direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C5×Q8⋊3Q8, C10.1222- 1+4, Q8⋊3(C5×Q8), (C5×Q8)⋊10Q8, C4⋊Q8.13C10, (C4×Q8).9C10, C4.19(Q8×C10), (Q8×C20).22C2, C20.125(C2×Q8), C42.51(C2×C10), C42.C2.6C10, C20.348(C4○D4), C10.65(C22×Q8), (C4×C20).292C22, (C2×C20).715C23, (C2×C10).377C24, C22.51(C23×C10), (Q8×C10).278C22, C2.14(C5×2- 1+4), C2.11(Q8×C2×C10), C4.46(C5×C4○D4), (C5×C4⋊Q8).28C2, C4⋊C4.77(C2×C10), C2.30(C10×C4○D4), C10.249(C2×C4○D4), (C2×Q8).66(C2×C10), (C5×C4⋊C4).402C22, (C2×C4).63(C22×C10), (C5×C42.C2).13C2, SmallGroup(320,1559)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5×Q8⋊3Q8
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, ece-1=b2c, ede-1=d-1 >
Subgroups: 234 in 200 conjugacy classes, 166 normal (20 characteristic)
C1, C2, C4, C4, C22, C5, C2×C4, C2×C4, Q8, Q8, C10, C42, C4⋊C4, C4⋊C4, C2×Q8, C2×Q8, C20, C20, C2×C10, C4×Q8, C4×Q8, C42.C2, C4⋊Q8, C2×C20, C2×C20, C5×Q8, C5×Q8, Q8⋊3Q8, C4×C20, C5×C4⋊C4, C5×C4⋊C4, Q8×C10, Q8×C10, Q8×C20, Q8×C20, C5×C42.C2, C5×C4⋊Q8, C5×Q8⋊3Q8
Quotients: C1, C2, C22, C5, Q8, C23, C10, C2×Q8, C4○D4, C24, C2×C10, C22×Q8, C2×C4○D4, 2- 1+4, C5×Q8, C22×C10, Q8⋊3Q8, Q8×C10, C5×C4○D4, C23×C10, Q8×C2×C10, C10×C4○D4, C5×2- 1+4, C5×Q8⋊3Q8
►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 47 35 41)(2 48 31 42)(3 49 32 43)(4 50 33 44)(5 46 34 45)(6 305 320 306)(7 301 316 307)(8 302 317 308)(9 303 318 309)(10 304 319 310)(11 311 17 297)(12 312 18 298)(13 313 19 299)(14 314 20 300)(15 315 16 296)(21 55 27 36)(22 51 28 37)(23 52 29 38)(24 53 30 39)(25 54 26 40)(56 87 75 81)(57 88 71 82)(58 89 72 83)(59 90 73 84)(60 86 74 85)(61 95 67 76)(62 91 68 77)(63 92 69 78)(64 93 70 79)(65 94 66 80)(96 121 115 127)(97 122 111 128)(98 123 112 129)(99 124 113 130)(100 125 114 126)(101 116 107 135)(102 117 108 131)(103 118 109 132)(104 119 110 133)(105 120 106 134)(136 161 155 167)(137 162 151 168)(138 163 152 169)(139 164 153 170)(140 165 154 166)(141 156 147 175)(142 157 148 171)(143 158 149 172)(144 159 150 173)(145 160 146 174)(176 207 195 201)(177 208 191 202)(178 209 192 203)(179 210 193 204)(180 206 194 205)(181 215 187 196)(182 211 188 197)(183 212 189 198)(184 213 190 199)(185 214 186 200)(216 247 235 241)(217 248 231 242)(218 249 232 243)(219 250 233 244)(220 246 234 245)(221 255 227 236)(222 251 228 237)(223 252 229 238)(224 253 230 239)(225 254 226 240)(256 281 275 287)(257 282 271 288)(258 283 272 289)(259 284 273 290)(260 285 274 286)(261 276 267 295)(262 277 268 291)(263 278 269 292)(264 279 270 293)(265 280 266 294)
