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## G = S3×C22×C20order 480 = 25·3·5

### Direct product of C22×C20 and S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — S3×C22×C20
 Chief series C1 — C3 — C6 — C30 — S3×C10 — S3×C2×C10 — S3×C22×C10 — S3×C22×C20
 Lower central C3 — S3×C22×C20
 Upper central C1 — C22×C20

Generators and relations for S3×C22×C20
G = < a,b,c,d,e | a2=b2=c20=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 836 in 472 conjugacy classes, 290 normal (22 characteristic)
C1, C2, C2 [×6], C2 [×8], C3, C4 [×4], C4 [×4], C22 [×7], C22 [×28], C5, S3 [×8], C6, C6 [×6], C2×C4 [×6], C2×C4 [×22], C23, C23 [×14], C10, C10 [×6], C10 [×8], Dic3 [×4], C12 [×4], D6 [×28], C2×C6 [×7], C15, C22×C4, C22×C4 [×13], C24, C20 [×4], C20 [×4], C2×C10 [×7], C2×C10 [×28], C4×S3 [×16], C2×Dic3 [×6], C2×C12 [×6], C22×S3 [×14], C22×C6, C5×S3 [×8], C30, C30 [×6], C23×C4, C2×C20 [×6], C2×C20 [×22], C22×C10, C22×C10 [×14], S3×C2×C4 [×12], C22×Dic3, C22×C12, S3×C23, C5×Dic3 [×4], C60 [×4], S3×C10 [×28], C2×C30 [×7], C22×C20, C22×C20 [×13], C23×C10, S3×C22×C4, S3×C20 [×16], C10×Dic3 [×6], C2×C60 [×6], S3×C2×C10 [×14], C22×C30, C23×C20, S3×C2×C20 [×12], Dic3×C2×C10, C22×C60, S3×C22×C10, S3×C22×C20
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C5, S3, C2×C4 [×28], C23 [×15], C10 [×15], D6 [×7], C22×C4 [×14], C24, C20 [×8], C2×C10 [×35], C4×S3 [×4], C22×S3 [×7], C5×S3, C23×C4, C2×C20 [×28], C22×C10 [×15], S3×C2×C4 [×6], S3×C23, S3×C10 [×7], C22×C20 [×14], C23×C10, S3×C22×C4, S3×C20 [×4], S3×C2×C10 [×7], C23×C20, S3×C2×C20 [×6], S3×C22×C10, S3×C22×C20

Smallest permutation representation of S3×C22×C20
On 240 points
Generators in S240
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 101)(11 102)(12 103)(13 104)(14 105)(15 106)(16 107)(17 108)(18 109)(19 110)(20 111)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 95)(42 96)(43 97)(44 98)(45 99)(46 100)(47 81)(48 82)(49 83)(50 84)(51 85)(52 86)(53 87)(54 88)(55 89)(56 90)(57 91)(58 92)(59 93)(60 94)(121 161)(122 162)(123 163)(124 164)(125 165)(126 166)(127 167)(128 168)(129 169)(130 170)(131 171)(132 172)(133 173)(134 174)(135 175)(136 176)(137 177)(138 178)(139 179)(140 180)(141 216)(142 217)(143 218)(144 219)(145 220)(146 201)(147 202)(148 203)(149 204)(150 205)(151 206)(152 207)(153 208)(154 209)(155 210)(156 211)(157 212)(158 213)(159 214)(160 215)(181 223)(182 224)(183 225)(184 226)(185 227)(186 228)(187 229)(188 230)(189 231)(190 232)(191 233)(192 234)(193 235)(194 236)(195 237)(196 238)(197 239)(198 240)(199 221)(200 222)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 100)(10 81)(11 82)(12 83)(13 84)(14 85)(15 86)(16 87)(17 88)(18 89)(19 90)(20 91)(21 187)(22 188)(23 189)(24 190)(25 191)(26 192)(27 193)(28 194)(29 195)(30 196)(31 197)(32 198)(33 199)(34 200)(35 181)(36 182)(37 183)(38 184)(39 185)(40 186)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(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)(121 145)(122 146)(123 147)(124 148)(125 149)(126 150)(127 151)(128 152)(129 153)(130 154)(131 155)(132 156)(133 157)(134 158)(135 159)(136 160)(137 141)(138 142)(139 143)(140 144)(161 220)(162 201)(163 202)(164 203)(165 204)(166 205)(167 206)(168 207)(169 208)(170 209)(171 210)(172 211)(173 212)(174 213)(175 214)(176 215)(177 216)(178 217)(179 218)(180 219)
(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)
(1 187 155)(2 188 156)(3 189 157)(4 190 158)(5 191 159)(6 192 160)(7 193 141)(8 194 142)(9 195 143)(10 196 144)(11 197 145)(12 198 146)(13 199 147)(14 200 148)(15 181 149)(16 182 150)(17 183 151)(18 184 152)(19 185 153)(20 186 154)(21 131 92)(22 132 93)(23 133 94)(24 134 95)(25 135 96)(26 136 97)(27 137 98)(28 138 99)(29 139 100)(30 140 81)(31 121 82)(32 122 83)(33 123 84)(34 124 85)(35 125 86)(36 126 87)(37 127 88)(38 128 89)(39 129 90)(40 130 91)(41 76 174)(42 77 175)(43 78 176)(44 79 177)(45 80 178)(46 61 179)(47 62 180)(48 63 161)(49 64 162)(50 65 163)(51 66 164)(52 67 165)(53 68 166)(54 69 167)(55 70 168)(56 71 169)(57 72 170)(58 73 171)(59 74 172)(60 75 173)(101 238 219)(102 239 220)(103 240 201)(104 221 202)(105 222 203)(106 223 204)(107 224 205)(108 225 206)(109 226 207)(110 227 208)(111 228 209)(112 229 210)(113 230 211)(114 231 212)(115 232 213)(116 233 214)(117 234 215)(118 235 216)(119 236 217)(120 237 218)
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 210)(22 211)(23 212)(24 213)(25 214)(26 215)(27 216)(28 217)(29 218)(30 219)(31 220)(32 201)(33 202)(34 203)(35 204)(36 205)(37 206)(38 207)(39 208)(40 209)(61 143)(62 144)(63 145)(64 146)(65 147)(66 148)(67 149)(68 150)(69 151)(70 152)(71 153)(72 154)(73 155)(74 156)(75 157)(76 158)(77 159)(78 160)(79 141)(80 142)(81 101)(82 102)(83 103)(84 104)(85 105)(86 106)(87 107)(88 108)(89 109)(90 110)(91 111)(92 112)(93 113)(94 114)(95 115)(96 116)(97 117)(98 118)(99 119)(100 120)(121 239)(122 240)(123 221)(124 222)(125 223)(126 224)(127 225)(128 226)(129 227)(130 228)(131 229)(132 230)(133 231)(134 232)(135 233)(136 234)(137 235)(138 236)(139 237)(140 238)(161 197)(162 198)(163 199)(164 200)(165 181)(166 182)(167 183)(168 184)(169 185)(170 186)(171 187)(172 188)(173 189)(174 190)(175 191)(176 192)(177 193)(178 194)(179 195)(180 196)

