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## G = (C2×C20).287D4order 320 = 26·5

### 261st non-split extension by C2×C20 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C10 — (C2×C20).287D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C10.D4 — (C2×C20).287D4
 Lower central C5 — C22×C10 — (C2×C20).287D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C20).287D4
G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=ab9, dcd-1=ab10c-1 >

Subgroups: 486 in 150 conjugacy classes, 61 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C4 [×11], C22 [×3], C22 [×4], C5, C2×C4 [×2], C2×C4 [×23], C23, C10 [×3], C10 [×4], C4⋊C4 [×8], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×7], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.81C23, C10.D4 [×6], C5×C4⋊C4 [×2], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42, C10.10C42 [×2], C2×C10.D4, C2×C10.D4 [×2], C10×C4⋊C4, (C2×C20).287D4
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, C5⋊D4 [×2], C22×D5, C23.81C23, C4○D20 [×2], D4×D5, D42D5, Q8×D5 [×2], C2×C5⋊D4, Dic5.Q8 [×2], D10⋊Q8 [×2], C23.23D10, C202D4, Dic5⋊Q8, (C2×C20).287D4

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

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

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

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

62 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4F 4G ··· 4N 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 ··· 4 4 ··· 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 4 ··· 4 20 ··· 20 2 2 2 ··· 2 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + - + + + + - - image C1 C2 C2 C2 D4 Q8 D4 D5 C4○D4 D10 C5⋊D4 C4○D20 D4×D5 D4⋊2D5 Q8×D5 kernel (C2×C20).287D4 C10.10C42 C2×C10.D4 C10×C4⋊C4 C2×Dic5 C2×Dic5 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C22 C22 C22 C22 # reps 1 3 3 1 2 4 2 2 6 6 8 16 2 2 4

Matrix representation of (C2×C20).287D4 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 22 0 0 0 0 13 0 0 0 0 0 0 0 0 1 0 0 0 0 40 7 0 0 0 0 0 0 20 23 0 0 0 0 18 35
,
 0 19 0 0 0 0 28 0 0 0 0 0 0 0 29 16 0 0 0 0 14 12 0 0 0 0 0 0 25 16 0 0 0 0 2 16
,
 30 22 0 0 0 0 28 11 0 0 0 0 0 0 15 38 0 0 0 0 20 26 0 0 0 0 0 0 6 6 0 0 0 0 1 35

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,13,0,0,0,0,22,0,0,0,0,0,0,0,0,40,0,0,0,0,1,7,0,0,0,0,0,0,20,18,0,0,0,0,23,35],[0,28,0,0,0,0,19,0,0,0,0,0,0,0,29,14,0,0,0,0,16,12,0,0,0,0,0,0,25,2,0,0,0,0,16,16],[30,28,0,0,0,0,22,11,0,0,0,0,0,0,15,20,0,0,0,0,38,26,0,0,0,0,0,0,6,1,0,0,0,0,6,35] >;

(C2×C20).287D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{287}D_4
% in TeX

G:=Group("(C2xC20).287D4");
// GroupNames label

G:=SmallGroup(320,607);
// by ID

G=gap.SmallGroup(320,607);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,232,254,387,100,12550]);
// Polycyclic

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

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