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## G = C2.(C4×D20)  order 320 = 26·5

### 1st central stem extension by C2 of C4×D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2.(C4×D20)
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4⋊Dic5 — C2.(C4×D20)
 Lower central C5 — C2×C10 — C2.(C4×D20)
 Upper central C1 — C23 — C2.C42

Generators and relations for C2.(C4×D20)
G = < a,b,c,d | a2=b4=c20=1, d2=a, cbc-1=dbd-1=ab=ba, ac=ca, ad=da, dcd-1=c-1 >

Subgroups: 478 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2, C4, C22, C5, C2×C4, C2×C4, C23, C10, C42, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2.C42, C2.C42, C2×C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.63C23, C4×Dic5, C10.D4, C4⋊Dic5, C22×Dic5, C22×C20, C10.10C42, C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C2.(C4×D20)
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4, D10, C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D5, D20, C22×D5, C23.63C23, C2×C4×D5, C2×D20, C4○D20, D42D5, Q8×D5, C4×D20, C23.11D10, C23.D10, C22.D20, Dic53Q8, Dic5.Q8, D102Q8, C2.(C4×D20)

Smallest permutation representation of C2.(C4×D20)
Regular action on 320 points
Generators in S320
(1 207)(2 208)(3 209)(4 210)(5 211)(6 212)(7 213)(8 214)(9 215)(10 216)(11 217)(12 218)(13 219)(14 220)(15 201)(16 202)(17 203)(18 204)(19 205)(20 206)(21 131)(22 132)(23 133)(24 134)(25 135)(26 136)(27 137)(28 138)(29 139)(30 140)(31 121)(32 122)(33 123)(34 124)(35 125)(36 126)(37 127)(38 128)(39 129)(40 130)(41 262)(42 263)(43 264)(44 265)(45 266)(46 267)(47 268)(48 269)(49 270)(50 271)(51 272)(52 273)(53 274)(54 275)(55 276)(56 277)(57 278)(58 279)(59 280)(60 261)(61 259)(62 260)(63 241)(64 242)(65 243)(66 244)(67 245)(68 246)(69 247)(70 248)(71 249)(72 250)(73 251)(74 252)(75 253)(76 254)(77 255)(78 256)(79 257)(80 258)(81 293)(82 294)(83 295)(84 296)(85 297)(86 298)(87 299)(88 300)(89 281)(90 282)(91 283)(92 284)(93 285)(94 286)(95 287)(96 288)(97 289)(98 290)(99 291)(100 292)(101 240)(102 221)(103 222)(104 223)(105 224)(106 225)(107 226)(108 227)(109 228)(110 229)(111 230)(112 231)(113 232)(114 233)(115 234)(116 235)(117 236)(118 237)(119 238)(120 239)(141 195)(142 196)(143 197)(144 198)(145 199)(146 200)(147 181)(148 182)(149 183)(150 184)(151 185)(152 186)(153 187)(154 188)(155 189)(156 190)(157 191)(158 192)(159 193)(160 194)(161 309)(162 310)(163 311)(164 312)(165 313)(166 314)(167 315)(168 316)(169 317)(170 318)(171 319)(172 320)(173 301)(174 302)(175 303)(176 304)(177 305)(178 306)(179 307)(180 308)
(1 74 265 84)(2 253 266 297)(3 76 267 86)(4 255 268 299)(5 78 269 88)(6 257 270 281)(7 80 271 90)(8 259 272 283)(9 62 273 92)(10 241 274 285)(11 64 275 94)(12 243 276 287)(13 66 277 96)(14 245 278 289)(15 68 279 98)(16 247 280 291)(17 70 261 100)(18 249 262 293)(19 72 263 82)(20 251 264 295)(21 189 232 307)(22 156 233 180)(23 191 234 309)(24 158 235 162)(25 193 236 311)(26 160 237 164)(27 195 238 313)(28 142 239 166)(29 197 240 315)(30 144 221 168)(31 199 222 317)(32 146 223 170)(33 181 224 319)(34 148 225 172)(35 183 226 301)(36 150 227 174)(37 185 228 303)(38 152 229 176)(39 187 230 305)(40 154 231 178)(41 81 204 71)(42 294 205 250)(43 83 206 73)(44 296 207 252)(45 85 208 75)(46 298 209 254)(47 87 210 77)(48 300 211 256)(49 89 212 79)(50 282 213 258)(51 91 214 61)(52 284 215 260)(53 93 216 63)(54 286 217 242)(55 95 218 65)(56 288 219 244)(57 97 220 67)(58 290 201 246)(59 99 202 69)(60 292 203 248)(101 167 139 143)(102 316 140 198)(103 169 121 145)(104 318 122 200)(105 171 123 147)(106 320 124 182)(107 173 125 149)(108 302 126 184)(109 175 127 151)(110 304 128 186)(111 177 129 153)(112 306 130 188)(113 179 131 155)(114 308 132 190)(115 161 133 157)(116 310 134 192)(117 163 135 159)(118 312 136 194)(119 165 137 141)(120 314 138 196)
(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 25 207 135)(2 24 208 134)(3 23 209 133)(4 22 210 132)(5 21 211 131)(6 40 212 130)(7 39 213 129)(8 38 214 128)(9 37 215 127)(10 36 216 126)(11 35 217 125)(12 34 218 124)(13 33 219 123)(14 32 220 122)(15 31 201 121)(16 30 202 140)(17 29 203 139)(18 28 204 138)(19 27 205 137)(20 26 206 136)(41 120 262 239)(42 119 263 238)(43 118 264 237)(44 117 265 236)(45 116 266 235)(46 115 267 234)(47 114 268 233)(48 113 269 232)(49 112 270 231)(50 111 271 230)(51 110 272 229)(52 109 273 228)(53 108 274 227)(54 107 275 226)(55 106 276 225)(56 105 277 224)(57 104 278 223)(58 103 279 222)(59 102 280 221)(60 101 261 240)(61 152 259 186)(62 151 260 185)(63 150 241 184)(64 149 242 183)(65 148 243 182)(66 147 244 181)(67 146 245 200)(68 145 246 199)(69 144 247 198)(70 143 248 197)(71 142 249 196)(72 141 250 195)(73 160 251 194)(74 159 252 193)(75 158 253 192)(76 157 254 191)(77 156 255 190)(78 155 256 189)(79 154 257 188)(80 153 258 187)(81 166 293 314)(82 165 294 313)(83 164 295 312)(84 163 296 311)(85 162 297 310)(86 161 298 309)(87 180 299 308)(88 179 300 307)(89 178 281 306)(90 177 282 305)(91 176 283 304)(92 175 284 303)(93 174 285 302)(94 173 286 301)(95 172 287 320)(96 171 288 319)(97 170 289 318)(98 169 290 317)(99 168 291 316)(100 167 292 315)

