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 300 320 314)(7 296 316 315)(8 297 317 311)(9 298 318 312)(10 299 319 313)(11 308 17 302)(12 309 18 303)(13 310 19 304)(14 306 20 305)(15 307 16 301)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)(56 95 75 76)(57 91 71 77)(58 92 72 78)(59 93 73 79)(60 94 74 80)(61 87 67 81)(62 88 68 82)(63 89 69 83)(64 90 70 84)(65 86 66 85)(96 116 115 135)(97 117 111 131)(98 118 112 132)(99 119 113 133)(100 120 114 134)(101 121 107 127)(102 122 108 128)(103 123 109 129)(104 124 110 130)(105 125 106 126)(136 156 155 175)(137 157 151 171)(138 158 152 172)(139 159 153 173)(140 160 154 174)(141 161 147 167)(142 162 148 168)(143 163 149 169)(144 164 150 170)(145 165 146 166)(176 215 195 196)(177 211 191 197)(178 212 192 198)(179 213 193 199)(180 214 194 200)(181 207 187 201)(182 208 188 202)(183 209 189 203)(184 210 190 204)(185 206 186 205)(216 255 235 236)(217 251 231 237)(218 252 232 238)(219 253 233 239)(220 254 234 240)(221 247 227 241)(222 248 228 242)(223 249 229 243)(224 250 230 244)(225 246 226 245)(256 276 275 295)(257 277 271 291)(258 278 272 292)(259 279 273 293)(260 280 274 294)(261 281 267 287)(262 282 268 288)(263 283 269 289)(264 284 270 290)(265 285 266 286)
(1 115 35 96)(2 111 31 97)(3 112 32 98)(4 113 33 99)(5 114 34 100)(6 240 320 254)(7 236 316 255)(8 237 317 251)(9 238 318 252)(10 239 319 253)(11 248 17 242)(12 249 18 243)(13 250 19 244)(14 246 20 245)(15 247 16 241)(21 107 27 101)(22 108 28 102)(23 109 29 103)(24 110 30 104)(25 106 26 105)(36 135 55 116)(37 131 51 117)(38 132 52 118)(39 133 53 119)(40 134 54 120)(41 127 47 121)(42 128 48 122)(43 129 49 123)(44 130 50 124)(45 126 46 125)(56 155 75 136)(57 151 71 137)(58 152 72 138)(59 153 73 139)(60 154 74 140)(61 147 67 141)(62 148 68 142)(63 149 69 143)(64 150 70 144)(65 146 66 145)(76 175 95 156)(77 171 91 157)(78 172 92 158)(79 173 93 159)(80 174 94 160)(81 167 87 161)(82 168 88 162)(83 169 89 163)(84 170 90 164)(85 166 86 165)(176 256 195 275)(177 257 191 271)(178 258 192 272)(179 259 193 273)(180 260 194 274)(181 261 187 267)(182 262 188 268)(183 263 189 269)(184 264 190 270)(185 265 186 266)(196 276 215 295)(197 277 211 291)(198 278 212 292)(199 279 213 293)(200 280 214 294)(201 281 207 287)(202 282 208 288)(203 283 209 289)(204 284 210 290)(205 285 206 286)(216 296 235 315)(217 297 231 311)(218 298 232 312)(219 299 233 313)(220 300 234 314)(221 301 227 307)(222 302 228 308)(223 303 229 309)(224 304 230 310)(225 305 226 306)
G:=sub<Sym(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), (1,67,21,75)(2,68,22,71)(3,69,23,72)(4,70,24,73)(5,66,25,74)(6,285,20,280)(7,281,16,276)(8,282,17,277)(9,283,18,278)(10,284,19,279)(11,291,317,288)(12,292,318,289)(13,293,319,290)(14,294,320,286)(15,295,316,287)(26,60,34,65)(27,56,35,61)(28,57,31,62)(29,58,32,63)(30,59,33,64)(36,87,41,95)(37,88,42,91)(38,89,43,92)(39,90,44,93)(40,86,45,94)(46,80,54,85)(47,76,55,81)(48,77,51,82)(49,78,52,83)(50,79,53,84)(96,147,101,155)(97,148,102,151)(98,149,103,152)(99,150,104,153)(100,146,105,154)(106,140,114,145)(107,136,115,141)(108,137,111,142)(109,138,112,143)(110,139,113,144)(116,167,121,175)(117,168,122,171)(118,169,123,172)(119,170,124,173)(120,166,125,174)(126,160,134,165)(127,156,