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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

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

Generators and relations
 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 >

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

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

Matrix representation ►G ⊆ GL5(𝔽31)

300000
01000
003000
000300
000030
,
10000
01000
003000
00010
00001
,
10000
01000
00100
000300
000030
,
300000
030000
00100
000270
000027
,
10000
01000
00100
000030
000130
,
10000
030000
003000
00001
00010

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] >;
 

240 conjugacy classes

class 1 2A···2O2P···2AE 3 5A5B5C5D6A···6O10A···10BH10BI···10DT15A15B15C15D30A···30BH
order12···22···2355556···610···1010···101515151530···30
size11···13···3211112···21···13···322222···2

240 irreducible representations

dim1111112222
type+++++
imageC1C2C2C5C10C10S3D6C5×S3S3×C10
kernelS3×C23×C10S3×C22×C10C23×C30S3×C24S3×C23C23×C6C23×C10C22×C10C24C23
# reps130141204115460

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
 

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