direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D6.D10, C30.11C24, C60.189C23, D30.35C23, Dic15.37C23, (C4×D5)⋊18D6, (C4×S3)⋊18D10, C30⋊5(C4○D4), C6⋊2(C4○D20), C15⋊Q8⋊18C22, C10⋊2(C4○D12), (C2×C20).343D6, C6.11(C23×D5), (S3×C20)⋊19C22, (C2×C12).347D10, (D5×C12)⋊19C22, (C4×D15)⋊25C22, C3⋊D20⋊20C22, C5⋊D12⋊21C22, C15⋊D4⋊21C22, C10.11(S3×C23), D6.24(C22×D5), (C22×D5).96D6, (C6×D5).38C23, (S3×C10).27C23, C20.186(C22×S3), (C2×C30).230C23, (C2×C60).245C22, (C2×Dic5).198D6, D10.40(C22×S3), (C22×S3).82D10, C12.186(C22×D5), (C2×Dic3).169D10, (C5×Dic3).28C23, Dic3.25(C22×D5), Dic5.42(C22×S3), (C3×Dic5).40C23, (C6×Dic5).227C22, (C2×Dic15).234C22, (C10×Dic3).209C22, (C22×D15).119C22, (S3×C2×C20)⋊9C2, (C2×C4×D5)⋊15S3, (D5×C2×C12)⋊9C2, (S3×C2×C4)⋊15D5, C15⋊5(C2×C4○D4), C5⋊2(C2×C4○D12), C3⋊2(C2×C4○D20), (C2×C4×D15)⋊30C2, (C2×C15⋊Q8)⋊26C2, C4.159(C2×S3×D5), (C2×C15⋊D4)⋊24C2, (C2×C3⋊D20)⋊24C2, (C2×C5⋊D12)⋊24C2, C22.99(C2×S3×D5), C2.15(C22×S3×D5), (C2×C4).248(S3×D5), (D5×C2×C6).115C22, (S3×C2×C10).100C22, (C2×C6).240(C22×D5), (C2×C10).240(C22×S3), SmallGroup(480,1083)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1564 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×4], C6, C6 [×2], C6 [×2], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×6], C2×C6, C2×C6 [×4], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×6], C2×C10, C2×C10 [×4], Dic6 [×4], C4×S3 [×4], C4×S3 [×4], D12 [×4], C2×Dic3, C2×Dic3, C3⋊D4 [×8], C2×C12, C2×C12 [×5], C22×S3, C22×S3, C22×C6, C5×S3 [×2], C3×D5 [×2], D15 [×2], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×4], C4×D5 [×4], D20 [×4], C2×Dic5, C2×Dic5, C5⋊D4 [×8], C2×C20, C2×C20 [×5], C22×D5, C22×D5, C22×C10, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, C5×Dic3 [×2], C3×Dic5 [×2], Dic15 [×2], C60 [×2], C6×D5 [×2], C6×D5 [×2], S3×C10 [×2], S3×C10 [×2], D30 [×2], D30 [×2], C2×C30, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×C4○D12, C15⋊D4 [×4], C3⋊D20 [×4], C5⋊D12 [×4], C15⋊Q8 [×4], D5×C12 [×4], C6×Dic5, S3×C20 [×4], C10×Dic3, C4×D15 [×4], C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, C2×C4○D20, D6.D10 [×8], C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, C2×C15⋊Q8, D5×C2×C12, S3×C2×C20, C2×C4×D15, C2×D6.D10
Quotients:
C1, C2 [×15], C22 [×35], S3, C23 [×15], D5, D6 [×7], C4○D4 [×2], C24, D10 [×7], C22×S3 [×7], C2×C4○D4, C22×D5 [×7], C4○D12 [×2], S3×C23, S3×D5, C4○D20 [×2], C23×D5, C2×C4○D12, C2×S3×D5 [×3], C2×C4○D20, D6.D10 [×2], C22×S3×D5, C2×D6.D10
Generators and relations
G = < a,b,c,d,e | a2=b6=c2=1, d10=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=d9 >
►On 240 points
Generators in S240
(1 58)(2 59)(3 60)(4 41)(5 42)(6 43)(7 44)(8 45)(9 46)(10 47)(11 48)(12 49)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 240)(22 221)(23 222)(24 223)(25 224)(26 225)(27 226)(28 227)(29 228)(30 229)(31 230)(32 231)(33 232)(34 233)(35 234)(36 235)(37 236)(38 237)(39 238)(40 239)(61 82)(62 83)(63 84)(64 85)(65 86)(66 87)(67 88)(68 89)(69 90)(70 91)(71 92)(72 93)(73 94)(74 95)(75 96)(76 97)(77 98)(78 99)(79 100)(80 81)(101 150)(102 151)(103 152)(104 153)(105 154)(106 155)(107 