(1 107 35 101)(2 108 31 102)(3 109 32 103)(4 110 33 104)(5 106 34 105)(6 245 320 246)(7 241 316 247)(8 242 317 248)(9 243 318 249)(10 244 319 250)(11 251 17 237)(12 252 18 238)(13 253 19 239)(14 254 20 240)(15 255 16 236)(21 115 27 96)(22 111 28 97)(23 112 29 98)(24 113 30 99)(25 114 26 100)(36 127 55 121)(37 128 51 122)(38 129 52 123)(39 130 53 124)(40 126 54 125)(41 135 47 116)(42 131 48 117)(43 132 49 118)(44 133 50 119)(45 134 46 120)(56 147 75 141)(57 148 71 142)(58 149 72 143)(59 150 73 144)(60 146 74 145)(61 155 67 136)(62 151 68 137)(63 152 69 138)(64 153 70 139)(65 154 66 140)(76 167 95 161)(77 168 91 162)(78 169 92 163)(79 170 93 164)(80 166 94 165)(81 175 87 156)(82 171 88 157)(83 172 89 158)(84 173 90 159)(85 174 86 160)(176 261 195 267)(177 262 191 268)(178 263 192 269)(179 264 193 270)(180 265 194 266)(181 256 187 275)(182 257 188 271)(183 258 189 272)(184 259 190 273)(185 260 186 274)(196 281 215 287)(197 282 211 288)(198 283 212 289)(199 284 213 290)(200 285 214 286)(201 276 207 295)(202 277 208 291)(203 278 209 292)(204 279 210 293)(205 280 206 294)(216 301 235 307)(217 302 231 308)(218 303 232 309)(219 304 233 310)(220 305 234 306)(221 296 227 315)(222 297 228 311)(223 298 229 312)(224 299 230 313)(225 300 226 314)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 286 20 294)(7 287 16 295)(8 288 17 291)(9 289 18 292)(10 290 19 293)(11 277 317 282)(12 278 318 283)(13 279 319 284)(14 280 320 285)(15 276 316 281)(26 74 34 66)(27 75 35 67)(28 71 31 68)(29 72 32 69)(30 73 33 70)(36 81 41 76)(37 82 42 77)(38 83 43 78)(39 84 44 79)(40 85 45 80)(46 94 54 86)(47 95 55 87)(48 91 51 88)(49 92 52 89)(50 93 53 90)(96 141 101 136)(97 142 102 137)(98 143 103 138)(99 144 104 139)(100 145 105 140)(106 154 114 146)(107 155 115 147)(108 151 111 148)(109 152 112 149)(110 153 113 150)(116 161 121 156)(117 162 122 157)(118 163 123 158)(119 164 124 159)(120 165 125 160)(126 174 134 166)(127 175 135 167)(128 171 131 168)(129 172 132 169)(130 173 133 170)(176 216 181 221)(177 217 182 222)(178 218 183 223)(179 219 184 224)(180 220 185 225)(186 226 194 234)(187 227 195 235)(188 228 191 231)(189 229 192 232)(190 230 193 233)(196 236 201 241)(197 237 202 242)(198 238 203 243)(199 239 204 244)(200 240 205 245)(206 246 214 254)(207 247 215 255)(208 248 211 251)(209 249 212 252)(210 250 213 253)(256 296 261 301)(257 297 262 302)(258 298 263 303)(259 299 264 304)(260 300 265 305)(266 306 274 314)(267 307 275 315)(268 308 271 311)(269 309 272 312)(270 310 273 313)
(1 181 21 176)(2 182 22 177)(3 183 23 178)(4 184 24 179)(5 185 25 180)(6 174 20 166)(7 175 16 167)(8 171 17 168)(9 172 18 169)(10 173 19 170)(11 162 317 157)(12 163 318 158)(13 164 319 159)(14 165 320 160)(15 161 316 156)(26 194 34 186)(27 195 35 187)(28 191 31 188)(29 192 32 189)(30 193 33 190)(36 201 41 196)(37 202 42 197)(38 203 43 198)(39 204 44 199)(40 205 45 200)(46 214 54 206)(47 215 55 207)(48 211 51 208)(49 212 52 209)(50 213 53 210)(56 221 61 216)(57 222 62 217)(58 223 63 218)(59 224 64 219)(60 225 65 220)(66 234 74 226)(67 235 75 227)(68 231 71 228)(69 232 72 229)(70 233 73 230)(76 241 81 236)(77 242 82 237)(78 243 83 238)(79 244 84 239)(80 245 85 240)(86 254 94 246)(87 255 95 247)(88 251 91 248)(89 252 92 249)(90 253 93 250)(96 261 101 256)(97 262 102 257)(98 263 103 258)(99 