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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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).28D4, (C2×C4).17D20, (C2×Dic5).3Q8, C22.41(Q8×D5), C22.78(C2×D20), (C22×C4).12D10, C2.7(D102Q8), C10.1(C4.4D4), C2.5(C4.D20), C10.23(C22⋊Q8), (C22×C20).9C22, C22.86(C4○D20), C10.13(C42.C2), C2.C42.14D5, C23.355(C22×D5), C10.10(C422C2), C22.84(D42D5), C10.10C42.7C2, (C22×C10).284C23, C52(C23.83C23), C10.6(C22.D4), C2.6(C22.D20), C2.9(C23.D10), (C22×Dic5).9C22, C2.10(Dic5.Q8), (C2×C10).94(C2×D4), (C2×C4⋊Dic5).9C2, (C2×C10).65(C2×Q8), (C2×C10).128(C4○D4), (C2×C10.D4).21C2, (C5×C2.C42).10C2, SmallGroup(320,286)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).28D4
C1C5C10C2×C10C22×C10C22×Dic5C10.10C42 — (C2×C20).28D4
C5C22×C10 — (C2×C20).28D4
C1C23C2.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 [×3], C2 [×4], C4 [×9], C22 [×3], C22 [×4], C5, C2×C4 [×2], C2×C4 [×21], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], Dic5 [×5], C20 [×4], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×4], C2×C4⋊C4 [×2], C2×Dic5 [×2], C2×Dic5 [×11], C2×C20 [×2], C2×C20 [×8], C22×C10, C23.83C23, C10.D4 [×2], C4⋊Dic5 [×2], C22×Dic5 [×2], C22×Dic5 [×2], C22×C20, C22×C20 [×2], C10.10C42 [×4], C5×C2.C42, C2×C10.D4, C2×C4⋊Dic5, (C2×C20).28D4
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], D20 [×2], C22×D5, C23.83C23, C2×D20, C4○D20 [×2], D42D5 [×3], Q8×D5, C4.D20, C23.D10 [×2], C22.D20, Dic5.Q8 [×2], D102Q8, (C2×C20).28D4

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

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

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

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

62 conjugacy classes

class 1 2A···2G4A···4F4G···4N5A5B10A···10N20A···20X
order12···24···44···45510···1020···20
size11···14···420···20222···24···4

62 irreducible representations

dim11111222222244
type+++++-++++--
imageC1C2C2C2C2Q8D4D5C4○D4D10D20C4○D20D42D5Q8×D5
kernel(C2×C20).28D4C10.10C42C5×C2.C42C2×C10.D4C2×C4⋊Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C22C22C22
# reps1411122210681662

Matrix representation of (C2×C20).28D4 in GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
4000000
010000
001000
000100
000080
00001736
,
0400000
4000000
000100
0040000
0000400
0000101
,
0320000
3200000
000100
001000
00003231
000009

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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