156)(108 157)(109 158)(110 159)(111 160)(112 141)(113 142)(114 143)(115 144)(116 145)(117 146)(118 147)(119 148)(120 149)(121 181)(122 182)(123 183)(124 184)(125 185)(126 186)(127 187)(128 188)(129 189)(130 190)(131 191)(132 192)(133 193)(134 194)(135 195)(136 196)(137 197)(138 198)(139 199)(140 200)(161 214)(162 215)(163 216)(164 217)(165 218)(166 219)(167 220)(168 201)(169 202)(170 203)(171 204)(172 205)(173 206)(174 207)(175 208)(176 209)(177 210)(178 211)(179 212)(180 213)
(1 122 177 11 132 167)(2 123 178 12 133 168)(3 124 179 13 134 169)(4 125 180 14 135 170)(5 126 161 15 136 171)(6 127 162 16 137 172)(7 128 163 17 138 173)(8 129 164 18 139 174)(9 130 165 19 140 175)(10 131 166 20 121 176)(21 159 100 31 149 90)(22 160 81 32 150 91)(23 141 82 33 151 92)(24 142 83 34 152 93)(25 143 84 35 153 94)(26 144 85 36 154 95)(27 145 86 37 155 96)(28 146 87 38 156 97)(29 147 88 39 157 98)(30 148 89 40 158 99)(41 185 213 51 195 203)(42 186 214 52 196 204)(43 187 215 53 197 205)(44 188 216 54 198 206)(45 189 217 55 199 207)(46 190 218 56 200 208)(47 191 219 57 181 209)(48 192 220 58 182 210)(49 193 201 59 183 211)(50 194 202 60 184 212)(61 232 102 71 222 112)(62 233 103 72 223 113)(63 234 104 73 224 114)(64 235 105 74 225 115)(65 236 106 75 226 116)(66 237 107 76 227 117)(67 238 108 77 228 118)(68 239 109 78 229 119)(69 240 110 79 230 120)(70 221 111 80 231 101)
(1 113)(2 114)(3 115)(4 116)(5 117)(6 118)(7 119)(8 120)(9 101)(10 102)(11 103)(12 104)(13 105)(14 106)(15 107)(16 108)(17 109)(18 110)(19 111)(20 112)(21 199)(22 200)(23 181)(24 182)(25 183)(26 184)(27 185)(28 186)(29 187)(30 188)(31 189)(32 190)(33 191)(34 192)(35 193)(36 194)(37 195)(38 196)(39 197)(40 198)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 141)(58 142)(59 143)(60 144)(61 166)(62 167)(63 168)(64 169)(65 170)(66 171)(67 172)(68 173)(69 174)(70 175)(71 176)(72 177)(73 178)(74 179)(75 180)(76 161)(77 162)(78 163)(79 164)(80 165)(81 218)(82 219)(83 220)(84 201)(85 202)(86 203)(87 204)(88 205)(89 206)(90 207)(91 208)(92 209)(93 210)(94 211)(95 212)(96 213)(97 214)(98 215)(99 216)(100 217)(121 222)(122 223)(123 224)(124 225)(125 226)(126 227)(127 228)(128 229)(129 230)(130 231)(131 232)(132 233)(133 234)(134 235)(135 236)(136 237)(137 238)(138 239)(139 240)(140 221)
(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 10 11 20)(2 19 12 9)(3 8 13 18)(4 17 14 7)(5 6 15 16)(21 36 31 26)(22 25 32 35)(23 34 33 24)(27 30 37 40)(28 39 38 29)(41 54 51 44)(42 43 52 53)(45 50 55 60)(46 59 56 49)(47 48 57 58)(61 72 71 62)(63 70 73 80)(64 79 74 69)(65 68 75 78)(66 77 76 67)(81 84 91 94)(82 93 92 83)(85 100 95 90)(86 89 96 99)(87 98 97 88)(101 104 111 114)(102 113 112 103)(105 120 115 110)(106 109 116 119)(107 118 117 108)(121 122 131 132)(123 140 133 130)(124 129 134 139)(125 138 135 128)(126 127 136 137)(141 152 151 142)(143 150 153 160)(144 159 154 149)(145 148 155 158)(146 157 156 147)(161 162 171 172)(163 180 173 170)(164 169 174 179)(165 178 175 168)(166 167 176 177)(181 182 191 192)(183 200 193 190)(184 189 194 199)(185 198 195 188)(186 187 196 197)(201 218 211 208)(202 207 212 217)(203 216 213 206)(204 205 214 215)(209 210 219 220)(221 224 231 234)(222 233 232 223)(225 240 235 230)(226 229 236 239)(227 238 237 228)