264 104 259)(100 265 105 260)(106 274 114 266)(107 275 115 267)(108 271 111 268)(109 272 112 269)(110 273 113 270)(116 281 121 276)(117 282 122 277)(118 283 123 278)(119 284 124 279)(120 285 125 280)(126 294 134 286)(127 295 135 287)(128 291 131 288)(129 292 132 289)(130 293 133 290)(136 301 141 296)(137 302 142 297)(138 303 143 298)(139 304 144 299)(140 305 145 300)(146 314 154 306)(147 315 155 307)(148 311 151 308)(149 312 152 309)(150 313 153 310)
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), (1,47,35,41)(2,48,31,42)(3,49,32,43)(4,50,33,44)(5,46,34,45)(6,305,320,306)(7,301,316,307)(8,302,317,308)(9,303,318,309)(10,304,319,310)(11,311,17,297)(12,312,18,298)(13,313,19,299)(14,314,20,300)(15,315,16,296)(21,55,27,36)(22,51,28,37)(23,52,29,38)(24,53,30,39)(25,54,26,40)(56,87,75,81)(57,88,71,82)(58,89,72,83)(59,90,73,84)(60,86,74,85)(61,95,67,76)(62,91,68,77)(63,92,69,78)(64,93,70,79)(65,94,66,80)(96,121,115,127)(97,122,111,128)(98,123,112,129)(99,124,113,130)(100,125,114,126)(101,116,107,135)(102,117,108,131)(103,118,109,132)(104,119,110,133)(105,120,106,134)(136,161,155,167)(137,162,151,168)(138,163,152,169)(139,164,153,170)(140,165,154,166)(141,156,147,175)(142,157,148,171)(143,158,149,172)(144,159,150,173)(145,160,146,174)(176,207,195,201)(177,208,191,202)(178,209,192,203)(179,210,193,204)(180,206,194,205)(181,215,187,196)(182,211,188,197)(183,212,189,198)(184,213,190,199)(185,214,186,200)(216,247,235,241)(217,248,231,242)(218,249,232,243)(219,250,233,244)(220,246,234,245)(221,255,227,236)(222,251,228,237)(223,252,229,238)(224,253,230,239)(225,254,226,240)(256,281,275,287)(257,282,271,288)(258,283,272,289)(259,284,273,290)(260,285,274,286)(261,276,267,295)(262,277,268,291)(263,278,269,292)(264,279,270,293)(265,280,266,294), (1,107,35,101)(2,108,31,102)(3,109,32,103)(4,110,33,104)(5,106,34,105)(6,245,320,246)(7,241,316,247)(8,242,317,248)(9,243,318,249)(10,244,319,250)(11,251,17,237)(12,252,18,238)(13,253,19,239)(14,254,20,240)(15,255,16,236)(21,115,27,96)(22,111,28,97)(23,112,29,98)(24,113,30,99)(25,114,26,100)(36,127,55,121)(37,128,51,122)(38,129,52,123)(39,130,53,124)(40,126,54,125)(41,135,47,116)(42,131,48,117)(43,132,49,118)(44,133,50,119)(45,134,46,120)(56,147,75,141)(57,148,71,142)(58,149,72,143)(59,150,73,144)(60,146,74,145)(61,155,67,136)(62,151,68,137)(63,152,69,138)(64,153,70,139)(65,154,66,140)(76,167,95,161)(77,168,91,162)(78,169,92,163)(79,170,93,164)(80,166,94,165)(81,175,87,156)(82,171,88,157)(83,172,89,158)(84,173,90,159)(85,174,86,160)(176,261,195,267)(177,262,191,268)(178,263,192,269)(179,264,193,270)(180,265,194,266)(181,256,187,275)(182,257,188,271)(183,258,189,272)(184,259,190,273)(185,260,186,274)(196,281,215,287)(197,282,211,288)(198,283,212,289)(199,284,213,290)(200,285,214,286)(201,276,207,295)(202,277,208,291)(203,278,209,292)(204,279,210,293)(205,280,206,294)(216,301,235,307)(217,302,231,308)(218,303,232,309)(219