135,161)(128,157,131,162)(129,158,132,163)(130,159,133,164)(176,235,181,227)(177,231,182,228)(178,232,183,229)(179,233,184,230)(180,234,185,226)(186,225,194,220)(187,221,195,216)(188,222,191,217)(189,223,192,218)(190,224,193,219)(196,255,201,247)(197,251,202,248)(198,252,203,249)(199,253,204,250)(200,254,205,246)(206,245,214,240)(207,241,215,236)(208,242,211,237)(209,243,212,238)(210,244,213,239)(256,315,261,307)(257,311,262,308)(258,312,263,309)(259,313,264,310)(260,314,265,306)(266,305,274,300)(267,301,275,296)(268,302,271,297)(269,303,272,298)(270,304,273,299), (1,207,21,215)(2,208,22,211)(3,209,23,212)(4,210,24,213)(5,206,25,214)(6,154,20,146)(7,155,16,147)(8,151,17,148)(9,152,18,149)(10,153,19,150)(11,142,317,137)(12,143,318,138)(13,144,319,139)(14,145,320,140)(15,141,316,136)(26,200,34,205)(27,196,35,201)(28,197,31,202)(29,198,32,203)(30,199,33,204)(36,181,41,176)(37,182,42,177)(38,183,43,178)(39,184,44,179)(40,185,45,180)(46,194,54,186)(47,195,55,187)(48,191,51,188)(49,192,52,189)(50,193,53,190)(56,247,61,255)(57,248,62,251)(58,249,63,252)(59,250,64,253)(60,246,65,254)(66,240,74,245)(67,236,75,241)(68,237,71,242)(69,238,72,243)(70,239,73,244)(76,221,81,216)(77,222,82,217)(78,223,83,218)(79,224,84,219)(80,225,85,220)(86,234,94,226)(87,235,95,227)(88,231,91,228)(89,232,92,229)(90,233,93,230)(96,281,101,276)(97,282,102,277)(98,283,103,278)(99,284,104,279)(100,285,105,280)(106,294,114,286)(107,295,115,287)(108,291,111,288)(109,292,112,289)(110,293,113,290)(116,267,121,275)(117,268,122,271)(118,269,123,272)(119,270,124,273)(120,266,125,274)(126,260,134,265)(127,256,135,261)(128,257,131,262)(129,258,132,263)(130,259,133,264)(156,307,161,315)(157,308,162,311)(158,309,163,312)(159,310,164,313)(160,306,165,314)(166,300,174,305)(167,296,175,301)(168,297,171,302)(169,298,172,303)(170,299,173,304), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,300,320,314)(7,296,316,315)(8,297,317,311)(9,298,318,312)(10,299,319,313)(11,308,17,302)(12,309,18,303)(13,310,19,304)(14,306,20,305)(15,307,16,301)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,87,67,81)(62,88,68,82)(63,89,69,83)(64,90,70,84)(65,86,66,85)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,107,127)(102,122,108,128)(103,123,109,129)(104,124,110,130)(105,125,106,126)(136,156,155,175)(137,157,151,171)(138,158,152,172)(139,159,153,173)(140,160,154,174)(141,161,147,167)(142,162,148,168)(143,163,149,169)(144,164,150,170)(145,165,146,166)(176,215,195,196)(177,211,191,197)(178,212,192,198)(179,213,193,199)(180,214,194,200)(181,207,187,201)(182,208,188,202)(183,209,189,203)(184,210,190,204)(185,206,186,205)(216,255,235,236)(217,251,231,237)(218,252,232,238)(219,253,233,239)(220,254,234,240)(221,247,227,241)(222,248,228,242)(223,249,229,243)(224,250,230,244)(225,246,226,245)(256,276,275,295)(257,277,271,291)(258,278,272,292)(259,279,273,293)(260,280,274,294)(261,281,267,287)(262,282,268,288)(263,283,269,289)(264,284,270,290)(265,285,266,286), (1,115,35,96)(2,111,31,97)(3,112,32,98)(4,113,33,99)(5,114,34,100)(6,240,320,254)(7,236,316,255)(8,237,317,251)(9,238,318,252)(10,239,319,253)(11,248,17,242)(12,249,18,243)(13,250,19,244)(14,246,20,245)(15,247,16,241)(21,107,27,101)(22,108,28,102)(23,109,29,103)(24,110,30,104)(25,106