G:=sub<Sym(240)| (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,240)(22,221)(23,222)(24,223)(25,224)(26,225)(27,226)(28,227)(29,228)(30,229)(31,230)(32,231)(33,232)(34,233)(35,234)(36,235)(37,236)(38,237)(39,238)(40,239)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,81)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)(137,197)(138,198)(139,199)(140,200)(161,214)(162,215)(163,216)(164,217)(165,218)(166,219)(167,220)(168,201)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213), (1,122,177,11,132,167)(2,123,178,12,133,168)(3,124,179,13,134,169)(4,125,180,14,135,170)(5,126,161,15,136,171)(6,127,162,16,137,172)(7,128,163,17,138,173)(8,129,164,18,139,174)(9,130,165,19,140,175)(10,131,166,20,121,176)(21,159,100,31,149,90)(22,160,81,32,150,91)(23,141,82,33,151,92)(24,142,83,34,152,93)(25,143,84,35,153,94)(26,144,85,36,154,95)(27,145,86,37,155,96)(28,146,87,38,156,97)(29,147,88,39,157,98)(30,148,89,40,158,99)(41,185,213,51,195,203)(42,186,214,52,196,204)(43,187,215,53,197,205)(44,188,216,54,198,206)(45,189,217,55,199,207)(46,190,218,56,200,208)(47,191,219,57,181,209)(48,192,220,58,182,210)(49,193,201,59,183,211)(50,194,202,60,184,212)(61,232,102,71,222,112)(62,233,103,72,223,113)(63,234,104,73,224,114)(64,235,105,74,225,115)(65,236,106,75,226,116)(66,237,107,76,227,117)(67,238,108,77,228,118)(68,239,109,78,229,119)(69,240,110,79,230,120)(70,221,111,80,231,101), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,199)(22,200)(23,181)(24,182)(25,183)(26,184)(27,185)(28,186)(29,187)(30,188)(31,189)(32,190)(33,191)(34,192)(35,193)(36,194)(37,195)(38,196)(39,197)(40,198)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,166)(62,167)(63,168)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,176)(72,177)(73,178)(74,179)(75,180)(76,161)(77,162)(78,163)(79,164)(80,165)(81,218)(82,219)(83,220)(84,201)(85,202)(86,203)(87,204)(88,205)(89,206)(90,207)(91,208)(92,209)(93,210)(94,211)(95,212)(96,213)(97,214)(98,215)(99,216)(100,217)(121,222)(122,223)(123,224)(124,225)(125,226)(126,227)(127,228)(128,229)(129,230)(130,231)(131,232)(132,233)(133,234)(134,235)(135,236)(136,237)(137,238)(138,239)(139,240)(140,221), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,54,51,44)(42,43,52,53)(45,50,55,60)(46,59,56,49)(47,48,57,58)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137)(141,152,151,142)(143,150,153,160)(144,159,154,149)(145,148,155,158)(146,157,156,147)(161,162,171,172)(163,180,173,170)(164,169,174,179)(165,178,175,168)(166,167,176,177)(181,182,191,192)(183,200,193,190)(184,189,194,199)(185,198,195,188)(186,187,196,197)(201,218,211,208)(202,207,212,217)(203,216,213,206)(204,205,214,215)(209,210,219,220)(221,224,231,234)(222,233,232,223)(225,240,235,230)(226,229,236,239)(227,238,237,228)>;