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

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

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

68 conjugacy classes

 class 1 2A ··· 2G 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 4S 4T 5A 5B 10A ··· 10N 20A ··· 20X order 1 2 ··· 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 4 4 5 5 10 ··· 10 20 ··· 20 size 1 1 ··· 1 2 2 2 2 4 4 4 4 10 ··· 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4

68 irreducible representations

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

Matrix representation of C2.(C4×D20) in GL6(𝔽41)

 1 0 0 0 0 0 0 1 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
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 21 0 0 0 0 0 40
,
 7 1 0 0 0 0 40 0 0 0 0 0 0 0 24 23 0 0 0 0 7 17 0 0 0 0 0 0 31 17 0 0 0 0 40 10
,
 26 38 0 0 0 0 20 15 0 0 0 0 0 0 1 0 0 0 0 0 30 40 0 0 0 0 0 0 33 30 0 0 0 0 32 8

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,21,40],[7,40,0,0,0,0,1,0,0,0,0,0,0,0,24,7,0,0,0,0,23,17,0,0,0,0,0,0,31,40,0,0,0,0,17,10],[26,20,0,0,0,0,38,15,0,0,0,0,0,0,1,30,0,0,0,0,0,40,0,0,0,0,0,0,33,32,0,0,0,0,30,8] >;

C2.(C4×D20) in GAP, Magma, Sage, TeX

C_2.(C_4\times D_{20})
% in TeX

G:=Group("C2.(C4xD20)");
// GroupNames label

G:=SmallGroup(320,280);
// by ID

G=gap.SmallGroup(320,280);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,120,387,58,12550]);
// Polycyclic

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

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