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G = C2×D205S3order 480 = 25·3·5

Direct product of C2 and D205S3

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

Aliases: C2×D205S3, D2027D6, C30.2C24, C60.110C23, Dic3028C22, Dic15.2C23, (C6×D20)⋊7C2, (C4×S3)⋊14D10, (C2×D20)⋊15S3, C63(C4○D20), C301(C4○D4), C6.2(C23×D5), (C2×C20).305D6, C15⋊D48C22, C10.2(S3×C23), C102(D42S3), (C6×D5).1C23, (S3×C20)⋊16C22, (C2×Dic30)⋊24C2, (C2×C12).159D10, (C3×D20)⋊22C22, (D5×Dic3)⋊5C22, (C22×D5).67D6, D6.22(C22×D5), D10.1(C22×S3), (S3×C10).24C23, (C2×C30).221C23, (C2×C60).124C22, C20.160(C22×S3), (C22×S3).80D10, C12.121(C22×D5), (C2×Dic3).189D10, Dic3.32(C22×D5), (C5×Dic3).25C23, (C2×Dic15).148C22, (C10×Dic3).206C22, (S3×C2×C4)⋊3D5, (S3×C2×C20)⋊4C2, C33(C2×C4○D20), C151(C2×C4○D4), C52(C2×D42S3), C4.109(C2×S3×D5), C2.6(C22×S3×D5), (C2×D5×Dic3)⋊19C2, (C2×C15⋊D4)⋊17C2, C22.93(C2×S3×D5), (C2×C4).115(S3×D5), (D5×C2×C6).58C22, (S3×C2×C10).97C22, (C2×C6).233(C22×D5), (C2×C10).233(C22×S3), SmallGroup(480,1074)

Series: Derived Chief Lower central Upper central

C1C30 — C2×D205S3
C1C5C15C30C6×D5D5×Dic3C2×D5×Dic3 — C2×D205S3
C15C30 — C2×D205S3
C1C22C2×C4

Generators and relations for C2×D205S3
 G = < a,b,c,d,e | a2=b20=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b10c, ede=d-1 >

Subgroups: 1468 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], Dic3 [×2], Dic3 [×4], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×8], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×4], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C5×S3 [×2], C3×D5 [×4], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×C10, C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], Dic15 [×4], C60 [×2], C6×D5 [×4], C6×D5 [×4], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×D42S3, D5×Dic3 [×8], C15⋊D4 [×8], C3×D20 [×4], S3×C20 [×4], C10×Dic3, Dic30 [×4], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], S3×C2×C10, C2×C4○D20, D205S3 [×8], C2×D5×Dic3 [×2], C2×C15⋊D4 [×2], C6×D20, S3×C2×C20, C2×Dic30, C2×D205S3
Quotients: C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], D42S3 [×2], S3×C23, S3×D5, C4○D20 [×2], C23×D5, C2×D42S3, C2×S3×D5 [×3], C2×C4○D20, D205S3 [×2], C22×S3×D5, C2×D205S3