G:=Group( (1,58)(2,59)(3,60)(4,41)(5,42)(6,43)(7,44)(8,45)(9,46)(10,47)(11,48)(12,49)(13,50)(14,51)(15,52)(16,53)(17,54)(18,55)(19,56)(20,57)(21,240)(22,221)(23,222)(24,223)(25,224)(26,225)(27,226)(28,227)(29,228)(30,229)(31,230)(32,231)(33,232)(34,233)(35,234)(36,235)(37,236)(38,237)(39,238)(40,239)(61,82)(62,83)(63,84)(64,85)(65,86)(66,87)(67,88)(68,89)(69,90)(70,91)(71,92)(72,93)(73,94)(74,95)(75,96)(76,97)(77,98)(78,99)(79,100)(80,81)(101,150)(102,151)(103,152)(104,153)(105,154)(106,155)(107,156)(108,157)(109,158)(110,159)(111,160)(112,141)(113,142)(114,143)(115,144)(116,145)(117,146)(118,147)(119,148)(120,149)(121,181)(122,182)(123,183)(124,184)(125,185)(126,186)(127,187)(128,188)(129,189)(130,190)(131,191)(132,192)(133,193)(134,194)(135,195)(136,196)(137,197)(138,198)(139,199)(140,200)(161,214)(162,215)(163,216)(164,217)(165,218)(166,219)(167,220)(168,201)(169,202)(170,203)(171,204)(172,205)(173,206)(174,207)(175,208)(176,209)(177,210)(178,211)(179,212)(180,213), (1,122,177,11,132,167)(2,123,178,12,133,168)(3,124,179,13,134,169)(4,125,180,14,135,170)(5,126,161,15,136,171)(6,127,162,16,137,172)(7,128,163,17,138,173)(8,129,164,18,139,174)(9,130,165,19,140,175)(10,131,166,20,121,176)(21,159,100,31,149,90)(22,160,81,32,150,91)(23,141,82,33,151,92)(24,142,83,34,152,93)(25,143,84,35,153,94)(26,144,85,36,154,95)(27,145,86,37,155,96)(28,146,87,38,156,97)(29,147,88,39,157,98)(30,148,89,40,158,99)(41,185,213,51,195,203)(42,186,214,52,196,204)(43,187,215,53,197,205)(44,188,216,54,198,206)(45,189,217,55,199,207)(46,190,218,56,200,208)(47,191,219,57,181,209)(48,192,220,58,182,210)(49,193,201,59,183,211)(50,194,202,60,184,212)(61,232,102,71,222,112)(62,233,103,72,223,113)(63,234,104,73,224,114)(64,235,105,74,225,115)(65,236,106,75,226,116)(66,237,107,76,227,117)(67,238,108,77,228,118)(68,239,109,78,229,119)(69,240,110,79,230,120)(70,221,111,80,231,101), (1,113)(2,114)(3,115)(4,116)(5,117)(6,118)(7,119)(8,120)(9,101)(10,102)(11,103)(12,104)(13,105)(14,106)(15,107)(16,108)(17,109)(18,110)(19,111)(20,112)(21,199)(22,200)(23,181)(24,182)(25,183)(26,184)(27,185)(28,186)(29,187)(30,188)(31,189)(32,190)(33,191)(34,192)(35,193)(36,194)(37,195)(38,196)(39,197)(40,198)(41,145)(42,146)(43,147)(44,148)(45,149)(46,150)(47,151)(48,152)(49,153)(50,154)(51,155)(52,156)(53,157)(54,158)(55,159)(56,160)(57,141)(58,142)(59,143)(60,144)(61,166)(62,167)(63,168)(64,169)(65,170)(66,171)(67,172)(68,173)(69,174)(70,175)(71,176)(72,177)(73,178)(74,179)(75,180)(76,161)(77,162)(78,163)(79,164)(80,165)(81,218)(82,219)(83,220)(84,201)(85,202)(86,203)(87,204)(88,205)(89,206)(90,207)(91,208)(92,209)(93,210)(94,211)(95,212)(96,213)(97,214)(98,215)(99,216)(100,217)(121,222)(122,223)(123,224)(124,225)(125,226)(126,227)(127,228)(128,229)(129,230)(130,231)(131,232)(132,233)(133,234)(134,235)(135,236)(136,237)(137,238)(138,239)(139,240)(140,221), (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), (1,10,11,20)(2,19,12,9)(3,8,13,18)(4,17,14,7)(5,6,15,16)(21,36,31,26)(22,25,32,35)(23,34,33,24)(27,30,37,40)(28,39,38,29)(41,54,51,44)(42,43,52,53)(45,50,55,60)(46,59,56,49)(47,48,57,58)(61,72,71,62)(63,70,73,80)(64,79,74,69)(65,68,75,78)(66,77,76,67)(81,84,91,94)(82,93,92,83)(85,100,95,90)(86,89,96,99)(87,98,97,88)(101,104,111,114)(102,113,112,103)(105,120,115,110)(106,109,116,119)(107,118,117,108)(121,122,131,132)(123,140