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