,304,233,310)(220,305,234,306)(221,296,227,315)(222,297,228,311)(223,298,229,312)(224,299,230,313)(225,300,226,314), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,286,20,294)(7,287,16,295)(8,288,17,291)(9,289,18,292)(10,290,19,293)(11,277,317,282)(12,278,318,283)(13,279,319,284)(14,280,320,285)(15,276,316,281)(26,74,34,66)(27,75,35,67)(28,71,31,68)(29,72,32,69)(30,73,33,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,94,54,86)(47,95,55,87)(48,91,51,88)(49,92,52,89)(50,93,53,90)(96,141,101,136)(97,142,102,137)(98,143,103,138)(99,144,104,139)(100,145,105,140)(106,154,114,146)(107,155,115,147)(108,151,111,148)(109,152,112,149)(110,153,113,150)(116,161,121,156)(117,162,122,157)(118,163,123,158)(119,164,124,159)(120,165,125,160)(126,174,134,166)(127,175,135,167)(128,171,131,168)(129,172,132,169)(130,173,133,170)(176,216,181,221)(177,217,182,222)(178,218,183,223)(179,219,184,224)(180,220,185,225)(186,226,194,234)(187,227,195,235)(188,228,191,231)(189,229,192,232)(190,230,193,233)(196,236,201,241)(197,237,202,242)(198,238,203,243)(199,239,204,244)(200,240,205,245)(206,246,214,254)(207,247,215,255)(208,248,211,251)(209,249,212,252)(210,250,213,253)(256,296,261,301)(257,297,262,302)(258,298,263,303)(259,299,264,304)(260,300,265,305)(266,306,274,314)(267,307,275,315)(268,308,271,311)(269,309,272,312)(270,310,273,313), (1,181,21,176)(2,182,22,177)(3,183,23,178)(4,184,24,179)(5,185,25,180)(6,174,20,166)(7,175,16,167)(8,171,17,168)(9,172,18,169)(10,173,19,170)(11,162,317,157)(12,163,318,158)(13,164,319,159)(14,165,320,160)(15,161,316,156)(26,194,34,186)(27,195,35,187)(28,191,31,188)(29,192,32,189)(30,193,33,190)(36,201,41,196)(37,202,42,197)(38,203,43,198)(39,204,44,199)(40,205,45,200)(46,214,54,206)(47,215,55,207)(48,211,51,208)(49,212,52,209)(50,213,53,210)(56,221,61,216)(57,222,62,217)(58,223,63,218)(59,224,64,219)(60,225,65,220)(66,234,74,226)(67,235,75,227)(68,231,71,228)(69,232,72,229)(70,233,73,230)(76,241,81,236)(77,242,82,237)(78,243,83,238)(79,244,84,239)(80,245,85,240)(86,254,94,246)(87,255,95,247)(88,251,91,248)(89,252,92,249)(90,253,93,250)(96,261,101,256)(97,262,102,257)(98,263,103,258)(99,264,104,259)(100,265,105,260)(106,274,114,266)(107,275,115,267)(108,271,111,268)(109,272,112,269)(110,273,113,270)(116,281,121,276)(117,282,122,277)(118,283,123,278)(119,284,124,279)(120,285,125,280)(126,294,134,286)(127,295,135,287)(128,291,131,288)(129,292,132,289)(130,293,133,290)(136,301,141,296)(137,302,142,297)(138,303,143,298)(139,304,144,299)(140,305,145,300)(146,314,154,306)(147,315,155,307)(148,311,151,308)(149,312,152,309)(150,313,153,310)>;