,26,105)(36,135,55,116)(37,131,51,117)(38,132,52,118)(39,133,53,119)(40,134,54,120)(41,127,47,121)(42,128,48,122)(43,129,49,123)(44,130,50,124)(45,126,46,125)(56,155,75,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,147,67,141)(62,148,68,142)(63,149,69,143)(64,150,70,144)(65,146,66,145)(76,175,95,156)(77,171,91,157)(78,172,92,158)(79,173,93,159)(80,174,94,160)(81,167,87,161)(82,168,88,162)(83,169,89,163)(84,170,90,164)(85,166,86,165)(176,256,195,275)(177,257,191,271)(178,258,192,272)(179,259,193,273)(180,260,194,274)(181,261,187,267)(182,262,188,268)(183,263,189,269)(184,264,190,270)(185,265,186,266)(196,276,215,295)(197,277,211,291)(198,278,212,292)(199,279,213,293)(200,280,214,294)(201,281,207,287)(202,282,208,288)(203,283,209,289)(204,284,210,290)(205,285,206,286)(216,296,235,315)(217,297,231,311)(218,298,232,312)(219,299,233,313)(220,300,234,314)(221,301,227,307)(222,302,228,308)(223,303,229,309)(224,304,230,310)(225,305,226,306)>;
G:=Group( (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), (1,67,21,75)(2,68,22,71)(3,69,23,72)(4,70,24,73)(5,66,25,74)(6,285,20,280)(7,281,16,276)(8,282,17,277)(9,283,18,278)(10,284,19,279)(11,291,317,288)(12,292,318,289)(13,293,319,290)(14,294,320,286)(15,295,316,287)(26,60,34,65)(27,56,35,61)(28,57,31,62)(29,58,32,63)(30,59,33,64)(36,87,41,95)(37,88,42,91)(38,89,43,92)(39,90,44,93)(40,86,45,94)(46,80,54,85)(47,76,55,81)(48,77,51,82)(49,78,52,83)(50,79,53,84)(96,147,101,155)(97,148,102,151)(98,149,103,152)(99,150,104,153)(100,146,105,154)(106,140,114,145)(107,136,115,141)(108,137,111,142)(109,138,112,143)(110,139,113,144)(116,167,121,175)(117,168,122,171)(118,169,123,172)(119,170,124,173)(120,166,125,174)(126,160,134,165)(127,156,135,161)(128,157,131,162)(129,158,132,163)(130,159,133,164)(176,235,181,227)(177,231,182,228)(178,232,183,229)(179,233,184,230)(180,234,185,226)(186,225,194,220)(187,221,195,216)(188,222,191,217)(189,223,192,218)(190,224,193,219)(196,255,201,247)(197,251,202,248)(198,252,203,249)(199,253,204,250)(200,254,205,246)(206,245,214,240)(207,241,215,236)(208,242,211,237)(209,243,212,238)(210,244,213,239)(256,315,261,307)(257,311,262,308)(258,312,263,309)(259,313,264,310)(260,314,265,306)(266,305,274,300)(267,301,275,296)(268,302,271,297)(269,303,272,298)(270,304,273,299), (1,207,21,215)(2,208,22,211)(3,209,23,212)(4,210,24,213)(5,206,25,214)(6,154,20,146)(7,155,16,147)(8,151,17,148)(9,152,18,149)(10,153,19,150)(11,142,317,137)(12,143,318,138)(13,144,319,139)(14,145,320,140)(15,141,316,136)(26,200,34,205)(27,196,35,201)(28,197,31,202)(29,198,32,203)(30,199,33,204)(36,181,41,176)(37,182,42,177)(38,183,43,178)(39,184,44,179)(40,185,45,180)(46,194,54,186)(47,195,55,187)(48,191,51,188)(49,192,52,189)(50,193,53,190)(56,247,61,255)(57,248,62,251)(58,249,63,252)(59,250,64,253)(60,246,65,254)(66,240,74,245)(67,236,75,241)(68,237,71,242)(69,238,72,243)(70,239,73,244)(76,221,81,216)(77,222,82,217)(78,223,83,218)(79,224,84,219)(80,225,85,220)(86,234,94,226)(87,235,95,227)(88,231,91,228)(89,232,92,229)(90,233,93,230)(96,281,101,276)(97,282,102,277)(98,283,103,278)(99,284,104,279)(100,285,105,280)(106,294,114,286)(107,295,115,287)(108,291,111,288)(109,292,112,289)(110,293,113,290)(116,267,121,275)(117,268,122,271)(118,269,123