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G = C2×D20⋊5S3order 480 = 25·3·5

Direct product of C2 and D20⋊5S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D20⋊5S3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C2×D20⋊5S3
 Lower central C15 — C30 — C2×D20⋊5S3
 Upper central C1 — C22 — C2×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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, C30, C30, C2×C4○D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×Dic6, S3×C2×C4, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, C5×Dic3, Dic15, C60, C6×D5, C6×D5, S3×C10, S3×C10, C2×C30, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C2×D42S3, D5×Dic3, C15⋊D4, C3×D20, S3×C20, C10×Dic3, Dic30, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C2×C4○D20, D205S3, C2×D5×Dic3, C2×C15⋊D4, C6×D20, S3×C2×C20, C2×Dic30, C2×D205S3
Quotients: C1, C2, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, D42S3, S3×C23, S3×D5, C4○D20, C23×D5, C2×D42S3, C2×S3×D5, C2×C4○D20, D205S3, C22×S3×D5, C2×D205S3

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 6A 6B 6C 6D 6E 6F 6G 10A ··· 10F 10G ··· 10N 12A 12B 15A 15B 20A ··· 20H 20I ··· 20P 30A ··· 30F 60A ··· 60H order 1 2 2 2 2 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 4 5 5 6 6 6 6 6 6 6 10 ··· 10 10 ··· 10 12 12 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 6 6 10 10 10 10 2 2 2 3 3 3 3 30 30 30 30 2 2 2 2 2 20 20 20 20 2 ··· 2 6 ··· 6 4 4 4 4 2 ··· 2 6 ··· 6 4 ··· 4 4 ··· 4

78 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 D10 C4○D20 D4⋊2S3 S3×D5 C2×S3×D5 C2×S3×D5 D20⋊5S3 kernel C2×D20⋊5S3 D20⋊5S3 C2×D5×Dic3 C2×C15⋊D4 C6×D20 S3×C2×C20 C2×Dic30 C2×D20 S3×C2×C4 D20 C2×C20 C22×D5 C30 C4×S3 C2×Dic3 C2×C12 C22×S3 C6 C10 C2×C4 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 1 2 4 8 2 2 2 16 2 2 4 2 8

Matrix representation of C2×D205S3 in GL6(𝔽61)

 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 1 0 0 0 0 60 0 0 0 0 0 0 0 31 23 0 0 0 0 51 30
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 18 1 0 0 0 0 43 43 0 0 0 0 0 0 31 23 0 0 0 0 6 30
,
 1 46 0 0 0 0 49 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 3 0 0 0 0 45 7 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 25 52 0 0 0 0 49 36

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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