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), (1,47,35,41)(2,48,31,42)(3,49,32,43)(4,50,33,44)(5,46,34,45)(6,305,320,306)(7,301,316,307)(8,302,317,308)(9,303,318,309)(10,304,319,310)(11,311,17,297)(12,312,18,298)(13,313,19,299)(14,314,20,300)(15,315,16,296)(21,55,27,36)(22,51,28,37)(23,52,29,38)(24,53,30,39)(25,54,26,40)(56,87,75,81)(57,88,71,82)(58,89,72,83)(59,90,73,84)(60,86,74,85)(61,95,67,76)(62,91,68,77)(63,92,69,78)(64,93,70,79)(65,94,66,80)(96,121,115,127)(97,122,111,128)(98,123,112,129)(99,124,113,130)(100,125,114,126)(101,116,107,135)(102,117,108,131)(103,118,109,132)(104,119,110,133)(105,120,106,134)(136,161,155,167)(137,162,151,168)(138,163,152,169)(139,164,153,170)(140,165,154,166)(141,156,147,175)(142,157,148,171)(143,158,149,172)(144,159,150,173)(145,160,146,174)(176,207,195,201)(177,208,191,202)(178,209,192,203)(179,210,193,204)(180,206,194,205)(181,215,187,196)(182,211,188,197)(183,212,189,198)(184,213,190,199)(185,214,186,200)(216,247,235,241)(217,248,231,242)(218,249,232,243)(219,250,233,244)(220,246,234,245)(221,255,227,236)(222,251,228,237)(223,252,229,238)(224,253,230,239)(225,254,226,240)(256,281,275,287)(257,282,271,288)(258,283,272,289)(259,284,273,290)(260,285,274,286)(261,276,267,295)(262,277,268,291)(263,278,269,292)(264,279,270,293)(265,280,266,294), (1,107,35,101)(2,108,31,102)(3,109,32,103)(4,110,33,104)(5,106,34,105)(6,245,320,246)(7,241,316,247)(8,242,317,248)(9,243,318,249)(10,244,319,250)(11,251,17,237)(12,252,18,238)(13,253,19,239)(14,254,20,240)(15,255,16,236)(21,115,27,96)(22,111,28,97)(23,112,29,98)(24,113,30,99)(25,114,26,100)(36,127,55,121)(37,128,51,122)(38,129,52,123)(39,130,53,124)(40,126,54,125)(41,135,47,116)(42,131,48,117)(43,132,49,118)(44,133,50,119)(45,134,46,120)(56,147,75,141)(57,148,71,142)(58,149,72,143)(59,150,73,144)(60,146,74,145)(61,155,67,136)(62,151,68,137)(63,152,69,138)(64,153,70,139)(65,154,66,140)(76,167,95,161)(77,168,91,162)(78,169,92,163)(79,170,93,164)(80,166,94,165)(81,175,87,156)(82,171,88,157)(83,172,89,158)(84,173,90,159)(85,174,86,160)(176,261,195,267)(177,262,191,268)(178,263,192,269)(179,264,193,270)(180,265,194,266)(181,256,187,275)(182,257,188,271)(183,258,189,272)(184,259,190,273)(185,260,186,274)(196,281,215,287)(197,282,211,288)(198,283,212,289)(199,284,213,290)(200,285,214,286)(201,276,207,295)(202,277,208,291)(203,278,209,292)(204,279,210,293)(205,280,206,294)(216,301,235,307)(217,302,231,308)(218,303,232,309)(219,304,233,310)(220,305,234,306)(221,296,227,315)(222,297,228,311)(223,298,229,312)(224,299,230,313)(225,300,226,314), (1,61,21,56)(2,62,22,57)(3,63,23,58)(4,64,24,59)(5,65,25,60)(6,286,20,294)(7,287,16,295)(8,288,17,291)(9,289,18,292)(10,290,19,293)(11,277,317,282)(12,278,318,283)(13,279,319,284)(14,280,320,285)(15,276,316,281)(26,74,34,66)(27,75,35,67)(28,71,31,68)(29,72,32,69)(30,73,33,70)(36,81,41,76)(37,82,42,77)(38,83,43,78)(39,84,44,79)(40,85,45,80)(46,94,54,86)(47,95,55,87)(48,91,51,88)(49,92,52,89)(50,93,53,90)(96,141,101,136)(97,142,102,137)(98,143,103,138)(99,144,104,139)(100,145,105,140)(106,154,114,146)(107,155,115,147)(108,151,111,148)(109,152,112,149)(110,153,113,150)(116,161,121,156)(117,162,122,157)(118,163,123,158)(119,164,124,159)(120,165