Smallest permutation representation of C2×D205S3
On 240 points
Generators in S240
(1 82)(2 83)(3 84)(4 85)(5 86)(6 87)(7 88)(8 89)(9 90)(10 91)(11 92)(12 93)(13 94)(14 95)(15 96)(16 97)(17 98)(18 99)(19 100)(20 81)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 101)(29 102)(30 103)(31 104)(32 105)(33 106)(34 107)(35 108)(36 109)(37 110)(38 111)(39 112)(40 113)(41 205)(42 206)(43 207)(44 208)(45 209)(46 210)(47 211)(48 212)(49 213)(50 214)(51 215)(52 216)(53 217)(54 218)(55 219)(56 220)(57 201)(58 202)(59 203)(60 204)(61 194)(62 195)(63 196)(64 197)(65 198)(66 199)(67 200)(68 181)(69 182)(70 183)(71 184)(72 185)(73 186)(74 187)(75 188)(76 189)(77 190)(78 191)(79 192)(80 193)(121 168)(122 169)(123 170)(124 171)(125 172)(126 173)(127 174)(128 175)(129 176)(130 177)(131 178)(132 179)(133 180)(134 161)(135 162)(136 163)(137 164)(138 165)(139 166)(140 167)(141 239)(142 240)(143 221)(144 222)(145 223)(146 224)(147 225)(148 226)(149 227)(150 228)(151 229)(152 230)(153 231)(154 232)(155 233)(156 234)(157 235)(158 236)(159 237)(160 238)
(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)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 58)(42 57)(43 56)(44 55)(45 54)(46 53)(47 52)(48 51)(49 50)(59 60)(61 70)(62 69)(63 68)(64 67)(65 66)(71 80)(72 79)(73 78)(74 77)(75 76)(81 82)(83 100)(84 99)(85 98)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 128)(122 127)(123 126)(124 125)(129 140)(130 139)(131 138)(132 137)(133 136)(134 135)(141 150)(142 149)(143 148)(144 147)(145 146)(151 160)(152 159)(153 158)(154 157)(155 156)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 196)(182 195)(183 194)(184 193)(185 192)(186 191)(187 190)(188 189)(197 200)(198 199)(201 206)(202 205)(203 204)(207 220)(208 219)(209 218)(210 217)(211 216)(212 215)(213 214)(221 226)(222 225)(223 224)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)
(1 76 162)(2 77 163)(3 78 164)(4 79 165)(5 80 166)(6 61 167)(7 62 168)(8 63 169)(9 64 170)(10 65 171)(11 66 172)(12 67 173)(13 68 174)(14 69 175)(15 70 176)(16 71 177)(17 72 178)(18 73 179)(19 74 180)(20 75 161)(21 211 143)(22 212 144)(23 213 145)(24 214 146)(25 215 147)(26 216 148)(27 217 149)(28 218 150)(29 219 151)(30 220 152)(31 201 153)(32 202 154)(33 203 155)(34 204 156)(35 205 157)(36 206 158)(37 207 159)(38 208 160)(39 209 141)(40 210 142)(41 235 108)(42 236 109)(43 237 110)(44 238 111)(45 239 112)(46 240 113)(47 221 114)(48 222 115)(49 223 116)(50 224 117)(51 225 118)(52 226 119)(53 227 120)(54 228 101)(55 229 102)(56 230 103)(57 231 104)(58 232 105)(59 233 106)(60 234 107)(81 188 134)(82 189 135)(83 190 136)(84 191 137)(85 192 138)(86 193 139)(87 194 140)(88 195 121)(89 196 122)(90 197 123)(91 198 124)(92 199 125)(93 200 126)(94 181 127)(95 182 128)(96 183 129)(97 184 130)(98 185 131)(99 186 132)(100 187 133)
(1 55)(2 56)(3 57)(4 58)(5 59)(6 60)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 181)(22 182)(23 183)(24 184)(25 185)(26 186)(27 187)(28 188)(29 189)(30 190)(31 191)(32 192)(33 193)(34 194)(35 195)(36 196)(37 197)(38 198)(39 199)(40 200)(61 107)(62 108)(63 109)(64 110)(65 111)(66 112)(67 113)(68 114)(69 115)(70 116)(71 117)(72 118)(73 119)(74 120)(75 101)(76 102)(77 103)(78 104)(79 105)(80 106)(81 218)(82 219)(83 220)(84 201)(85 202)(86 203)(87 204)(88 205)(89 206)(90 207)(91 208)(92 209)(93 210)(94 211)(95 212)(96 213)(97 214)(98 215)(99 216)(100 217)(121 157)(122 158)(123 159)(124 160)(125 141)(126 142)(127 143)(128 144)(129 145)(130 146)(131 147)(132 148)(133 149)(134 150)(135 151)(136 152)(137 153)(138 154)(139 155)(140 156)(161 228)(162 229)(163 230)(164 231)(165 232)(166 233)(167 234)(168 235)(169 236)(170 237)(171 238)(172 239)(173 240)(174 221)(175 222)(176 223)(177 224)(178 225)(179 226)(180 227)

