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G = (C2×C20).54D4order 320 = 26·5

28th non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).54D4, (C2×C20).39Q8, C10.22(C4⋊Q8), C2.15(C20⋊Q8), (C2×Dic5).8Q8, C22.49(Q8×D5), (C2×Dic5).68D4, (C2×C4).12Dic10, C22.246(D4×D5), C10.90(C4⋊D4), C2.7(D103Q8), (C22×C4).101D10, C10.59(C22⋊Q8), C2.8(C4.Dic10), (C22×C20).65C22, C10.23(C42.C2), C22.48(C2×Dic10), C23.378(C22×D5), C2.12(Dic5⋊D4), C2.10(C20.48D4), C22.106(C4○D20), (C22×C10).348C23, C55(C23.81C23), C22.49(Q82D5), C2.19(D10.13D4), C22.102(D42D5), C10.10C42.20C2, C2.14(Dic5.Q8), C10.50(C22.D4), (C22×Dic5).56C22, (C2×C4⋊C4).21D5, (C10×C4⋊C4).23C2, (C2×C10).37(C2×Q8), (C2×C10).332(C2×D4), (C2×C4).39(C5⋊D4), (C2×C4⋊Dic5).20C2, (C2×C10).85(C4○D4), C22.138(C2×C5⋊D4), (C2×C10.D4).17C2, SmallGroup(320,611)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).54D4
C1C5C10C2×C10C22×C10C22×Dic5C2×C10.D4 — (C2×C20).54D4
C5C22×C10 — (C2×C20).54D4
C1C23C2×C4⋊C4

Generators and relations for (C2×C20).54D4
 G = < a,b,c,d | a2=b20=c4=1, d2=ab10, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd-1=ab-1, dcd-1=b10c-1 >

Subgroups: 486 in 150 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C4 [×11], C22 [×7], C5, C2×C4 [×4], C2×C4 [×21], C23, C10 [×7], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], Dic5 [×6], C20 [×5], C2×C10 [×7], C2.C42 [×3], C2×C4⋊C4, C2×C4⋊C4 [×3], C2×Dic5 [×4], C2×Dic5 [×10], C2×C20 [×4], C2×C20 [×7], C22×C10, C23.81C23, C10.D4 [×4], C4⋊Dic5 [×2], C5×C4⋊C4 [×2], C22×Dic5 [×4], C22×C20 [×3], C10.10C42 [×3], C2×C10.D4 [×2], C2×C4⋊Dic5, C10×C4⋊C4, (C2×C20).54D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], Q8 [×4], C23, D5, C2×D4 [×2], C2×Q8 [×2], C4○D4 [×3], D10 [×3], C4⋊D4, C22⋊Q8 [×2], C22.D4, C42.C2 [×2], C4⋊Q8, Dic10 [×2], C5⋊D4 [×2], C22×D5, C23.81C23, C2×Dic10, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, C20⋊Q8, Dic5.Q8, C4.Dic10, D10.13D4, C20.48D4, Dic5⋊D4, D103Q8, (C2×C20).54D4

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

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

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

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

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

dim1111122222222224444
type++++++-+-++-+--+
imageC1C2C2C2C2D4Q8D4Q8D5C4○D4D10Dic10C5⋊D4C4○D20D4×D5D42D5Q8×D5Q82D5
kernel(C2×C20).54D4C10.10C42C2×C10.D4C2×C4⋊Dic5C10×C4⋊C4C2×Dic5C2×Dic5C2×C20C2×C20C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22C22C22
# reps1321122222668882222

Matrix representation of (C2×C20).54D4 in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
3200000
20380000
00273900
00161100
00002716
00001614
,
0400000
100000
0011200
00223000
000010
000001
,
4000000
0400000
0093200
0003200
0000040
000010

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[3,20,0,0,0,0,20,38,0,0,0,0,0,0,27,16,0,0,0,0,39,11,0,0,0,0,0,0,27,16,0,0,0,0,16,14],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,11,22,0,0,0,0,2,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,9,0,0,0,0,0,32,32,0,0,0,0,0,0,0,1,0,0,0,0,40,0] >;

(C2×C20).54D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{54}D_4
% in TeX

G:=Group("(C2xC20).54D4");
// GroupNames label

G:=SmallGroup(320,611);
// by ID

G=gap.SmallGroup(320,611);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,253,344,254,387,184,12550]);
// 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=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

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