,125,160)(126,174,134,166)(127,175,135,167)(128,171,131,168)(129,172,132,169)(130,173,133,170)(176,216,181,221)(177,217,182,222)(178,218,183,223)(179,219,184,224)(180,220,185,225)(186,226,194,234)(187,227,195,235)(188,228,191,231)(189,229,192,232)(190,230,193,233)(196,236,201,241)(197,237,202,242)(198,238,203,243)(199,239,204,244)(200,240,205,245)(206,246,214,254)(207,247,215,255)(208,248,211,251)(209,249,212,252)(210,250,213,253)(256,296,261,301)(257,297,262,302)(258,298,263,303)(259,299,264,304)(260,300,265,305)(266,306,274,314)(267,307,275,315)(268,308,271,311)(269,309,272,312)(270,310,273,313), (1,181,21,176)(2,182,22,177)(3,183,23,178)(4,184,24,179)(5,185,25,180)(6,174,20,166)(7,175,16,167)(8,171,17,168)(9,172,18,169)(10,173,19,170)(11,162,317,157)(12,163,318,158)(13,164,319,159)(14,165,320,160)(15,161,316,156)(26,194,34,186)(27,195,35,187)(28,191,31,188)(29,192,32,189)(30,193,33,190)(36,201,41,196)(37,202,42,197)(38,203,43,198)(39,204,44,199)(40,205,45,200)(46,214,54,206)(47,215,55,207)(48,211,51,208)(49,212,52,209)(50,213,53,210)(56,221,61,216)(57,222,62,217)(58,223,63,218)(59,224,64,219)(60,225,65,220)(66,234,74,226)(67,235,75,227)(68,231,71,228)(69,232,72,229)(70,233,73,230)(76,241,81,236)(77,242,82,237)(78,243,83,238)(79,244,84,239)(80,245,85,240)(86,254,94,246)(87,255,95,247)(88,251,91,248)(89,252,92,249)(90,253,93,250)(96,261,101,256)(97,262,102,257)(98,263,103,258)(99,264,104,259)(100,265,105,260)(106,274,114,266)(107,275,115,267)(108,271,111,268)(109,272,112,269)(110,273,113,270)(116,281,121,276)(117,282,122,277)(118,283,123,278)(119,284,124,279)(120,285,125,280)(126,294,134,286)(127,295,135,287)(128,291,131,288)(129,292,132,289)(130,293,133,290)(136,301,141,296)(137,302,142,297)(138,303,143,298)(139,304,144,299)(140,305,145,300)(146,314,154,306)(147,315,155,307)(148,311,151,308)(149,312,152,309)(150,313,153,310) );
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)], [(1,47,35,41),(2,48,31,42),(3,49,32,43),(4,50,33,44),(5,46,34,45),(6,305,320,306),(7,301,316,307),(8,302,317,308),(9,303,318,309),(10,304,319,310),(11,311,17,297),(12,312,18,298),(13,313,19,299),(14,314,20,300),(15,315,16,296),(21,55,27,36),(22,51,28,37),(23,52,29,38),(24,53,30,39),(25,54,26,40),(56,87,75,81),(57,88,71,82),(58,89,72,83),(59,90,73,84),(60,86,74,85),(61,95,67,76),(62,91,68,77),(63,92,69,78),(64,93,70,79),(65,94,66,80),(96,121,115,127),(97,122,111,128),(98,123,112,129),(99,124,113,130),(100,125,114,126),(101,116,107,135),(102,117,108,131),(103,118,109,132),(104,119,110,133),(105,120,106,134),(136,161,155,167),(137,162,151,168),(138,163,152,169),(139,164,153,170),(140,165,154,166),(141,156,147,175),(142,157,148,171),(143,158,149,172),(144,159,150,173),(145,160,146,174),(176,207,195,201),(177,208,191,202),(178,209,192,203),(179,210,193,204),(180,206,194,205),(181,215,187,196),(182,211,188,197),(183,212,189,198),(184,213,190,199),(185,214,186,200),(216,247,235,