,133,130)(124,129,134,139)(125,138,135,128)(126,127,136,137)(141,152,151,142)(143,150,153,160)(144,159,154,149)(145,148,155,158)(146,157,156,147)(161,162,171,172)(163,180,173,170)(164,169,174,179)(165,178,175,168)(166,167,176,177)(181,182,191,192)(183,200,193,190)(184,189,194,199)(185,198,195,188)(186,187,196,197)(201,218,211,208)(202,207,212,217)(203,216,213,206)(204,205,214,215)(209,210,219,220)(221,224,231,234)(222,233,232,223)(225,240,235,230)(226,229,236,239)(227,238,237,228) );
G=PermutationGroup([(1,58),(2,59),(3,60),(4,41),(5,42),(6,43),(7,44),(8,45),(9,46),(10,47),(11,48),(12,49),(13,50),(14,51),(15,52),(16,53),(17,54),(18,55),(19,56),(20,57),(21,240),(22,221),(23,222),(24,223),(25,224),(26,225),(27,226),(28,227),(29,228),(30,229),(31,230),(32,231),(33,232),(34,233),(35,234),(36,235),(37,236),(38,237),(39,238),(40,239),(61,82),(62,83),(63,84),(64,85),(65,86),(66,87),(67,88),(68,89),(69,90),(70,91),(71,92),(72,93),(73,94),(74,95),(75,96),(76,97),(77,98),(78,99),(79,100),(80,81),(101,150),(102,151),(103,152),(104,153),(105,154),(106,155),(107,156),(108,157),(109,158),(110,159),(111,160),(112,141),(113,142),(114,143),(115,144),(116,145),(117,146),(118,147),(119,148),(120,149),(121,181),(122,182),(123,183),(124,184),(125,185),(126,186),(127,187),(128,188),(129,189),(130,190),(131,191),(132,192),(133,193),(134,194),(135,195),(136,196),(137,197),(138,198),(139,199),(140,200),(161,214),(162,215),(163,216),(164,217),(165,218),(166,219),(167,220),(168,201),(169,202),(170,203),(171,204),(172,205),(173,206),(174,207),(175,208),(176,209),(177,210),(178,211),(179,212),(180,213)], [(1,122,177,11,132,167),(2,123,178,12,133,168),(3,124,179,13,134,169),(4,125,180,14,135,170),(5,126,161,15,136,171),(6,127,162,16,137,172),(7,128,163,17,138,173),(8,129,164,18,139,174),(9,130,165,19,140,175),(10,131,166,20,121,176),(21,159,100,31,149,90),(22,160,81,32,150,91),(23,141,82,33,151,92),(24,142,83,34,152,93),(25,143,84,35,153,94),(26,144,85,36,154,95),(27,145,86,37,155,96),(28,146,87,38,156,97),(29,147,88,39,157,98),(30,148,89,40,158,99),(41,185,213,51,195,203),(42,186,214,52,196,204),(43,187,215,53,197,205),(44,188,216,54,198,206),(45,189,217,55,199,207),(46,190,218,56,200,208),(47,191,219,57,181,209),(48,192,220,58,182,210),(49,193,201,59,183,211),(50,194,202,60,184,212),(61,232,102,71,222,112),(62,233,103,72,223,113),(63,234,104,73,224,114),(64,235,105,74,225,115),(65,236,106,75,226,116),(66,237,107,76,227,117),(67,238,108,77,228,118),(68,239,109,78,229,119),(69,240,110,79,230,120),(70,221,111,80,231,101)], [(1,113),(2,114),(3,115),(4,116),(5,117),(6,118),(7,119),(8,120),(9,101),(10,102),(11,103),(12,104),(13,105),(14,106),(15,107),(16,108),(17,109),(18,110),(19,111),(20,112),(21,199),(22,200),(23,181),(24,182),(25,183),(26,184),(27,185),(28,186),(29,187),(30,188),(31,189),(32,190),(33,191),(34,192),(35,193),(36,194),(37,195),(38,196),(39,197),(40,198),(41,145),(42,146),(43,147),(44,148),(45,149),(46,150),(47,151),(48,152),(49,153),(50,154),(51,155),(52,156),(53,157),(54,158),(55,159),(56,160),(57,141),(58,142),(59,143),(60,144),(61,166),(62,167),(63,168),(64,169),(65