G:=sub<Sym(240)| (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,201)(58,202)(59,203)(60,204)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(141,239)(142,240)(143,221)(144,222)(145,223)(146,224)(147,225)(148,226)(149,227)(150,228)(151,229)(152,230)(153,231)(154,232)(155,233)(156,234)(157,235)(158,236)(159,237)(160,238), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(141,150)(142,149)(143,148)(144,147)(145,146)(151,160)(152,159)(153,158)(154,157)(155,156)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,196)(182,195)(183,194)(184,193)(185,192)(186,191)(187,190)(188,189)(197,200)(198,199)(201,206)(202,205)(203,204)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,76,162)(2,77,163)(3,78,164)(4,79,165)(5,80,166)(6,61,167)(7,62,168)(8,63,169)(9,64,170)(10,65,171)(11,66,172)(12,67,173)(13,68,174)(14,69,175)(15,70,176)(16,71,177)(17,72,178)(18,73,179)(19,74,180)(20,75,161)(21,211,143)(22,212,144)(23,213,145)(24,214,146)(25,215,147)(26,216,148)(27,217,149)(28,218,150)(29,219,151)(30,220,152)(31,201,153)(32,202,154)(33,203,155)(34,204,156)(35,205,157)(36,206,158)(37,207,159)(38,208,160)(39,209,141)(40,210,142)(41,235,108)(42,236,109)(43,237,110)(44,238,111)(45,239,112)(46,240,113)(47,221,114)(48,222,115)(49,223,116)(50,224,117)(51,225,118)(52,226,119)(53,227,120)(54,228,101)(55,229,102)(56,230,103)(57,231,104)(58,232,105)(59,233,106)(60,234,107)(81,188,134)(82,189,135)(83,190,136)(84,191,137)(85,192,138)(86,193,139)(87,194,140)(88,195,121)(89,196,122)(90,197,123)(91,198,124)(92,199,125)(93,200,126)(94,181,127)(95,182,128)(96,183,129)(97,184,130)(98,185,131)(99,186,132)(100,187,133), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,197)(38,198)(39,199)(40,200)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,218)(82,219)(83,220)(84,201)(85,202)(86,203)(87,204)(88,205)(89,206)(90,207)(91,208)(92,209)(93,210)(94,211)(95,212)(96,213)(97,214)(98,215)(99,216)(100,217)(121,157)(122,158)(123,159)(124,160)(125,141)(126,142)(127,143)(128,144)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)(137,153)(138,154)(139,155)(140,156)(161,228)(162,229)(163,230)(164,231)(165,232)(166,233)(167,234)(168,235)(169,236)(170,237)(171,238)(172,239)(173,240)(174,221)(175,222)(176,223)(177,224)(178,225)(179,226)(180,227)>;