241),(217,248,231,242),(218,249,232,243),(219,250,233,244),(220,246,234,245),(221,255,227,236),(222,251,228,237),(223,252,229,238),(224,253,230,239),(225,254,226,240),(256,281,275,287),(257,282,271,288),(258,283,272,289),(259,284,273,290),(260,285,274,286),(261,276,267,295),(262,277,268,291),(263,278,269,292),(264,279,270,293),(265,280,266,294)], [(1,107,35,101),(2,108,31,102),(3,109,32,103),(4,110,33,104),(5,106,34,105),(6,245,320,246),(7,241,316,247),(8,242,317,248),(9,243,318,249),(10,244,319,250),(11,251,17,237),(12,252,18,238),(13,253,19,239),(14,254,20,240),(15,255,16,236),(21,115,27,96),(22,111,28,97),(23,112,29,98),(24,113,30,99),(25,114,26,100),(36,127,55,121),(37,128,51,122),(38,129,52,123),(39,130,53,124),(40,126,54,125),(41,135,47,116),(42,131,48,117),(43,132,49,118),(44,133,50,119),(45,134,46,120),(56,147,75,141),(57,148,71,142),(58,149,72,143),(59,150,73,144),(60,146,74,145),(61,155,67,136),(62,151,68,137),(63,152,69,138),(64,153,70,139),(65,154,66,140),(76,167,95,161),(77,168,91,162),(78,169,92,163),(79,170,93,164),(80,166,94,165),(81,175,87,156),(82,171,88,157),(83,172,89,158),(84,173,90,159),(85,174,86,160),(176,261,195,267),(177,262,191,268),(178,263,192,269),(179,264,193,270),(180,265,194,266),(181,256,187,275),(182,257,188,271),(183,258,189,272),(184,259,190,273),(185,260,186,274),(196,281,215,287),(197,282,211,288),(198,283,212,289),(199,284,213,290),(200,285,214,286),(201,276,207,295),(202,277,208,291),(203,278,209,292),(204,279,210,293),(205,280,206,294),(216,301,235,307),(217,302,231,308),(218,303,232,309),(219,304,233,310),(220,305,234,306),(221,296,227,315),(222,297,228,311),(223,298,229,312),(224,299,230,313),(225,300,226,314)], [(1,61,21,56),(2,62,22,57),(3,63,23,58),(4,64,24,59),(5,65,25,60),(6,286,20,294),(7,287,16,295),(8,288,17,291),(9,289,18,292),(10,290,19,293),(11,277,317,282),(12,278,318,283),(13,279,319,284),(14,280,320,285),(15,276,316,281),(26,74,34,66),(27,75,35,67),(28,71,31,68),(29,72,32,69),(30,73,33,70),(36,81,41,76),(37,82,42,77),(38,83,43,78),(39,84,44,79),(40,85,45,80),(46,94,54,86),(47,95,55,87),(48,91,51,88),(49,92,52,89),(50,93,53,90),(96,141,101,136),(97,142,102,137),(98,143,103,138),(99,144,104,139),(100,145,105,140),(106,154,114,146),(107,155,115,147),(108,151,111,148),(109,152,112,149),(110,153,113,150),(116,161,121,156),(117,162,122,157),(118,163,123,158),(119,164,124,159),(120,165,125,160),(126,174,134,166),(127,175,135,167),(128,171,131,168),(129,172,132,169),(130,173,133,170),(176,216,181,221),(177,217,182,222),(178,218,183,223),(179,219,184,224),(180,220,185,225),(186,226,194,234),(187,227,195,235),(188,228,191,231),(189,229,192,232),(190,230,193,233),(196,236,201,241),(197,237,202,242),(198,238,203,243),(199,239,204,244),(200,240,205,245),(206,246,214,254),(207,247,215,255),(208,248,211,251),(209,249,212,252),(210,250,213,253),(256,296,261,301),(257,297,262,302),(258,298,263,303),(259,299,264,304