,170),(66,171),(67,172),(68,173),(69,174),(70,175),(71,176),(72,177),(73,178),(74,179),(75,180),(76,161),(77,162),(78,163),(79,164),(80,165),(81,218),(82,219),(83,220),(84,201),(85,202),(86,203),(87,204),(88,205),(89,206),(90,207),(91,208),(92,209),(93,210),(94,211),(95,212),(96,213),(97,214),(98,215),(99,216),(100,217),(121,222),(122,223),(123,224),(124,225),(125,226),(126,227),(127,228),(128,229),(129,230),(130,231),(131,232),(132,233),(133,234),(134,235),(135,236),(136,237),(137,238),(138,239),(139,240),(140,221)], [(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)], [(1,10,11,20),(2,19,12,9),(3,8,13,18),(4,17,14,7),(5,6,15,16),(21,36,31,26),(22,25,32,35),(23,34,33,24),(27,30,37,40),(28,39,38,29),(41,54,51,44),(42,43,52,53),(45,50,55,60),(46,59,56,49),(47,48,57,58),(61,72,71,62),(63,70,73,80),(64,79,74,69),(65,68,75,78),(66,77,76,67),(81,84,91,94),(82,93,92,83),(85,100,95,90),(86,89,96,99),(87,98,97,88),(101,104,111,114),(102,113,112,103),(105,120,115,110),(106,109,116,119),(107,118,117,108),(121,122,131,132),(123,140,133,130),(124,129,134,139),(125,138,135,128),(126,127,136,137),(141,152,151,142),(143,150,153,160),(144,159,154,149),(145,148,155,158),(146,157,156,147),(161,162,171,172),(163,180,173,170),(164,169,174,179),(165,178,175,168),(166,167,176,177),(181,182,191,192),(183,200,193,190),(184,189,194,199),(185,198,195,188),(186,187,196,197),(201,218,211,208),(202,207,212,217),(203,216,213,206),(204,205,214,215),(209,210,219,220),(221,224,231,234),(222,233,232,223),(225,240,235,230),(226,229,236,239),(227,238,237,228)])
Matrix representation ►G ⊆ GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 48 | 46 |
| 0 | 0 | 0 | 0 | 14 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 36 | 39 |
| 0 | 0 | 0 | 45 | 25 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 44 | 0 | 0 |
| 0 | 17 | 17 | 0 | 0 |
| 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 50 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 44 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 50 | 42 |
| 0 | 0 | 0 | 0 | 11 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,48,0,0,0,0,46,14],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,36,45,0,0,0,39,25],[60,0,0,0,0,0,1,17,0,0,0,44,17,0,0,0,0,0,50,0,0,0,0,0,50],[1,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,50,0,0,0,0,42,11] >;
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | C4○D12 | C4○D20 | S3×D5 | C2×S3×D5 | C2×S3×D5 | D6.D10 |
| kernel | C2×D6.D10 | D6.D10 | C2×C15⋊D4 | C2×C3⋊D20 | C2×C5⋊D12 | C2×C15⋊Q8 | D5×C2×C12 | S3×C2×C20 | C2×C4×D15 | C2×C4×D5 | S3×C2×C4 | C4×D5 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C10 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 1 | 1 | 4 | 8 | 2 | 2 | 2 | 8 | 16 | 2 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_6.D_{10} % in TeX
G:=Group("C2xD6.D10"); // GroupNames label
G:=SmallGroup(480,1083);
// by ID
G=gap.SmallGroup(480,1083);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^10=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=d^9>;
// generators/relations
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