G:=Group( (1,82)(2,83)(3,84)(4,85)(5,86)(6,87)(7,88)(8,89)(9,90)(10,91)(11,92)(12,93)(13,94)(14,95)(15,96)(16,97)(17,98)(18,99)(19,100)(20,81)(21,114)(22,115)(23,116)(24,117)(25,118)(26,119)(27,120)(28,101)(29,102)(30,103)(31,104)(32,105)(33,106)(34,107)(35,108)(36,109)(37,110)(38,111)(39,112)(40,113)(41,205)(42,206)(43,207)(44,208)(45,209)(46,210)(47,211)(48,212)(49,213)(50,214)(51,215)(52,216)(53,217)(54,218)(55,219)(56,220)(57,201)(58,202)(59,203)(60,204)(61,194)(62,195)(63,196)(64,197)(65,198)(66,199)(67,200)(68,181)(69,182)(70,183)(71,184)(72,185)(73,186)(74,187)(75,188)(76,189)(77,190)(78,191)(79,192)(80,193)(121,168)(122,169)(123,170)(124,171)(125,172)(126,173)(127,174)(128,175)(129,176)(130,177)(131,178)(132,179)(133,180)(134,161)(135,162)(136,163)(137,164)(138,165)(139,166)(140,167)(141,239)(142,240)(143,221)(144,222)(145,223)(146,224)(147,225)(148,226)(149,227)(150,228)(151,229)(152,230)(153,231)(154,232)(155,233)(156,234)(157,235)(158,236)(159,237)(160,238), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,26)(22,25)(23,24)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(41,58)(42,57)(43,56)(44,55)(45,54)(46,53)(47,52)(48,51)(49,50)(59,60)(61,70)(62,69)(63,68)(64,67)(65,66)(71,80)(72,79)(73,78)(74,77)(75,76)(81,82)(83,100)(84,99)(85,98)(86,97)(87,96)(88,95)(89,94)(90,93)(91,92)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,140)(130,139)(131,138)(132,137)(133,136)(134,135)(141,150)(142,149)(143,148)(144,147)(145,146)(151,160)(152,159)(153,158)(154,157)(155,156)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,196)(182,195)(183,194)(184,193)(185,192)(186,191)(187,190)(188,189)(197,200)(198,199)(201,206)(202,205)(203,204)(207,220)(208,219)(209,218)(210,217)(211,216)(212,215)(213,214)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,76,162)(2,77,163)(3,78,164)(4,79,165)(5,80,166)(6,61,167)(7,62,168)(8,63,169)(9,64,170)(10,65,171)(11,66,172)(12,67,173)(13,68,174)(14,69,175)(15,70,176)(16,71,177)(17,72,178)(18,73,179)(19,74,180)(20,75,161)(21,211,143)(22,212,144)(23,213,145)(24,214,146)(25,215,147)(26,216,148)(27,217,149)(28,218,150)(29,219,151)(30,220,152)(31,201,153)(32,202,154)(33,203,155)(34,204,156)(35,205,157)(36,206,158)(37,207,159)(38,208,160)(39,209,141)(40,210,142)(41,235,108)(42,236,109)(43,237,110)(44,238,111)(45,239,112)(46,240,113)(47,221,114)(48,222,115)(49,223,116)(50,224,117)(51,225,118)(52,226,119)(53,227,120)(54,228,101)(55,229,102)(56,230,103)(57,231,104)(58,232,105)(59,233,106)(60,234,107)(81,188,134)(82,189,135)(83,190,136)(84,191,137)(85,192,138)(86,193,139)(87,194,140)(88,195,121)(89,196,122)(90,197,123)(91,198,124)(92,199,125)(93,200,126)(94,181,127)(95,182,128)(96,183,129)(97,184,130)(98,185,131)(99,186,132)(100,187,133), (1,55)(2,56)(3,57)(4,58)(5,59)(6,60)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,181)(22,182)(23,183)(24,184)(25,185)(26,186)(27,187)(28,188)(29,189)(30,190)(31,191)(32,192)(33,193)(34,194)(35,195)(36,196)(37,197)(38,198)(39,199)(40,200)(61,107)(62,108)(63,109)(64,110)(65,111)(66,112)(67,113)(68,114)(69,115)(70,116)(71,117)(72,118)(73,119)(74,120)(75,101)(76,102)(77,103)(78,104)(79,105)(80,106)(81,218)(82,219)(83,220)(84,201)(85,202)(86,203)(87,204)(88,205)(89,206)(90,207)(91,208)(92,209)(93,210)(94,211)(95,212)(96,213)(97,214)(98,215)(99,216)(100,217)(121,157)(122,158)(123,159)(124,160)(125,141)(126,142)(127,143)(128,144)(129,145)(130,146)(131,147)(132,148)(133,149)(134,150)(135,151)(136,152)(137,153)(138,154)(139,155)(140,156)(161,228)(162,229)(163,230)(164,231)(165,232)(166,233)(167,234)(168,235)(169,236)(170,237)(171,238)(172,239)(173,240)(174,221)(175,222)(176,223)(177,224)(178,225)(179,226)(180,227) );

