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## G = C2×D20⋊S3order 480 = 25·3·5

### Direct product of C2 and D20⋊S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C30 — C2×D20⋊S3
 Chief series C1 — C5 — C15 — C30 — C6×D5 — D5×Dic3 — C2×D5×Dic3 — C2×D20⋊S3
 Lower central C15 — C30 — C2×D20⋊S3
 Upper central C1 — C22 — C2×C4

Generators and relations for C2×D20⋊S3
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, ebe=b9, cd=dc, ece=b18c, ede=d-1 >

Subgroups: 1596 in 328 conjugacy classes, 116 normal (24 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 [×6], C10, C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×8], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×2], C20 [×4], D10 [×4], D10 [×8], C2×C10, Dic6 [×4], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×9], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C3×D5 [×4], D15 [×2], C30, C30 [×2], C2×C4○D4, C4×D5 [×12], D20 [×4], D20 [×8], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5 [×2], C22×D5, C2×Dic6, S3×C2×C4, D42S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×4], Dic15 [×2], C60 [×2], C6×D5 [×4], C6×D5 [×4], D30 [×2], D30 [×2], C2×C30, C2×C4×D5 [×3], C2×D20, C2×D20 [×2], Q82D5 [×8], Q8×C10, C2×D42S3, D5×Dic3 [×8], C3⋊D20 [×8], C3×D20 [×4], C5×Dic6 [×4], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, D5×C2×C6 [×2], C22×D15, C2×Q82D5, D20⋊S3 [×8], C2×D5×Dic3 [×2], C2×C3⋊D20 [×2], C6×D20, C10×Dic6, C2×C4×D15, C2×D20⋊S3
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, Q82D5 [×2], C23×D5, C2×D42S3, C2×S3×D5 [×3], C2×Q82D5, D20⋊S3 [×2], C22×S3×D5, C2×D20⋊S3

Smallest permutation representation of C2×D20⋊S3
On 240 points
Generators in S240
(1 135)(2 136)(3 137)(4 138)(5 139)(6 140)(7 121)(8 122)(9 123)(10 124)(11 125)(12 126)(13 127)(14 128)(15 129)(16 130)(17 131)(18 132)(19 133)(20 134)(21 176)(22 177)(23 178)(24 179)(25 180)(26 161)(27 162)(28 163)(29 164)(30 165)(31 166)(32 167)(33 168)(34 169)(35 170)(36 171)(37 172)(38 173)(39 174)(40 175)(41 158)(42 159)(43 160)(44 141)(45 142)(46 143)(47 144)(48 145)(49 146)(50 147)(51 148)(52 149)(53 150)(54 151)(55 152)(56 153)(57 154)(58 155)(59 156)(60 157)(61 192)(62 193)(63 194)(64 195)(65 196)(66 197)(67 198)(68 199)(69 200)(70 181)(71 182)(72 183)(73 184)(74 185)(75 186)(76 187)(77 188)(78 189)(79 190)(80 191)(81 221)(82 222)(83 223)(84 224)(85 225)(86 226)(87 227)(88 228)(89 229)(90 230)(91 231)(92 232)(93 233)(94 234)(95 235)(96 236)(97 237)(98 238)(99 239)(100 240)(101 212)(102 213)(103 214)(104 215)(105 216)(106 217)(107 218)(108 219)(109 220)(110 201)(111 202)(112 203)(113 204)(114 205)(115 206)(116 207)(117 208)(118 209)(119 210)(120 211)
(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 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)(41 56)(42 55)(43 54)(44 53)(45 52)(46 51)(47 50)(48 49)(57 60)(58 59)(61 74)(62 73)(63 72)(64 71)(65 70)(66 69)(67 68)(75 80)(76 79)(77 78)(81 84)(82 83)(85 100)(86 99)(87 98)(88 97)(89 96)(90 95)(91 94)(92 93)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(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)(161 176)(162 175)(163 174)(164 173)(165 172)(166 171)(167 170)(168 169)(177 180)(178 179)(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)(221 224)(222 223)(225 240)(226 239)(227 238)(228 237)(229 236)(230 235)(231 234)(232 233)
(1 113 34)(2 114 35)(3 115 36)(4 116 37)(5 117 38)(6 118 39)(7 119 40)(8 120 21)(9 101 22)(10 102 23)(11 103 24)(12 104 25)(13 105 26)(14 106 27)(15 107 28)(16 108 29)(17 109 30)(18 110 31)(19 111 32)(20 112 33)(41 235 181)(42 236 182)(43 237 183)(44 238 184)(45 239 185)(46 240 186)(47 221 187)(48 222 188)(49 223 189)(50 224 190)(51 225 191)(52 226 192)(53 227 193)(54 228 194)(55 229 195)(56 230 196)(57 231 197)(58 232 198)(59 233 199)(60 234 200)(61 149 86)(62 150 87)(63 151 88)(64 152 89)(65 153 90)(66 154 91)(67 155 92)(68 156 93)(69 157 94)(70 158 95)(71 159 96)(72 160 97)(73 141 98)(74 142 99)(75 143 100)(76 144 81)(77 145 82)(78 146 83)(79 147 84)(80 148 85)(121 210 175)(122 211 176)(123 212 177)(124 213 178)(125 214 179)(126 215 180)(127 216 161)(128 217 162)(129 218 163)(130 219 164)(131 220 165)(132 201 166)(133 202 167)(134 203 168)(135 204 169)(136 205 170)(137 206 171)(138 207 172)(139 208 173)(140 209 174)
(1 63)(2 72)(3 61)(4 70)(5 79)(6 68)(7 77)(8 66)(9 75)(10 64)(11 73)(12 62)(13 71)(14 80)(15 69)(16 78)(17 67)(18 76)(19 65)(20 74)(21 154)(22 143)(23 152)(24 141)(25 150)(26 159)(27 148)(28 157)(29 146)(30 155)(31 144)(32 153)(33 142)(34 151)(35 160)(36 149)(37 158)(38 147)(39 156)(40 145)(41 172)(42 161)(43 170)(44 179)(45 168)(46 177)(47 166)(48 175)(49 164)(50 173)(51 162)(52 171)(53 180)(54 169)(55 178)(56 167)(57 176)(58 165)(59 174)(60 163)(81 110)(82 119)(83 108)(84 117)(85 106)(86 115)(87 104)(88 113)(89 102)(90 111)(91 120)(92 109)(93 118)(94 107)(95 116)(96 105)(97 114)(98 103)(99 112)(100 101)(121 188)(122 197)(123 186)(124 195)(125 184)(126 193)(127 182)(128 191)(129 200)(130 189)(131 198)(132 187)(133 196)(134 185)(135 194)(136 183)(137 192)(138 181)(139 190)(140 199)(201 221)(202 230)(203 239)(204 228)(205 237)(206 226)(207 235)(208 224)(209 233)(210 222)(211 231)(212 240)(213 229)(214 238)(215 227)(216 236)(217 225)(218 234)(219 223)(220 232)

