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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, C23.39(S3×C10), C10.70(S3×C23), D6.8(C22×C10), (S3×C10).44C23, (C2×C30).442C23, Dic3⋊3(C22×C10), (C5×Dic3)⋊11C23, (C22×C10).153D6, (C22×Dic3)⋊10C10, (C10×Dic3)⋊39C22, (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, (C2×C6).63(C22×C10), (C22×C6).44(C2×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

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

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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

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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