,272)(119,270,124,273)(120,266,125,274)(126,260,134,265)(127,256,135,261)(128,257,131,262)(129,258,132,263)(130,259,133,264)(156,307,161,315)(157,308,162,311)(158,309,163,312)(159,310,164,313)(160,306,165,314)(166,300,174,305)(167,296,175,301)(168,297,171,302)(169,298,172,303)(170,299,173,304), (1,55,35,36)(2,51,31,37)(3,52,32,38)(4,53,33,39)(5,54,34,40)(6,300,320,314)(7,296,316,315)(8,297,317,311)(9,298,318,312)(10,299,319,313)(11,308,17,302)(12,309,18,303)(13,310,19,304)(14,306,20,305)(15,307,16,301)(21,47,27,41)(22,48,28,42)(23,49,29,43)(24,50,30,44)(25,46,26,45)(56,95,75,76)(57,91,71,77)(58,92,72,78)(59,93,73,79)(60,94,74,80)(61,87,67,81)(62,88,68,82)(63,89,69,83)(64,90,70,84)(65,86,66,85)(96,116,115,135)(97,117,111,131)(98,118,112,132)(99,119,113,133)(100,120,114,134)(101,121,107,127)(102,122,108,128)(103,123,109,129)(104,124,110,130)(105,125,106,126)(136,156,155,175)(137,157,151,171)(138,158,152,172)(139,159,153,173)(140,160,154,174)(141,161,147,167)(142,162,148,168)(143,163,149,169)(144,164,150,170)(145,165,146,166)(176,215,195,196)(177,211,191,197)(178,212,192,198)(179,213,193,199)(180,214,194,200)(181,207,187,201)(182,208,188,202)(183,209,189,203)(184,210,190,204)(185,206,186,205)(216,255,235,236)(217,251,231,237)(218,252,232,238)(219,253,233,239)(220,254,234,240)(221,247,227,241)(222,248,228,242)(223,249,229,243)(224,250,230,244)(225,246,226,245)(256,276,275,295)(257,277,271,291)(258,278,272,292)(259,279,273,293)(260,280,274,294)(261,281,267,287)(262,282,268,288)(263,283,269,289)(264,284,270,290)(265,285,266,286), (1,115,35,96)(2,111,31,97)(3,112,32,98)(4,113,33,99)(5,114,34,100)(6,240,320,254)(7,236,316,255)(8,237,317,251)(9,238,318,252)(10,239,319,253)(11,248,17,242)(12,249,18,243)(13,250,19,244)(14,246,20,245)(15,247,16,241)(21,107,27,101)(22,108,28,102)(23,109,29,103)(24,110,30,104)(25,106,26,105)(36,135,55,116)(37,131,51,117)(38,132,52,118)(39,133,53,119)(40,134,54,120)(41,127,47,121)(42,128,48,122)(43,129,49,123)(44,130,50,124)(45,126,46,125)(56,155,75,136)(57,151,71,137)(58,152,72,138)(59,153,73,139)(60,154,74,140)(61,147,67,141)(62,148,68,142)(63,149,69,143)(64,150,70,144)(65,146,66,145)(76,175,95,156)(77,171,91,157)(78,172,92,158)(79,173,93,159)(80,174,94,160)(81,167,87,161)(82,168,88,162)(83,169,89,163)(84,170,90,164)(85,166,86,165)(176,256,195,275)(177,257,191,271)(178,258,192,272)(179,259,193,273)(180,260,194,274)(181,261,187,267)(182,262,188,268)(183,263,189,269)(184,264,190,270)(185,265,186,266)(196,276,215,295)(197,277,211,291)(198,278,212,292)(199,279,213,293)(200,280,214,294)(201,281,207,287)(202,282,208,288)(203,283,209,289)(204,284,210,290)(205,285,206,286)(216,296,235,315)(217,297,231,311)(218,298,232,312)(219,299,233,313)(220,300,234,314)(221,301,227,307)(222,302,228,308)(223,303,229,309)(224,304,230,310)(225,305,226,306) );
