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G = (C2×C20).53D4order 320 = 26·5

27th non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).53D4, (C2×C4).44D20, C10.27(C4⋊Q8), (C2×Dic5).7Q8, C22.48(Q8×D5), C2.9(C207D4), C10.61(C4⋊D4), C22.128(C2×D20), (C22×C4).100D10, C10.47(C22⋊Q8), C2.16(D102Q8), C2.7(Dic5⋊Q8), (C22×C20).64C22, C10.22(C42.C2), C23.377(C22×D5), C22.105(C4○D20), (C22×C10).347C23, C54(C23.81C23), C22.101(D42D5), C10.10C42.28C2, C2.13(Dic5.Q8), C10.75(C22.D4), C2.9(C23.18D10), (C22×Dic5).55C22, (C2×C4⋊C4).20D5, (C10×C4⋊C4).22C2, (C2×C10).82(C2×Q8), (C2×C10).151(C2×D4), (C2×C4).38(C5⋊D4), (C2×C4⋊Dic5).19C2, (C2×C10).84(C4○D4), C22.137(C2×C5⋊D4), (C2×C10.D4).34C2, SmallGroup(320,610)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).53D4
C1C5C10C2×C10C22×C10C22×Dic5C2×C4⋊Dic5 — (C2×C20).53D4
C5C22×C10 — (C2×C20).53D4
C1C23C2×C4⋊C4

Generators and relations for (C2×C20).53D4
 G = < a,b,c,d | a2=b20=c4=1, d2=a, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 486 in 150 conjugacy classes, 63 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22 [×3], C22 [×4], C5, C2×C4 [×4], C2×C4 [×21], C23, C10 [×3], C10 [×4], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], Dic5 [×6], C20 [×5], C2×C10 [×3], C2×C10 [×4], C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.81C23, C10.D4 [×4], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42, C10.10C42 [×2], C2×C10.D4 [×2], C2×C4⋊Dic5, C10×C4⋊C4, (C2×C20).53D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D5, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D10 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, D20 [×2], C5⋊D4 [×2], C22×D5, C23.81C23, C2×D20, C4○D20, D42D5 [×2], Q8×D5 [×2], C2×C5⋊D4, Dic5.Q8 [×2], D102Q8 [×2], C207D4, C23.18D10, Dic5⋊Q8, (C2×C20).53D4

