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## G = S3×C23×C10order 480 = 25·3·5

### Direct product of C23×C10 and S3

Aliases: S3×C23×C10, C154C25, C304C24, 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 [×15], C2 [×16], C3, C22 [×35], C22 [×120], C5, S3 [×16], C6 [×15], C23 [×15], C23 [×140], C10 [×15], C10 [×16], D6 [×120], C2×C6 [×35], C15, C24, C24 [×30], C2×C10 [×35], C2×C10 [×120], C22×S3 [×140], C22×C6 [×15], C5×S3 [×16], C30 [×15], C25, C22×C10 [×15], C22×C10 [×140], S3×C23 [×30], C23×C6, S3×C10 [×120], C2×C30 [×35], C23×C10, C23×C10 [×30], S3×C24, S3×C2×C10 [×140], C22×C30 [×15], C24×C10, S3×C22×C10 [×30], C23×C30, S3×C23×C10
Quotients: C1, C2 [×31], C22 [×155], C5, S3, C23 [×155], C10 [×31], D6 [×15], C24 [×31], C2×C10 [×155], C22×S3 [×35], C5×S3, C25, C22×C10 [×155], S3×C23 [×15], S3×C10 [×15], C23×C10 [×31], S3×C24, S3×C2×C10 [×35], C24×C10, S3×C22×C10 [×15], 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 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(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)(11 161)(12 162)(13 163)(14 164)(15 165)(16 166)(17 167)(18 168)(19 169)(20 170)(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)(11 226)(12 227)(13 228)(14 229)(15 230)(16 221)(17 222)(18 223)(19 224)(20 225)(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 45 39)(2 46 40)(3 47 31)(4 48 32)(5 49 33)(6 50 34)(7 41 35)(8 42 36)(9 43 37)(10 44 38)(11 26 237)(12 27 238)(13 28 239)(14 29 240)(15 30 231)(16 21 232)(17 22 233)(18 23 234)(19 24 235)(20 25 236)(51 80 64)(52 71 65)(53 72 66)(54 73 67)(55 74 68)(56 75 69)(57 76 70)(58 77 61)(59 78 62)(60 79 63)(81 110 94)(82 101 95)(83 102 96)(84 103 97)(85 104 98)(86 105 99)(87 106 100)(88 107 91)(89 108 92)(90 109 93)(111 140 124)(112 131 125)(113 132 126)(114 133 127)(115 134 128)(116 135 129)(117 136 130)(118 137 121)(119 138 122)(120 139 123)(141 170 154)(142 161 155)(143 162 156)(144 163 157)(145 164 158)(146 165 159)(147 166 160)(148 167 151)(149 168 152)(150 169 153)(171 200 184)(172 191 185)(173 192 186)(174 193 187)(175 194 188)(176 195 189)(177 196 190)(178 197 181)(179 198 182)(180 199 183)(201 230 214)(202 221 215)(203 222 216)(204 223 217)(205 224 218)(206 225 219)(207 226 220)(208 227 211)(209 228 212)(210 229 213)
(1 176)(2 177)(3 178)(4 179)(5 180)(6 171)(7 172)(8 173)(9 174)(10 175)(11 100)(12 91)(13 92)(14 93)(15 94)(16 95)(17 96)(18 97)(19 98)(20 99)(21 101)(22 102)(23 103)(24 104)(25 105)(26 106)(27 107)(28 108)(29 109)(30 110)(31 197)(32 198)(33 199)(34 200)(35 191)(36 192)(37 193)(38 194)(39 195)(40 196)(41 185)(42 186)(43 187)(44 188)(45 189)(46 190)(47 181)(48 182)(49 183)(50 184)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(61 167)(62 168)(63 169)(64 170)(65 161)(66 162)(67 163)(68 164)(69 165)(70 166)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 151)(78 152)(79 153)(80 154)(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)(121 227)(122 228)(123 229)(124 230)(125 221)(126 222)(127 223)(128 224)(129 225)(130 226)(131 215)(132 216)(133 217)(134 218)(135 219)(136 220)(137 211)(138 212)(139 213)(140 214)

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

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

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

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

׿
×
𝔽