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G = C2×C20.44D4order 320 = 26·5

Direct product of C2 and C20.44D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20.44D4, C23.55D20, C22.5Dic20, (C2×C4).92D20, C10.9(C2×Q16), (C2×C10).9Q16, (C22×C8).7D5, (C2×C8).292D10, C20.409(C2×D4), (C2×C20).385D4, C2.2(C2×Dic20), C103(Q8⋊C4), Dic1025(C2×C4), (C2×Dic10)⋊18C4, (C22×C40).10C2, C10.14(C2×SD16), (C2×C10).20SD16, C22.49(C2×D20), C20.97(C22⋊C4), C20.170(C22×C4), (C2×C40).352C22, (C2×C20).762C23, (C22×C10).134D4, (C22×C4).422D10, C4.25(D10⋊C4), C22.10(C40⋊C2), C4⋊Dic5.278C22, (C22×Dic10).6C2, (C22×C20).514C22, C22.47(D10⋊C4), (C2×Dic10).224C22, C4.69(C2×C4×D5), C54(C2×Q8⋊C4), C2.2(C2×C40⋊C2), (C2×C4).114(C4×D5), C4.102(C2×C5⋊D4), (C2×C20).400(C2×C4), (C2×C10).152(C2×D4), C10.90(C2×C22⋊C4), (C2×C4⋊Dic5).22C2, C2.22(C2×D10⋊C4), (C2×C4).252(C5⋊D4), (C2×C4).709(C22×D5), (C2×C10).124(C22⋊C4), SmallGroup(320,730)

Series: Derived Chief Lower central Upper central

C1C20 — C2×C20.44D4
C1C5C10C20C2×C20C4⋊Dic5C2×C4⋊Dic5 — C2×C20.44D4
C5C10C20 — C2×C20.44D4
C1C23C22×C4C22×C8

Generators and relations for C2×C20.44D4
 G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c-1 >

Subgroups: 574 in 162 conjugacy classes, 79 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], Q8 [×10], C23, C10 [×3], C10 [×4], C4⋊C4 [×3], C2×C8 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], C2×Q8 [×9], Dic5 [×6], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], Q8⋊C4 [×4], C2×C4⋊C4, C22×C8, C22×Q8, C40 [×2], Dic10 [×4], Dic10 [×6], C2×Dic5 [×10], C2×C20 [×2], C2×C20 [×4], C22×C10, C2×Q8⋊C4, C4⋊Dic5 [×2], C4⋊Dic5, C2×C40 [×2], C2×C40 [×2], C2×Dic10 [×6], C2×Dic10 [×3], C22×Dic5 [×2], C22×C20, C20.44D4 [×4], C2×C4⋊Dic5, C22×C40, C22×Dic10, C2×C20.44D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], SD16 [×2], Q16 [×2], C22×C4, C2×D4 [×2], D10 [×3], Q8⋊C4 [×4], C2×C22⋊C4, C2×SD16, C2×Q16, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×Q8⋊C4, C40⋊C2 [×2], Dic20 [×2], D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C20.44D4 [×4], C2×C40⋊C2, C2×Dic20, C2×D10⋊C4, C2×C20.44D4

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

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

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

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

92 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L5A5B8A···8H10A···10N20A···20P40A···40AF
order12···244444···4558···810···1020···2040···40
size11···1222220···20222···22···22···22···2

92 irreducible representations

dim1111112222222222222
type++++++++-++++-
imageC1C2C2C2C2C4D4D4D5SD16Q16D10D10C4×D5D20C5⋊D4D20C40⋊C2Dic20
kernelC2×C20.44D4C20.44D4C2×C4⋊Dic5C22×C40C22×Dic10C2×Dic10C2×C20C22×C10C22×C8C2×C10C2×C10C2×C8C22×C4C2×C4C2×C4C2×C4C23C22C22
# reps141118312444284841616

Matrix representation of C2×C20.44D4 in GL5(𝔽41)

400000
01000
00100
000400
000040
,
10000
032000
00900
000250
000023
,
10000
00100
040000
00001
000400
,
10000
001400
038000
00009
000320

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,25,0,0,0,0,0,23],[1,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,1,0],[1,0,0,0,0,0,0,38,0,0,0,14,0,0,0,0,0,0,0,32,0,0,0,9,0] >;

C2×C20.44D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{44}D_4
% in TeX

G:=Group("C2xC20.44D4");
// GroupNames label

G:=SmallGroup(320,730);
// by ID

G=gap.SmallGroup(320,730);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,254,142,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^-1>;
// generators/relations

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