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

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

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

66 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 12A 12B 15A 15B 20A 20B 20C 20D 20E ··· 20L 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 12 12 15 15 20 20 20 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 1 1 10 10 10 10 30 30 2 2 2 6 6 6 6 15 15 15 15 2 2 2 2 2 20 20 20 20 2 ··· 2 4 4 4 4 4 4 4 4 12 ··· 12 4 ··· 4 4 ··· 4

66 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D5 D6 D6 D6 C4○D4 D10 D10 D10 D4⋊2S3 S3×D5 Q8⋊2D5 C2×S3×D5 C2×S3×D5 D20⋊S3 kernel C2×D20⋊S3 D20⋊S3 C2×D5×Dic3 C2×C3⋊D20 C6×D20 C10×Dic6 C2×C4×D15 C2×D20 C2×Dic6 D20 C2×C20 C22×D5 C30 Dic6 C2×Dic3 C2×C12 C10 C2×C4 C6 C4 C22 C2 # reps 1 8 2 2 1 1 1 1 2 4 1 2 4 8 4 2 2 2 4 4 2 8

Matrix representation of C2×D20⋊S3 in GL6(𝔽61)

 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60 0 0 0 0 0 0 60
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 60 18 0 0 0 0 0 0 45 46 0 0 0 0 9 16
,
 60 0 0 0 0 0 0 60 0 0 0 0 0 0 43 1 0 0 0 0 43 18 0 0 0 0 0 0 45 46 0 0 0 0 17 16
,
 0 1 0 0 0 0 60 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
,
 39 48 0 0 0 0 9 22 0 0 0 0 0 0 43 18 0 0 0 0 60 18 0 0 0 0 0 0 54 43 0 0 0 0 23 7

G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,18,0,0,0,0,0,0,45,9,0,0,0,0,46,16],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,45,17,0,0,0,0,46,16],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[39,9,0,0,0,0,48,22,0,0,0,0,0,0,43,60,0,0,0,0,18,18,0,0,0,0,0,0,54,23,0,0,0,0,43,7] >;

C2×D20⋊S3 in GAP, Magma, Sage, TeX

C_2\times D_{20}\rtimes S_3
% in TeX

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

G:=SmallGroup(480,1075);
// by ID

G=gap.SmallGroup(480,1075);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,346,80,1356,18822]);
// 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,e*b*e=b^9,c*d=d*c,e*c*e=b^18*c,e*d*e=d^-1>;
// generators/relations

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