G=PermutationGroup([(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)], [(1,67,21,75),(2,68,22,71),(3,69,23,72),(4,70,24,73),(5,66,25,74),(6,285,20,280),(7,281,16,276),(8,282,17,277),(9,283,18,278),(10,284,19,279),(11,291,317,288),(12,292,318,289),(13,293,319,290),(14,294,320,286),(15,295,316,287),(26,60,34,65),(27,56,35,61),(28,57,31,62),(29,58,32,63),(30,59,33,64),(36,87,41,95),(37,88,42,91),(38,89,43,92),(39,90,44,93),(40,86,45,94),(46,80,54,85),(47,76,55,81),(48,77,51,82),(49,78,52,83),(50,79,53,84),(96,147,101,155),(97,148,102,151),(98,149,103,152),(99,150,104,153),(100,146,105,154),(106,140,114,145),(107,136,115,141),(108,137,111,142),(109,138,112,143),(110,139,113,144),(116,167,121,175),(117,168,122,171),(118,169,123,172),(119,170,124,173),(120,166,125,174),(126,160,134,165),(127,156,135,161),(128,157,131,162),(129,158,132,163),(130,159,133,164),(176,235,181,227),(177,231,182,228),(178,232,183,229),(179,233,184,230),(180,234,185,226),(186,225,194,220),(187,221,195,216),(188,222,191,217),(189,223,192,218),(190,224,193,219),(196,255,201,247),(197,251,202,248),(198,252,203,249),(199,253,204,250),(200,254,205,246),(206,245,214,240),(207,241,215,236),(208,242,211,237),(209,243,212,238),(210,244,213,239),(256,315,261,307),(257,311,262,308),(258,312,263,309),(259,313,264,310),(260,314,265,306),(266,305,274,300),(267,301,275,296),(268,302,271,297),(269,303,272,298),(270,304,273,299)], [(1,207,21,215),(2,208,22,211),(3,209,23,212),(4,210,24,213),(5,206,25,214),(6,154,20,146),(7,155,16,147),(8,151,17,148),(9,152,18,149),(10,153,19,150),(11,142,317,137),(12,143,318,138),(13,144,319,139),(14,145,320,140),(15,141,316,136),(26,200,34,205),(27,196,35,201),(28,197,31,202),(29,198,32,203),(30,199,33,204),(36,181,41,176),(37,182,42,177),(38,183,43,178),(39,184,44,179),(40,185,45,180),(46,194,54,186),(47,195,55,187),(48,191,51,188),(49,192,52,189),(50,193,53,190),(56,247,61,255),(57,248,62,251),(58,249,63,252),(59,250,64,253),(60,246,65,254),(66,240,74,245),(67,236,75,241),(68,237,71,242),(69,238,72,243),(70,239,73,244),(76,221,81,216),(77,222,82,217),(78,223,83,218),(79,224,84,219),(80,225,85,220),(86,234,94,226),(87,235,95,227),(88,231,91,228),(89,232,92,229),(90,233,93,230),(96,281,101,276),(97,282,102,277),(98,283,103,278),(99,284,104,279),(100,285,105,280),(106,294,114,286),(107,295,115,287),(108,291,111,288),(109,292,112,289),(110,293,113,290),(116,267,121,275),(117,268,122,271),(118,269,123,272),(119,270,124,273),(120,266,125,274),(126,260,134,265),(127,256,135,261),(128,257,131,262),(129,258,132,263),(130,259,133,264),(156,307,161,315),(157,308,162,311),(158,309,163,312),(159,310,164,313),(160,306,165,314),(166,300,174,305),(167,296,175,301),(168,297,171,302),(169,298,172,303),(170,299,173,304)], [(1,55,35,36),(2,51,31,37),(3,52,32,38),(4,53,33,39),(5,54,34,40),(6,300,320,314),(7,296,316,315),(8,297,317,311),(9,298,318,312),(10,299,319,313),(11,308,17,302),(12,309,18,303),(13,310,19,304),(14,306,20,305),(15,307,16,301),(21,47,27,41),(22,48,28,42),(23,49,29,43),(24,50,30,44),(25,46,26,45),(56,95,75,76),(57,91,71,77),(58,92,72,78),(59,93,73,79),(60,94,74,80),(61,87,67,81),(62,88,68,82),(63,89,69,83),(64,90,70,84),(65,86,66,85),(96,116,115,135),(97,117,111,131),(98,118,112,132),(99,119,113,133),(100,120,114,134),(101,121,107,127),(102,122,108,128),(103,123,109,129),(104,124,110,130),(105,125,106,126),(136,156,155,175),(137,157,151,171),(138,158,152,172),(139,159,153,173),(140,160,154,174),(141,161,147,167),(142,162,148,168),(143,163,149,169),(144,164,150,170),(145,165,146,166),(176,215,195,196),(177,211,191,197),(178,212,192,198),(179,213,193,199),(180,214,194,200),(181,207,187,201),(182,208,188,202),