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G = (C2×C20)⋊10Q8order 320 = 26·5

1st semidirect product of C2×C20 and Q8 acting via Q8/C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20)⋊10Q8, (C2×C4)⋊8Dic10, (C2×C4).89D20, C10.7(C4⋊Q8), C10.18(C4×Q8), (C2×C20).470D4, (C2×C42).14D5, (C2×Dic10)⋊17C4, C2.2(C202Q8), C2.11(C4×Dic10), C22.34(C2×D20), C20.93(C22⋊C4), C2.1(C4.D20), (C22×C4).395D10, C10.53(C22⋊Q8), C4.22(D10⋊C4), C2.2(C20.48D4), C10.10(C4.4D4), C22.40(C4○D20), (C22×Dic10).4C2, C22.17(C2×Dic10), C23.262(C22×D5), (C22×C20).470C22, (C22×C10).304C23, C53(C23.67C23), C10.10C42.10C2, (C22×Dic5).27C22, (C2×C4×C20).10C2, (C2×C4).109(C4×D5), (C2×C10).24(C2×Q8), C22.117(C2×C4×D5), (C2×C20).397(C2×C4), C2.5(C2×D10⋊C4), (C2×C10).424(C2×D4), C10.72(C2×C22⋊C4), (C2×C4⋊Dic5).15C2, C22.39(C2×C5⋊D4), (C2×C10).65(C4○D4), (C2×C4).238(C5⋊D4), (C2×Dic5).27(C2×C4), (C2×C10).195(C22×C4), SmallGroup(320,556)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C20)⋊10Q8
C1C5C10C2×C10C22×C10C22×Dic5C22×Dic10 — (C2×C20)⋊10Q8
C5C2×C10 — (C2×C20)⋊10Q8
C1C23C2×C42

Generators and relations for (C2×C20)⋊10Q8
 G = < a,b,c,d | a2=b20=c4=1, d2=c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=ab9, dcd-1=c-1 >

Subgroups: 606 in 186 conjugacy classes, 87 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×4], C5, C2×C4 [×10], C2×C4 [×18], Q8 [×8], C23, C10 [×3], C10 [×4], C42 [×2], C4⋊C4 [×2], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×Q8 [×8], Dic5 [×6], C20 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42 [×4], C2×C42, C2×C4⋊C4, C22×Q8, Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×10], C2×C20 [×4], C22×C10, C23.67C23, C4⋊Dic5 [×2], C4×C20 [×2], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20, C22×C20 [×2], C10.10C42 [×4], C2×C4⋊Dic5, C2×C4×C20, C22×Dic10, (C2×C20)⋊10Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], Q8 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×2], D10 [×3], C2×C22⋊C4, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4, C4⋊Q8, Dic10 [×4], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.67C23, D10⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5, C2×D20, C4○D20 [×2], C2×C5⋊D4, C4×Dic10 [×2], C202Q8, C4.D20, C20.48D4 [×2], C2×D10⋊C4, (C2×C20)⋊10Q8

