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## G = (C2×C20).28D4order 320 = 26·5

### 2nd 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).28D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×C10 — C22×Dic5 — C10.10C42 — (C2×C20).28D4
 Lower central C5 — C22×C10 — (C2×C20).28D4
 Upper central C1 — C23 — C2.C42

Generators and relations for (C2×C20).28D4
G = < a,b,c,d | a2=b20=c4=1, d2=ab10, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=ab-1, dcd-1=b10c-1 >

Subgroups: 454 in 134 conjugacy classes, 57 normal (25 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C22×C4, C22×C4, C22×C4, Dic5, C20, C2×C10, C2×C10, C2.C42, C2.C42, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×C10, C23.83C23, C10.D4, C4⋊Dic5, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C10.10C42, C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, (C2×C20).28D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, D20, C22×D5, C23.83C23, C2×D20, C4○D20, D42D5, Q8×D5, C4.D20, C23.D10, C22.D20, Dic5.Q8, D102Q8, (C2×C20).28D4

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

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

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

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

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 4 4 type + + + + + - + + + + - - image C1 C2 C2 C2 C2 Q8 D4 D5 C4○D4 D10 D20 C4○D20 D4⋊2D5 Q8×D5 kernel (C2×C20).28D4 C10.10C42 C5×C2.C42 C2×C10.D4 C2×C4⋊Dic5 C2×Dic5 C2×C20 C2.C42 C2×C10 C22×C4 C2×C4 C22 C22 C22 # reps 1 4 1 1 1 2 2 2 10 6 8 16 6 2

Matrix representation of (C2×C20).28D4 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 1 0 0 0 0 0 0 1
,
 40 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 8 0 0 0 0 0 17 36
,
 0 40 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 40 0 0 0 0 0 10 1
,
 0 32 0 0 0 0 32 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 32 31 0 0 0 0 0 9

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,17,0,0,0,0,0,36],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,10,0,0,0,0,0,1],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,32,0,0,0,0,0,31,9] >;

(C2×C20).28D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{28}D_4
% in TeX

G:=Group("(C2xC20).28D4");
// GroupNames label

G:=SmallGroup(320,286);
// by ID

G=gap.SmallGroup(320,286);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,253,64,254,387,268,12550]);
// Polycyclic

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

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