(183,209,189,203),(184,210,190,204),(185,206,186,205),(216,255,235,236),(217,251,231,237),(218,252,232,238),(219,253,233,239),(220,254,234,240),(221,247,227,241),(222,248,228,242),(223,249,229,243),(224,250,230,244),(225,246,226,245),(256,276,275,295),(257,277,271,291),(258,278,272,292),(259,279,273,293),(260,280,274,294),(261,281,267,287),(262,282,268,288),(263,283,269,289),(264,284,270,290),(265,285,266,286)], [(1,115,35,96),(2,111,31,97),(3,112,32,98),(4,113,33,99),(5,114,34,100),(6,240,320,254),(7,236,316,255),(8,237,317,251),(9,238,318,252),(10,239,319,253),(11,248,17,242),(12,249,18,243),(13,250,19,244),(14,246,20,245),(15,247,16,241),(21,107,27,101),(22,108,28,102),(23,109,29,103),(24,110,30,104),(25,106,26,105),(36,135,55,116),(37,131,51,117),(38,132,52,118),(39,133,53,119),(40,134,54,120),(41,127,47,121),(42,128,48,122),(43,129,49,123),(44,130,50,124),(45,126,46,125),(56,155,75,136),(57,151,71,137),(58,152,72,138),(59,153,73,139),(60,154,74,140),(61,147,67,141),(62,148,68,142),(63,149,69,143),(64,150,70,144),(65,146,66,145),(76,175,95,156),(77,171,91,157),(78,172,92,158),(79,173,93,159),(80,174,94,160),(81,167,87,161),(82,168,88,162),(83,169,89,163),(84,170,90,164),(85,166,86,165),(176,256,195,275),(177,257,191,271),(178,258,192,272),(179,259,193,273),(180,260,194,274),(181,261,187,267),(182,262,188,268),(183,263,189,269),(184,264,190,270),(185,265,186,266),(196,276,215,295),(197,277,211,291),(198,278,212,292),(199,279,213,293),(200,280,214,294),(201,281,207,287),(202,282,208,288),(203,283,209,289),(204,284,210,290),(205,285,206,286),(216,296,235,315),(217,297,231,311),(218,298,232,312),(219,299,233,313),(220,300,234,314),(221,301,227,307),(222,302,228,308),(223,303,229,309),(224,304,230,310),(225,305,226,306)])
Matrix representation ►G ⊆ GL4(𝔽41) generated by
| 16 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 |
| 0 | 0 | 16 | 0 |
| 0 | 0 | 0 | 16 |
| 40 | 39 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 34 | 14 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 1 | 39 |
| 0 | 0 | 1 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 4 | 4 |
| 0 | 0 | 6 | 37 |
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[40,1,0,0,39,1,0,0,0,0,1,0,0,0,0,1],[34,14,0,0,14,7,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,1,0,0,39,40],[40,0,0,0,0,40,0,0,0,0,4,6,0,0,4,37] >;
125 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | ··· | 4L | 4M | ··· | 4U | 5A | 5B | 5C | 5D | 10A | ··· | 10L | 20A | ··· | 20AV | 20AW | ··· | 20CF |
| order | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
125 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C5 | C10 | C10 | Q8 | C5×Q8 | 2+ (1+4) | C5×2+ (1+4) |
| kernel | C5×Q82 | Q8×C20 | C5×C4⋊Q8 | Q82 | C4×Q8 | C4⋊Q8 | C5×Q8 | Q8 | C10 | C2 |
| # reps | 1 | 6 | 9 | 4 | 24 | 36 | 8 | 32 | 1 | 4 |
In GAP, Magma, Sage, TeX
C_5\times Q_8^2
% in TeX
G:=Group("C5xQ8^2"); // GroupNames label
G:=SmallGroup(320,1560);
// by ID
G=gap.SmallGroup(320,1560);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1680,1149,568,3446,856,1242,304]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^4=d^4=1,c^2=b^2,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,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
○
≀