Smallest permutation representation of (C2×C20).53D4
Regular action on 320 points
Generators in S320
(1 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 290)(22 291)(23 292)(24 293)(25 294)(26 295)(27 296)(28 297)(29 298)(30 299)(31 300)(32 281)(33 282)(34 283)(35 284)(36 285)(37 286)(38 287)(39 288)(40 289)(61 155)(62 156)(63 157)(64 158)(65 159)(66 160)(67 141)(68 142)(69 143)(70 144)(71 145)(72 146)(73 147)(74 148)(75 149)(76 150)(77 151)(78 152)(79 153)(80 154)(81 168)(82 169)(83 170)(84 171)(85 172)(86 173)(87 174)(88 175)(89 176)(90 177)(91 178)(92 179)(93 180)(94 161)(95 162)(96 163)(97 164)(98 165)(99 166)(100 167)(101 252)(102 253)(103 254)(104 255)(105 256)(106 257)(107 258)(108 259)(109 260)(110 241)(111 242)(112 243)(113 244)(114 245)(115 246)(116 247)(117 248)(118 249)(119 250)(120 251)(121 183)(122 184)(123 185)(124 186)(125 187)(126 188)(127 189)(128 190)(129 191)(130 192)(131 193)(132 194)(133 195)(134 196)(135 197)(136 198)(137 199)(138 200)(139 181)(140 182)(201 275)(202 276)(203 277)(204 278)(205 279)(206 280)(207 261)(208 262)(209 263)(210 264)(211 265)(212 266)(213 267)(214 268)(215 269)(216 270)(217 271)(218 272)(219 273)(220 274)(221 320)(222 301)(223 302)(224 303)(225 304)(226 305)(227 306)(228 307)(229 308)(230 309)(231 310)(232 311)(233 312)(234 313)(235 314)(236 315)(237 316)(238 317)(239 318)(240 319)
(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)
(1 286 205 104)(2 36 206 254)(3 284 207 102)(4 34 208 252)(5 282 209 120)(6 32 210 250)(7 300 211 118)(8 30 212 248)(9 298 213 116)(10 28 214 246)(11 296 215 114)(12 26 216 244)(13 294 217 112)(14 24 218 242)(15 292 219 110)(16 22 220 260)(17 290 201 108)(18 40 202 258)(19 288 203 106)(20 38 204 256)(21 275 259 48)(23 273 241 46)(25 271 243 44)(27 269 245 42)(29 267 247 60)(31 265 249 58)(33 263 251 56)(35 261 253 54)(37 279 255 52)(39 277 257 50)(41 297 268 115)(43 295 270 113)(45 293 272 111)(47 291 274 109)(49 289 276 107)(51 287 278 105)(53 285 280 103)(55 283 262 101)(57 281 264 119)(59 299 266 117)(61 230 90 135)(62 308 91 196)(63 228 92 133)(64 306 93 194)(65 226 94 131)(66 304 95 192)(67 224 96 129)(68 302 97 190)(69 222 98 127)(70 320 99 188)(71 240 100 125)(72 318 81 186)(73 238 82 123)(74 316 83 184)(75 236 84 121)(76 314 85 182)(77 234 86 139)(78 312 87 200)(79 232 88 137)(80 310 89 198)(122 148 237 170)(124 146 239 168)(126 144 221 166)(128 142 223 164)(130 160 225 162)(132 158 227 180)(134 156 229 178)(136 154 231 176)(138 152 233 174)(140 150 235 172)(141 303 163 191)(143 301 165 189)(145 319 167 187)(147 317 169 185)(149 315 171 183)(151 313 173 181)(153 311 175 199)(155 309 177 197)(157 307 179 195)(159 305 161 193)
(1 236 52 315)(2 235 53 314)(3 234 54 313)(4 233 55 312)(5 232 56 311)(6 231 57 310)(7 230 58 309)(8 229 59 308)(9 228 60 307)(10 227 41 306)(11 226 42 305)(12 225 43 304)(13 224 44 303)(14 223 45 302)(15 222 46 301)(16 221 47 320)(17 240 48 319)(18 239 49 318)(19 238 50 317)(20 237 51 316)(21 145 290 71)(22 144 291 70)(23 143 292 69)(24 142 293 68)(25 141 294 67)(26 160 295 66)(27 159 296 65)(28 158 297 64)(29 157 298 63)(30 156 299 62)(31 155 300 61)(32 154 281 80)(33 153 282 79)(34 152 283 78)(35 151 284 77)(36 150 285 76)(37 149 286 75)(38 148 287 74)(39 147 288 73)(40 146 289 72)(81 258 168 107)(82 257 169 106)(83 256 170 105)(84 255 171 104)(85 254 172 103)(86 253 173 102)(87 252 174 101)(88 251 175 120)(89 250 176 119)(90 249 177 118)(91 248 178 117)(92 247 179 116)(93 246 180 115)(94 245 161 114)(95 244 162 113)(96 243 163 112)(97 242 164 111)(98 241 165 110)(99 260 166 109)(100 259 167 108)(121 279 183 205)(122 278 184 204)(123 277 185 203)(124 276 186 202)(125 275 187 201)(126 274 188 220)(127 273 189 219)(128 272 190 218)(129 271 191 217)(130 270 192 216)(131 269 193 215)(132 268 194 214)(133 267 195 213)(134 266 196 212)(135 265 197 211)(136 264 198 210)(137 263 199 209)(138 262 200 208)(139 261 181 207)(140 280 182 206)

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim111112222222244
type+++++-++++--
imageC1C2C2C2C2Q8D4D5C4○D4D10D20C5⋊D4C4○D20D42D5Q8×D5
kernel(C2×C20).53D4C10.10C42C2×C10.D4C2×C4⋊Dic5C10×C4⋊C4C2×Dic5C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22
# reps132114426688844

Matrix representation of (C2×C20).53D4 in GL8(𝔽41)

10000000
01000000
00100000
00010000
00001000
00000100
000000400
000000040
,
10000000
01000000
0035400000
0036400000
000003200
000032000
00000009
00000090
,
2612000000
2915000000
0035400000
003560000
0000343800
00003700
000000400
000000040
,
2915000000
2612000000
00610000
006350000
00003700
0000343800
00000001
000000400

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,35,36,0,0,0,0,0,0,40,40,0,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0],[26,29,0,0,0,0,0,0,12,15,0,0,0,0,0,0,0,0,35,35,0,0,0,0,0,0,40,6,0,0,0,0,0,0,0,0,34,3,0,0,0,0,0,0,38,7,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[29,26,0,0,0,0,0,0,15,12,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,3,34,0,0,0,0,0,0,7,38,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0] >;

(C2×C20).53D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{53}D_4
% in TeX

G:=Group("(C2xC20).53D4");
// GroupNames label

G:=SmallGroup(320,610);
// by ID

G=gap.SmallGroup(320,610);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,254,387,184,12550]);
// Polycyclic

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

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