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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, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.83C23, C4⋊Dic5, C5×C4⋊C4, C22×Dic5, C22×C20, C10.10C42, C10.10C42, C2×C4⋊Dic5, C10×C4⋊C4, (C2×C20).55D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, Dic10, C5⋊D4, C22×D5, C23.83C23, C2×Dic10, C4○D20, D42D5, Q82D5, C2×C5⋊D4, C4.Dic10, C4⋊C4⋊D5, C20.48D4, C23.18D10, C20.23D4, (C2×C20).55D4

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

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

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

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

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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