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## G = (C2×C20).54D4order 320 = 26·5

### 28th 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).54D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C10.D4 — (C2×C20).54D4
 Lower central C5 — C22×C10 — (C2×C20).54D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C20).54D4
G = < a,b,c,d | a2=b20=c4=1, d2=ab10, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b10c-1 >

Subgroups: 486 in 150 conjugacy classes, 63 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, 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, C2×C10.D4, C2×C4⋊Dic5, C10×C4⋊C4, (C2×C20).54D4
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, Dic10, C5⋊D4, C22×D5, C23.81C23, C2×Dic10, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, C20⋊Q8, Dic5.Q8, C4.Dic10, D10.13D4, C20.48D4, Dic5⋊D4, D103Q8, (C2×C20).54D4

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

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

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

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

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 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + - + - + + - + - - + image C1 C2 C2 C2 C2 D4 Q8 D4 Q8 D5 C4○D4 D10 Dic10 C5⋊D4 C4○D20 D4×D5 D4⋊2D5 Q8×D5 Q8⋊2D5 kernel (C2×C20).54D4 C10.10C42 C2×C10.D4 C2×C4⋊Dic5 C10×C4⋊C4 C2×Dic5 C2×Dic5 C2×C20 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 C22 C22 # reps 1 3 2 1 1 2 2 2 2 2 6 6 8 8 8 2 2 2 2

Matrix representation of (C2×C20).54D4 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
,
 3 20 0 0 0 0 20 38 0 0 0 0 0 0 27 39 0 0 0 0 16 11 0 0 0 0 0 0 27 16 0 0 0 0 16 14
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 11 2 0 0 0 0 22 30 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 9 32 0 0 0 0 0 32 0 0 0 0 0 0 0 40 0 0 0 0 1 0

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],[3,20,0,0,0,0,20,38,0,0,0,0,0,0,27,16,0,0,0,0,39,11,0,0,0,0,0,0,27,16,0,0,0,0,16,14],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,11,22,0,0,0,0,2,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,32,32,0,0,0,0,0,0,0,1,0,0,0,0,40,0] >;

(C2×C20).54D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{54}D_4
% in TeX

G:=Group("(C2xC20).54D4");
// GroupNames label

G:=SmallGroup(320,611);
// by ID

G=gap.SmallGroup(320,611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,387,184,12550]);
// Polycyclic

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

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