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G = (C2×C20).53D4  order 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(C20⋊7D4), C10.61(C4⋊D4), C22.128(C2×D20), (C22×C4).100D10, C10.47(C22⋊Q8), C2.16(D10⋊2Q8), C2.7(Dic5⋊Q8), (C22×C20).64C22, C10.22(C42.C2), C23.377(C22×D5), C22.105(C4○D20), (C22×C10).347C23, C5⋊4(C23.81C23), C22.101(D4⋊2D5), 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

Derived series
C1 — C22×C10 — (C2×C20).53D4
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4⋊Dic5 — (C2×C20).53D4
Lower central
C5 — C22×C10 — (C2×C20).53D4
Upper central
C1 — C23 — C2×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, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.81C23, C10.D4, C4⋊Dic5, C5×C4⋊C4, C22×Dic5, C22×C20, C10.10C42, C10.10C42, C2×C10.D4, C2×C4⋊Dic5, C10×C4⋊C4, (C2×C20).53D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C4⋊D4, C22⋊Q8, C22.D4, C42.C2, C4⋊Q8, D20, C5⋊D4, C22×D5, C23.81C23, C2×D20, C4○D20, D4⋊2D5, Q8×D5, C2×C5⋊D4, Dic5.Q8, D10⋊2Q8, C20⋊7D4, C23.18D10, Dic5⋊Q8, (C2×C20).53D4

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

(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)

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

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

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○D20D4⋊2D5Q8×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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