G=PermutationGroup([(1,82),(2,83),(3,84),(4,85),(5,86),(6,87),(7,88),(8,89),(9,90),(10,91),(11,92),(12,93),(13,94),(14,95),(15,96),(16,97),(17,98),(18,99),(19,100),(20,81),(21,114),(22,115),(23,116),(24,117),(25,118),(26,119),(27,120),(28,101),(29,102),(30,103),(31,104),(32,105),(33,106),(34,107),(35,108),(36,109),(37,110),(38,111),(39,112),(40,113),(41,205),(42,206),(43,207),(44,208),(45,209),(46,210),(47,211),(48,212),(49,213),(50,214),(51,215),(52,216),(53,217),(54,218),(55,219),(56,220),(57,201),(58,202),(59,203),(60,204),(61,194),(62,195),(63,196),(64,197),(65,198),(66,199),(67,200),(68,181),(69,182),(70,183),(71,184),(72,185),(73,186),(74,187),(75,188),(76,189),(77,190),(78,191),(79,192),(80,193),(121,168),(122,169),(123,170),(124,171),(125,172),(126,173),(127,174),(128,175),(129,176),(130,177),(131,178),(132,179),(133,180),(134,161),(135,162),(136,163),(137,164),(138,165),(139,166),(140,167),(141,239),(142,240),(143,221),(144,222),(145,223),(146,224),(147,225),(148,226),(149,227),(150,228),(151,229),(152,230),(153,231),(154,232),(155,233),(156,234),(157,235),(158,236),(159,237),(160,238)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,26),(22,25),(23,24),(27,40),(28,39),(29,38),(30,37),(31,36),(32,35),(33,34),(41,58),(42,57),(43,56),(44,55),(45,54),(46,53),(47,52),(48,51),(49,50),(59,60),(61,70),(62,69),(63,68),(64,67),(65,66),(71,80),(72,79),(73,78),(74,77),(75,76),(81,82),(83,100),(84,99),(85,98),(86,97),(87,96),(88,95),(89,94),(90,93),(91,92),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117),(121,128),(122,127),(123,126),(124,125),(129,140),(130,139),(131,138),(132,137),(133,136),(134,135),(141,150),(142,149),(143,148),(144,147),(145,146),(151,160),(152,159),(153,158),(154,157),(155,156),(161,162),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172),(181,196),(182,195),(183,194),(184,193),(185,192),(186,191),(187,190),(188,189),(197,200),(198,199),(201,206),(202,205),(203,204),(207,220),(208,219),(209,218),(210,217),(211,216),(212,215),(213,214),(221,226),(222,225),(223,224),(227,240),(228,239),(229,238),(230,237),(231,236),(232,235),(233,234)], [(1,76,162),(2,77,163),(3,78,164),(4,79,165),(5,80,166),(6,61,167),(7,62,168),(8,63,169),(9,64,170),(10,65,171),(11,66,172),(12,67,173),(13,68,174),(14,69,175),(15,70,176),(16,71,177),(17,72,178),(18,73,179),(19,74,180),(20,75,161),(21,211,143),(22,212,144),(23,213,145),(24,214,146),(25,215,147),(26,216,148),(27,217,149),(28,218,150),(29,219,151),(30,220,152),(31,201,153),(32,202,154),(33,203,155),(34,204,156),(35,205,157),(36,206,158),(37,207,159),(38,208,160),(39,209,141),(40,210,142),(41,235,108),(42,236,109),(43,237,110),(44,238,111),(45,239,112),(46,240,113),(47,221,114),(48,222,115),(49,223,116),(50,224,117),(51,225,118),(52,226,119),(53,227,120),(54,228,101),(55,229,102),(56,230,103),(57,231,104),(58,232,105),(59,233,106),(60,234,107),(81,188,134),(82,189,135),(83,190,136),(84,191,137),(85,192,138),(86,193,139),(87,194,140),(88,195,121),(89,196,122),(90,197,123),(91,198,124),(92,199,125),(93,200,126),(94,181,127),(95,182,128),(96,183,129),(97,184,130),(98,185,131),(99,186,132),(100,187,133)], [(1,55),(2,56),(3,57),(4,58),(5,59),(6,60),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,181),(22,182),(23,183),(24,184),(25,185),(26,186),(27,187),(28,188),(29,189),(30,190),(31,191),(32,192),(33,193),(34,194),(35,195),(36,196),(37,197),(38,198),(39,199),(40,200),(61,107),(62,108),(63,109),(64,110),(65,111),(66,112),(67,113),(68,114),(69,115),(70,116),(71,117),(72,118),(73,119),(74,120),(75,101),(76,102),(77,103),(78,104),(79,105),(80,106),(81,218),(82,219),(83,220),(84,201),(85,202),(86,203),(87,204),(88,205),(89,206),(90,207),(91,208),(92,209),(93,210),(94,211),(95,212),(96,213),(97,214),(98,215),(99,216),(100,217),(121,157),(122,158),(123,159),(124,160),(125,141),(126,142),(127,143),(128,144),(129,145),(130,146),(131,147),(132,148),(133,149),(134,150),(135,151),(136,152),(137,153),(138,154),(139,155),(140,156),(161,228),(162,229),(163,230),(164,231),(165,232),(166,233),(167,234),(168,235),(169,236),(170,237),(171,238),(172,239),(173,240),(174,221),(175,222),(176,223),(177,224),(178,225),(179,226),(180,227)])