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

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

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

240 conjugacy classes

 class 1 2A ··· 2G 2H ··· 2O 3 4A ··· 4H 4I ··· 4P 5A 5B 5C 5D 6A ··· 6G 10A ··· 10AB 10AC ··· 10BH 12A ··· 12H 15A 15B 15C 15D 20A ··· 20AF 20AG ··· 20BL 30A ··· 30AB 60A ··· 60AF order 1 2 ··· 2 2 ··· 2 3 4 ··· 4 4 ··· 4 5 5 5 5 6 ··· 6 10 ··· 10 10 ··· 10 12 ··· 12 15 15 15 15 20 ··· 20 20 ··· 20 30 ··· 30 60 ··· 60 size 1 1 ··· 1 3 ··· 3 2 1 ··· 1 3 ··· 3 1 1 1 1 2 ··· 2 1 ··· 1 3 ··· 3 2 ··· 2 2 2 2 2 1 ··· 1 3 ··· 3 2 ··· 2 2 ··· 2

240 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + + image C1 C2 C2 C2 C2 C4 C5 C10 C10 C10 C10 C20 S3 D6 D6 C4×S3 C5×S3 S3×C10 S3×C10 S3×C20 kernel S3×C22×C20 S3×C2×C20 Dic3×C2×C10 C22×C60 S3×C22×C10 S3×C2×C10 S3×C22×C4 S3×C2×C4 C22×Dic3 C22×C12 S3×C23 C22×S3 C22×C20 C2×C20 C22×C10 C2×C10 C22×C4 C2×C4 C23 C22 # reps 1 12 1 1 1 16 4 48 4 4 4 64 1 6 1 8 4 24 4 32

Matrix representation of S3×C22×C20 in GL4(𝔽61) generated by

 1 0 0 0 0 1 0 0 0 0 60 0 0 0 0 60
,
 1 0 0 0 0 60 0 0 0 0 60 0 0 0 0 60
,
 28 0 0 0 0 60 0 0 0 0 53 0 0 0 0 53
,
 1 0 0 0 0 1 0 0 0 0 60 60 0 0 1 0
,
 60 0 0 0 0 1 0 0 0 0 1 0 0 0 60 60
G:=sub<GL(4,GF(61))| [1,0,0,0,0,1,0,0,0,0,60,0,0,0,0,60],[1,0,0,0,0,60,0,0,0,0,60,0,0,0,0,60],[28,0,0,0,0,60,0,0,0,0,53,0,0,0,0,53],[1,0,0,0,0,1,0,0,0,0,60,1,0,0,60,0],[60,0,0,0,0,1,0,0,0,0,1,60,0,0,0,60] >;

S3×C22×C20 in GAP, Magma, Sage, TeX

S_3\times C_2^2\times C_{20}
% in TeX

G:=Group("S3xC2^2xC20");
// GroupNames label

G:=SmallGroup(480,1151);
// by ID

G=gap.SmallGroup(480,1151);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-3,304,15686]);
// Polycyclic

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

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𝔽