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G = (C2×C20).55D4order 320 = 26·5

29th 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).55D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C2×C4⋊Dic5 — (C2×C20).55D4
 Lower central C5 — C22×C10 — (C2×C20).55D4
 Upper central C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C20).55D4
G = < a,b,c,d | a2=b20=c4=1, d2=ab10, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 438 in 134 conjugacy classes, 59 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×9], C22 [×3], C22 [×4], C5, C2×C4 [×4], C2×C4 [×19], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4 [×3], C22×C4 [×4], Dic5 [×4], C20 [×5], C2×C10 [×3], C2×C10 [×4], C2.C42 [×5], C2×C4⋊C4, C2×C4⋊C4, C2×Dic5 [×12], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.83C23, C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42, C10.10C42 [×4], C2×C4⋊Dic5, C10×C4⋊C4, (C2×C20).55D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4 [×5], D10 [×3], C22⋊Q8, C22.D4, C4.4D4, C42.C2 [×2], C422C2 [×2], Dic10 [×2], C5⋊D4 [×2], C22×D5, C23.83C23, C2×Dic10, C4○D20, D42D5 [×2], Q82D5 [×2], C2×C5⋊D4, C4.Dic10 [×2], C4⋊C4⋊D5 [×2], C20.48D4, C23.18D10, C20.23D4, (C2×C20).55D4

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

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

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

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

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 type + + + + + - + + - - + image C1 C2 C2 C2 D4 Q8 D5 C4○D4 D10 Dic10 C5⋊D4 C4○D20 D4⋊2D5 Q8⋊2D5 kernel (C2×C20).55D4 C10.10C42 C2×C4⋊Dic5 C10×C4⋊C4 C2×C20 C2×C20 C2×C4⋊C4 C2×C10 C22×C4 C2×C4 C2×C4 C22 C22 C22 # reps 1 5 1 1 2 2 2 10 6 8 8 8 4 4

Matrix representation of (C2×C20).55D4 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
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 27 39 0 0 0 0 16 11 0 0 0 0 0 0 0 1 0 0 0 0 40 0
,
 11 1 0 0 0 0 1 30 0 0 0 0 0 0 34 1 0 0 0 0 34 7 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 30 40 0 0 0 0 40 11 0 0 0 0 0 0 3 18 0 0 0 0 4 38 0 0 0 0 0 0 40 0 0 0 0 0 0 1

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],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,27,16,0,0,0,0,39,11,0,0,0,0,0,0,0,40,0,0,0,0,1,0],[11,1,0,0,0,0,1,30,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[30,40,0,0,0,0,40,11,0,0,0,0,0,0,3,4,0,0,0,0,18,38,0,0,0,0,0,0,40,0,0,0,0,0,0,1] >;

(C2×C20).55D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{55}D_4
% in TeX

G:=Group("(C2xC20).55D4");
// GroupNames label

G:=SmallGroup(320,613);
// by ID

G=gap.SmallGroup(320,613);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,344,254,387,184,12550]);
// 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=a*b^-1,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

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