78 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J5A5B6A6B6C6D6E6F6G10A···10F10G···10N12A12B15A15B20A···20H20I···20P30A···30F60A···60H
order12222222223444444444455666666610···1010···101212151520···2020···2030···3060···60
size1111661010101022233333030303022222202020202···26···644442···26···64···44···4

78 irreducible representations

dim11111112222222222244444
type++++++++++++++++-+++-
imageC1C2C2C2C2C2C2S3D5D6D6D6C4○D4D10D10D10D10C4○D20D42S3S3×D5C2×S3×D5C2×S3×D5D205S3
kernelC2×D205S3D205S3C2×D5×Dic3C2×C15⋊D4C6×D20S3×C2×C20C2×Dic30C2×D20S3×C2×C4D20C2×C20C22×D5C30C4×S3C2×Dic3C2×C12C22×S3C6C10C2×C4C4C22C2
# reps182211112412482221622428

Matrix representation of C2×D205S3 in GL6(𝔽61)

6000000
0600000
001000
000100
000010
000001
,
6000000
0600000
0018100
0060000
00003123
00005130
,
6000000
0600000
0018100
00434300
00003123
0000630
,
1460000
49590000
001000
000100
000010
000001
,
5430000
4570000
001000
000100
00002552
00004936

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,60,0,0,0,0,1,0,0,0,0,0,0,0,31,51,0,0,0,0,23,30],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,43,0,0,0,0,1,43,0,0,0,0,0,0,31,6,0,0,0,0,23,30],[1,49,0,0,0,0,46,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[54,45,0,0,0,0,3,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,49,0,0,0,0,52,36] >;

C2×D205S3 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes_5S_3
% in TeX

G:=Group("C2xD20:5S3");
// GroupNames label

G:=SmallGroup(480,1074);
// by ID

G=gap.SmallGroup(480,1074);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,80,1356,18822]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^20=c^2=d^3=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^10*c,e*d*e=d^-1>;
// generators/relations

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