Smallest permutation representation of (C2×C20)⋊10Q8
Regular action on 320 points
Generators in S320
(1 283)(2 284)(3 285)(4 286)(5 287)(6 288)(7 289)(8 290)(9 291)(10 292)(11 293)(12 294)(13 295)(14 296)(15 297)(16 298)(17 299)(18 300)(19 281)(20 282)(21 85)(22 86)(23 87)(24 88)(25 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 97)(34 98)(35 99)(36 100)(37 81)(38 82)(39 83)(40 84)(41 159)(42 160)(43 141)(44 142)(45 143)(46 144)(47 145)(48 146)(49 147)(50 148)(51 149)(52 150)(53 151)(54 152)(55 153)(56 154)(57 155)(58 156)(59 157)(60 158)(61 174)(62 175)(63 176)(64 177)(65 178)(66 179)(67 180)(68 161)(69 162)(70 163)(71 164)(72 165)(73 166)(74 167)(75 168)(76 169)(77 170)(78 171)(79 172)(80 173)(101 200)(102 181)(103 182)(104 183)(105 184)(106 185)(107 186)(108 187)(109 188)(110 189)(111 190)(112 191)(113 192)(114 193)(115 194)(116 195)(117 196)(118 197)(119 198)(120 199)(121 266)(122 267)(123 268)(124 269)(125 270)(126 271)(127 272)(128 273)(129 274)(130 275)(131 276)(132 277)(133 278)(134 279)(135 280)(136 261)(137 262)(138 263)(139 264)(140 265)(201 238)(202 239)(203 240)(204 221)(205 222)(206 223)(207 224)(208 225)(209 226)(210 227)(211 228)(212 229)(213 230)(214 231)(215 232)(216 233)(217 234)(218 235)(219 236)(220 237)(241 315)(242 316)(243 317)(244 318)(245 319)(246 320)(247 301)(248 302)(249 303)(250 304)(251 305)(252 306)(253 307)(254 308)(255 309)(256 310)(257 311)(258 312)(259 313)(260 314)
(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)
(1 171 263 55)(2 172 264 56)(3 173 265 57)(4 174 266 58)(5 175 267 59)(6 176 268 60)(7 177 269 41)(8 178 270 42)(9 179 271 43)(10 180 272 44)(11 161 273 45)(12 162 274 46)(13 163 275 47)(14 164 276 48)(15 165 277 49)(16 166 278 50)(17 167 279 51)(18 168 280 52)(19 169 261 53)(20 170 262 54)(21 213 307 111)(22 214 308 112)(23 215 309 113)(24 216 310 114)(25 217 311 115)(26 218 312 116)(27 219 313 117)(28 220 314 118)(29 201 315 119)(30 202 316 120)(31 203 317 101)(32 204 318 102)(33 205 319 103)(34 206 320 104)(35 207 301 105)(36 208 302 106)(37 209 303 107)(38 210 304 108)(39 211 305 109)(40 212 306 110)(61 121 156 286)(62 122 157 287)(63 123 158 288)(64 124 159 289)(65 125 160 290)(66 126 141 291)(67 127 142 292)(68 128 143 293)(69 129 144 294)(70 130 145 295)(71 131 146 296)(72 132 147 297)(73 133 148 298)(74 134 149 299)(75 135 150 300)(76 136 151 281)(77 137 152 282)(78 138 153 283)(79 139 154 284)(80 140 155 285)(81 226 249 186)(82 227 250 187)(83 228 251 188)(84 229 252 189)(85 230 253 190)(86 231 254 191)(87 232 255 192)(88 233 256 193)(89 234 257 194)(90 235 258 195)(91 236 259 196)(92 237 260 197)(93 238 241 198)(94 239 242 199)(95 240 243 200)(96 221 244 181)(97 222 245 182)(98 223 246 183)(99 224 247 184)(100 225 248 185)
(1 187 263 227)(2 117 264 219)(3 185 265 225)(4 115 266 217)(5 183 267 223)(6 113 268 215)(7 181 269 221)(8 111 270 213)(9 199 271 239)(10 109 272 211)(11 197 273 237)(12 107 274 209)(13 195 275 235)(14 105 276 207)(15 193 277 233)(16 103 278 205)(17 191 279 231)(18 101 280 203)(19 189 261 229)(20 119 262 201)(21 178 307 42)(22 74 308 149)(23 176 309 60)(24 72 310 147)(25 174 311 58)(26 70 312 145)(27 172 313 56)(28 68 314 143)(29 170 315 54)(30 66 316 141)(31 168 317 52)(32 64 318 159)(33 166 319 50)(34 62 320 157)(35 164 301 48)(36 80 302 155)(37 162 303 46)(38 78 304 153)(39 180 305 44)(40 76 306 151)(41 96 177 244)(43 94 179 242)(45 92 161 260)(47 90 163 258)(49 88 165 256)(51 86 167 254)(53 84 169 252)(55 82 171 250)(57 100 173 248)(59 98 175 246)(61 257 156 89)(63 255 158 87)(65 253 160 85)(67 251 142 83)(69 249 144 81)(71 247 146 99)(73 245 148 97)(75 243 150 95)(77 241 152 93)(79 259 154 91)(102 124 204 289)(104 122 206 287)(106 140 208 285)(108 138 210 283)(110 136 212 281)(112 134 214 299)(114 132 216 297)(116 130 218 295)(118 128 220 293)(120 126 202 291)(121 234 286 194)(123 232 288 192)(125 230 290 190)(127 228 292 188)(129 226 294 186)(131 224 296 184)(133 222 298 182)(135 240 300 200)(137 238 282 198)(139 236 284 196)

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

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

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

92 conjugacy classes

class 1 2A···2G4A···4L4M···4T5A5B10A···10N20A···20AV
order12···24···44···45510···1020···20
size11···12···220···20222···22···2

92 irreducible representations

dim1111112222222222
type++++++-++-+
imageC1C2C2C2C2C4D4Q8D5C4○D4D10Dic10C4×D5D20C5⋊D4C4○D20
kernel(C2×C20)⋊10Q8C10.10C42C2×C4⋊Dic5C2×C4×C20C22×Dic10C2×Dic10C2×C20C2×C20C2×C42C2×C10C22×C4C2×C4C2×C4C2×C4C2×C4C22
# reps141118442461688816

Matrix representation of (C2×C20)⋊10Q8 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
2290000
3200000
0030200
00251400
00002711
00003032
,
100000
010000
00302800
00221100
000010
000001
,
2020000
26210000
00163100
00382500
00002621
0000315

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],[22,32,0,0,0,0,9,0,0,0,0,0,0,0,30,25,0,0,0,0,2,14,0,0,0,0,0,0,27,30,0,0,0,0,11,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,30,22,0,0,0,0,28,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[20,26,0,0,0,0,2,21,0,0,0,0,0,0,16,38,0,0,0,0,31,25,0,0,0,0,0,0,26,3,0,0,0,0,21,15] >;

(C2×C20)⋊10Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_{10}Q_8
% in TeX

G:=Group("(C2xC20):10Q8");
// GroupNames label

G:=SmallGroup(320,556);
// by ID

G=gap.SmallGroup(320,556);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,120,758,58,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a*b^9,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