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G = S3×C22×C20  order 480 = 25·3·5

Direct product of C22×C20 and S3

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

Aliases: S3×C22×C20, C60⋊14C23, C30.85C24, C3⋊1(C23×C20), C6⋊1(C22×C20), C15⋊10(C23×C4), (C2×C60)⋊52C22, C12⋊3(C22×C10), (C22×C60)⋊24C2, C30⋊10(C22×C4), C6.2(C23×C10), (C22×C12)⋊10C10, (S3×C23).3C10, C10.70(S3×C23), C23.39(S3×C10), D6.8(C22×C10), (S3×C10).44C23, (C2×C30).442C23, (C5×Dic3)⋊11C23, Dic3⋊3(C22×C10), (C22×C10).153D6, (C10×Dic3)⋊39C22, (C22×Dic3)⋊10C10, (C22×C30).182C22, (C2×C6)⋊6(C2×C20), (C2×C30)⋊38(C2×C4), (C2×C12)⋊14(C2×C10), C2.1(S3×C22×C10), (Dic3×C2×C10)⋊21C2, (S3×C22×C10).6C2, C22.29(S3×C2×C10), (C2×Dic3)⋊12(C2×C10), (S3×C2×C10).127C22, (C22×C6).44(C2×C10), (C2×C6).63(C22×C10), (C22×S3).35(C2×C10), (C2×C10).376(C22×S3), SmallGroup(480,1151)

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, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, C23, C23, C10, C10, C10, Dic3, C12, D6, C2×C6, C15, C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C4×S3, C2×Dic3, C2×C12, C22×S3, C22×C6, C5×S3, C30, C30, C23×C4, C2×C20, C2×C20, C22×C10, C22×C10, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C5×Dic3, C60, S3×C10, C2×C30, C22×C20, C22×C20, C23×C10, S3×C22×C4, S3×C20, C10×Dic3, C2×C60, S3×C2×C10, C22×C30, C23×C20, S3×C2×C20, Dic3×C2×C10, C22×C60, S3×C22×C10, S3×C22×C20
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, C23, C10, D6, C22×C4, C24, C20, C2×C10, C4×S3, C22×S3, C5×S3, C23×C4, C2×C20, C22×C10, S3×C2×C4, S3×C23, S3×C10, C22×C20, C23×C10, S3×C22×C4, S3×C20, S3×C2×C10, C23×C20, S3×C2×C20, S3×C22×C10, S3×C22×C20

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

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

(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)

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

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

240 conjugacy classes

class 1 2A···2G2H···2O 3 4A···4H4I···4P5A5B5C5D6A···6G10A···10AB10AC···10BH12A···12H15A15B15C15D20A···20AF20AG···20BL30A···30AB60A···60AF
order12···22···234···44···455556···610···1010···1012···121515151520···2020···2030···3060···60
size11···13···321···13···311112···21···13···32···222221···13···32···22···2

240 irreducible representations

dim11111111111122222222
type++++++++
imageC1C2C2C2C2C4C5C10C10C10C10C20S3D6D6C4×S3C5×S3S3×C10S3×C10S3×C20
kernelS3×C22×C20S3×C2×C20Dic3×C2×C10C22×C60S3×C22×C10S3×C2×C10S3×C22×C4S3×C2×C4C22×Dic3C22×C12S3×C23C22×S3C22×C20C2×C20C22×C10C2×C10C22×C4C2×C4C23C22
# reps11211116448444641618424432

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

1000
0100
00600
00060
,
1000
06000
00600
00060
,
28000
06000
00530
00053
,
1000
0100
006060
0010
,
60000
0100
0010
006060
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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