S3xC2^3xC10 - GroupNames

G = S₃×C₂³×C₁₀ order 480 = 2⁵·3·5

Direct product of C₂³×C₁₀ and S₃

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

Aliases: S₃×C₂³×C₁₀, C₁₅⋊₄C₂⁵, C₃₀⋊₄C₂⁴, C₃⋊(C₂⁴×C₁₀), C₆⋊(C₂³×C₁₀), (C₂³×C₆)⋊₇C₁₀, (C₂×C₃₀)⋊₁₅C₂³, (C₂³×C₃₀)⋊₁₁C₂, (C₂²×C₃₀)⋊₂₄C₂², (C₂²×C₆)⋊₈(C₂×C₁₀), (C₂×C₆)⋊₄(C₂²×C₁₀), SmallGroup(480,1211)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃ — S₃×C₂³×C₁₀

Chief series

C₁ — C₃ — C₁₅ — C₅×S₃ — S₃×C₁₀ — S₃×C₂×C₁₀ — S₃×C₂²×C₁₀ — S₃×C₂³×C₁₀

Lower central

C₃ — S₃×C₂³×C₁₀

Upper central

C₁ — C₂³×C₁₀

Generators and relations for S₃×C₂³×C₁₀
G = < a,b,c,d,e,f | a²=b²=c²=d¹⁰=e³=f²=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 2724 in 1496 conjugacy classes, 882 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₂², C₂², C₅, S₃, C₆, C₂³, C₂³, C₁₀, C₁₀, D₆, C₂×C₆, C₁₅, C₂⁴, C₂⁴, C₂×C₁₀, C₂×C₁₀, C₂²×S₃, C₂²×C₆, C₅×S₃, C₃₀, C₂⁵, C₂²×C₁₀, C₂²×C₁₀, S₃×C₂³, C₂³×C₆, S₃×C₁₀, C₂×C₃₀, C₂³×C₁₀, C₂³×C₁₀, S₃×C₂⁴, S₃×C₂×C₁₀, C₂²×C₃₀, C₂⁴×C₁₀, S₃×C₂²×C₁₀, C₂³×C₃₀, S₃×C₂³×C₁₀
Quotients: C₁, C₂, C₂², C₅, S₃, C₂³, C₁₀, D₆, C₂⁴, C₂×C₁₀, C₂²×S₃, C₅×S₃, C₂⁵, C₂²×C₁₀, S₃×C₂³, S₃×C₁₀, C₂³×C₁₀, S₃×C₂⁴, S₃×C₂×C₁₀, C₂⁴×C₁₀, S₃×C₂²×C₁₀, S₃×C₂³×C₁₀

Smallest permutation representation of S₃×C₂³×C₁₀
►On 240 points

Generators in S₂₄₀

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

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

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

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

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

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

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

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

240 conjugacy classes

class	1	2A	···	2O	2P	···	2AE	3	5A	5B	5C	5D	6A	···	6O	10A	···	10BH	10BI	···	10DT	15A	15B	15C	15D	30A	···	30BH
order	1	2	···	2	2	···	2	3	5	5	5	5	6	···	6	10	···	10	10	···	10	15	15	15	15	30	···	30
size	1	1	···	1	3	···	3	2	1	1	1	1	2	···	2	1	···	1	3	···	3	2	2	2	2	2	···	2

240 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2
type	+	+	+				+	+
image	C₁	C₂	C₂	C₅	C₁₀	C₁₀	S₃	D₆	C₅×S₃	S₃×C₁₀
kernel	S₃×C₂³×C₁₀	S₃×C₂²×C₁₀	C₂³×C₃₀	S₃×C₂⁴	S₃×C₂³	C₂³×C₆	C₂³×C₁₀	C₂²×C₁₀	C₂⁴	C₂³
# reps	1	30	1	4	120	4	1	15	4	60

Matrix representation of S₃×C₂³×C₁₀ ►in GL₅(𝔽₃₁)

30	0	0	0	0
0	1	0	0	0
0	0	30	0	0
0	0	0	30	0
0	0	0	0	30

1	0	0	0	0
0	1	0	0	0
0	0	30	0	0
0	0	0	1	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	30	0
0	0	0	0	30

30	0	0	0	0
0	30	0	0	0
0	0	1	0	0
0	0	0	27	0
0	0	0	0	27

1	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	0	30
0	0	0	1	30

1	0	0	0	0
0	30	0	0	0
0	0	30	0	0
0	0	0	0	1
0	0	0	1	0

G:=sub<GL(5,GF(31))| [30,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,30],[1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,30,0,0,0,0,0,30],[30,0,0,0,0,0,30,0,0,0,0,0,1,0,0,0,0,0,27,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,30,30],[1,0,0,0,0,0,30,0,0,0,0,0,30,0,0,0,0,0,0,1,0,0,0,1,0] >;

S₃×C₂³×C₁₀ in GAP, Magma, Sage, TeX

S_3\times C_2^3\times C_{10}

% in TeX

G:=Group("S3xC2^3xC10");

// GroupNames label

G:=SmallGroup(480,1211);

// by ID

G=gap.SmallGroup(480,1211);

# by ID

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

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^10=e^3=f^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;

// generators/relations

G = S3×C23×C10 order 480 = 25·3·5

Direct product of C23×C10 and S3

G = S₃×C₂³×C₁₀ order 480 = 2⁵·3·5

Direct product of C₂³×C₁₀ and S₃