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G = C2.(C4×D20)  order 320 = 26·5

1st central stem extension by C2 of C4×D20

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2.4(C4×D20), C4⋊Dic514C4, C10.31(C4×D4), C10.24(C4×Q8), (C2×C20).233D4, (C2×C4).111D20, (C22×C4).9D10, C22.14(Q8×D5), (C2×Dic5).18Q8, C22.23(C2×D20), C2.1(D102Q8), C10.19(C22⋊Q8), C10.8(C42.C2), C2.C42.9D5, C10.8(C422C2), C2.6(Dic53Q8), C22.30(C4○D20), (C22×C20).42C22, C23.250(C22×D5), C10.37(C42⋊C2), C22.32(D42D5), (C22×C10).278C23, C2.4(Dic5.Q8), C53(C23.63C23), C10.2(C22.D4), C2.3(C23.D10), C2.1(C22.D20), (C22×Dic5).4C22, C10.10C42.22C2, C2.7(C23.11D10), (C2×C4).24(C4×D5), C22.85(C2×C4×D5), (C2×C10).92(C2×D4), (C2×C4⋊Dic5).4C2, (C2×C10).61(C2×Q8), (C2×C4×Dic5).27C2, (C2×C20).205(C2×C4), (C2×Dic5).87(C2×C4), (C2×C10).125(C4○D4), (C2×C10).145(C22×C4), (C2×C10.D4).28C2, (C5×C2.C42).16C2, SmallGroup(320,280)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2.(C4×D20)
C1C5C10C2×C10C22×C10C22×Dic5C2×C4⋊Dic5 — C2.(C4×D20)
C5C2×C10 — C2.(C4×D20)
C1C23C2.C42

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

Subgroups: 478 in 154 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C4 [×12], C22 [×7], C5, C2×C4 [×4], C2×C4 [×22], C23, C10 [×7], C42 [×2], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×4], Dic5 [×7], C20 [×5], C2×C10 [×7], C2.C42, C2.C42 [×3], C2×C42, C2×C4⋊C4 [×2], C2×Dic5 [×6], C2×Dic5 [×9], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.63C23, C4×Dic5 [×2], C10.D4 [×2], C4⋊Dic5 [×4], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×3], C5×C2.C42, C2×C4×Dic5, C2×C10.D4, C2×C4⋊Dic5, C2.(C4×D20)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C22×C4, C2×D4, C2×Q8, C4○D4 [×4], D10 [×3], C42⋊C2, C4×D4, C4×Q8, C22⋊Q8, C22.D4, C42.C2, C422C2, C4×D5 [×2], D20 [×2], C22×D5, C23.63C23, C2×C4×D5, C2×D20, C4○D20, D42D5 [×3], Q8×D5, C4×D20, C23.11D10, C23.D10, C22.D20, Dic53Q8, Dic5.Q8, D102Q8, C2.(C4×D20)

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

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

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

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

68 conjugacy classes

class 1 2A···2G4A4B4C4D4E4F4G4H4I···4P4Q4R4S4T5A5B10A···10N20A···20X
order12···2444444444···444445510···1020···20
size11···12222444410···1020202020222···24···4

68 irreducible representations

dim11111112222222244
type++++++-++++--
imageC1C2C2C2C2C2C4Q8D4D5C4○D4D10C4×D5D20C4○D20D42D5Q8×D5
kernelC2.(C4×D20)C10.10C42C5×C2.C42C2×C4×Dic5C2×C10.D4C2×C4⋊Dic5C4⋊Dic5C2×Dic5C2×C20C2.C42C2×C10C22×C4C2×C4C2×C4C22C22C22
# reps13111182228688862

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

100000
010000
001000
000100
0000400
0000040
,
900000
090000
001000
000100
0000121
0000040
,
710000
4000000
00242300
0071700
00003117
00004010
,
26380000
20150000
001000
00304000
00003330
0000328

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

C2.(C4×D20) in GAP, Magma, Sage, TeX

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

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

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

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

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

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

׿
×
𝔽