),(260,300,265,305),(266,306,274,314),(267,307,275,315),(268,308,271,311),(269,309,272,312),(270,310,273,313)], [(1,181,21,176),(2,182,22,177),(3,183,23,178),(4,184,24,179),(5,185,25,180),(6,174,20,166),(7,175,16,167),(8,171,17,168),(9,172,18,169),(10,173,19,170),(11,162,317,157),(12,163,318,158),(13,164,319,159),(14,165,320,160),(15,161,316,156),(26,194,34,186),(27,195,35,187),(28,191,31,188),(29,192,32,189),(30,193,33,190),(36,201,41,196),(37,202,42,197),(38,203,43,198),(39,204,44,199),(40,205,45,200),(46,214,54,206),(47,215,55,207),(48,211,51,208),(49,212,52,209),(50,213,53,210),(56,221,61,216),(57,222,62,217),(58,223,63,218),(59,224,64,219),(60,225,65,220),(66,234,74,226),(67,235,75,227),(68,231,71,228),(69,232,72,229),(70,233,73,230),(76,241,81,236),(77,242,82,237),(78,243,83,238),(79,244,84,239),(80,245,85,240),(86,254,94,246),(87,255,95,247),(88,251,91,248),(89,252,92,249),(90,253,93,250),(96,261,101,256),(97,262,102,257),(98,263,103,258),(99,264,104,259),(100,265,105,260),(106,274,114,266),(107,275,115,267),(108,271,111,268),(109,272,112,269),(110,273,113,270),(116,281,121,276),(117,282,122,277),(118,283,123,278),(119,284,124,279),(120,285,125,280),(126,294,134,286),(127,295,135,287),(128,291,131,288),(129,292,132,289),(130,293,133,290),(136,301,141,296),(137,302,142,297),(138,303,143,298),(139,304,144,299),(140,305,145,300),(146,314,154,306),(147,315,155,307),(148,311,151,308),(149,312,152,309),(150,313,153,310)]])
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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | - | - | ||||||||
| image | C1 | C2 | C2 | C2 | C5 | C10 | C10 | C10 | Q8 | C4○D4 | C5×Q8 | C5×C4○D4 | 2- 1+4 | C5×2- 1+4 |
| kernel | C5×Q8⋊3Q8 | Q8×C20 | C5×C42.C2 | C5×C4⋊Q8 | Q8⋊3Q8 | C4×Q8 | C42.C2 | C4⋊Q8 | C5×Q8 | C20 | Q8 | C4 | C10 | C2 |
| # reps | 1 | 6 | 6 | 3 | 4 | 24 | 24 | 12 | 4 | 4 | 16 | 16 | 1 | 4 |
Matrix representation of C5×Q8⋊3Q8 ►in GL4(𝔽41) generated by
| 37 | 0 | 0 | 0 |
| 0 | 37 | 0 | 0 |
| 0 | 0 | 37 | 0 |
| 0 | 0 | 0 | 37 |
| 1 | 39 | 0 | 0 |
| 1 | 40 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 7 | 14 | 0 | 0 |
| 14 | 34 | 0 | 0 |
| 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 40 |
| 40 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 40 | 0 |
| 32 | 18 | 0 | 0 |
| 32 | 9 | 0 | 0 |
| 0 | 0 | 0 | 32 |
| 0 | 0 | 32 | 0 |
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[1,1,0,0,39,40,0,0,0,0,40,0,0,0,0,40],[7,14,0,0,14,34,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[32,32,0,0,18,9,0,0,0,0,0,32,0,0,32,0] >;
C5×Q8⋊3Q8 in GAP, Magma, Sage, TeX
C_5\times Q_8\rtimes_3Q_8
% in TeX
G:=Group("C5xQ8:3Q8"); // GroupNames label
G:=SmallGroup(320,1559);
// by ID
G=gap.SmallGroup(320,1559);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,568,3446,856,1242,304]);
// 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,e*c*e^-1=b^2*c,e*d*e^-1=d^-1>;
// generators/relations