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G = C2×C20.8Q8order 320 = 26·5

Direct product of C2 and C20.8Q8

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20.8Q8, C104(C4⋊C8), (C2×Dic5)⋊5C8, Dic55(C2×C8), C20.77(C4⋊C4), (C2×C20).68Q8, C20.85(C2×Q8), (C22×C8).5D5, (C2×C8).290D10, C20.432(C2×D4), (C2×C20).497D4, (C22×C40).8C2, C22.16(C8×D5), C23.61(C4×D5), C10.42(C22×C8), (C4×Dic5).23C4, C4.50(C2×Dic10), (C2×C4).58Dic10, (C2×C40).350C22, (C2×C20).854C23, (C22×C4).461D10, C22.9(C8⋊D5), C10.44(C2×M4(2)), (C2×C10).29M4(2), C4.33(C10.D4), (C22×Dic5).17C4, (C22×C20).557C22, (C4×Dic5).310C22, C22.25(C10.D4), C56(C2×C4⋊C8), C2.18(D5×C2×C8), C10.65(C2×C4⋊C4), C2.3(C2×C8⋊D5), C22.57(C2×C4×D5), (C2×C10).46(C2×C8), (C2×C4).182(C4×D5), C4.122(C2×C5⋊D4), (C2×C4×Dic5).38C2, (C2×C10).75(C4⋊C4), (C2×C20).425(C2×C4), C2.2(C2×C10.D4), (C2×C4).275(C5⋊D4), (C22×C52C8).19C2, (C2×C4).796(C22×D5), (C22×C10).157(C2×C4), (C2×C10).225(C22×C4), (C2×C52C8).325C22, (C2×Dic5).152(C2×C4), SmallGroup(320,726)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C20.8Q8
C1C5C10C20C2×C20C4×Dic5C2×C4×Dic5 — C2×C20.8Q8
C5C10 — C2×C20.8Q8
C1C22×C4C22×C8

Generators and relations for C2×C20.8Q8
 G = < a,b,c,d | a2=b20=1, c4=b10, d2=b15c2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b9, dcd-1=b15c3 >

Subgroups: 334 in 138 conjugacy classes, 87 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], C23, C10 [×3], C10 [×4], C42 [×4], C2×C8 [×2], C2×C8 [×6], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5 [×2], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], C4⋊C8 [×4], C2×C42, C22×C8, C22×C8, C52C8 [×2], C40 [×2], C2×Dic5 [×8], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×C4⋊C8, C2×C52C8 [×2], C2×C52C8 [×2], C4×Dic5 [×4], C2×C40 [×2], C2×C40 [×2], C22×Dic5 [×2], C22×C20, C20.8Q8 [×4], C22×C52C8, C2×C4×Dic5, C22×C40, C2×C20.8Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C2×C8 [×6], M4(2) [×2], C22×C4, C2×D4, C2×Q8, D10 [×3], C4⋊C8 [×4], C2×C4⋊C4, C22×C8, C2×M4(2), Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C2×C4⋊C8, C8×D5 [×2], C8⋊D5 [×2], C10.D4 [×4], C2×Dic10, C2×C4×D5, C2×C5⋊D4, C20.8Q8 [×4], D5×C2×C8, C2×C8⋊D5, C2×C10.D4, C2×C20.8Q8

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

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

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

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

104 conjugacy classes

class 1 2A···2G4A···4H4I···4P5A5B8A···8H8I···8P10A···10N20A···20P40A···40AF
order12···24···44···4558···88···810···1020···2040···40
size11···11···110···10222···210···102···22···22···2

104 irreducible representations

dim11111111222222222222
type++++++-+++-
imageC1C2C2C2C2C4C4C8D4Q8D5M4(2)D10D10Dic10C4×D5C5⋊D4C4×D5C8×D5C8⋊D5
kernelC2×C20.8Q8C20.8Q8C22×C52C8C2×C4×Dic5C22×C40C4×Dic5C22×Dic5C2×Dic5C2×C20C2×C20C22×C8C2×C10C2×C8C22×C4C2×C4C2×C4C2×C4C23C22C22
# reps14111441622244284841616

Matrix representation of C2×C20.8Q8 in GL5(𝔽41)

400000
040000
004000
00010
00001
,
400000
00900
0321300
00009
0003219
,
90000
038000
003800
0003327
000148
,
10000
013200
0392800
0002914
0001612

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,0,32,0,0,0,9,13,0,0,0,0,0,0,32,0,0,0,9,19],[9,0,0,0,0,0,38,0,0,0,0,0,38,0,0,0,0,0,33,14,0,0,0,27,8],[1,0,0,0,0,0,13,39,0,0,0,2,28,0,0,0,0,0,29,16,0,0,0,14,12] >;

C2×C20.8Q8 in GAP, Magma, Sage, TeX

C_2\times C_{20}._8Q_8
% in TeX

G:=Group("C2xC20.8Q8");
// GroupNames label

G:=SmallGroup(320,726);
// by ID

G=gap.SmallGroup(320,726);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,58,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=b^10,d^2=b^15*c^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^9,d*c*d^-1=b^15*c^3>;
// generators/relations

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