direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×S3×Dic10, C30.6C24, C60.112C23, Dic30⋊29C22, Dic15.5C23, C10⋊2(S3×Q8), C30⋊2(C2×Q8), (S3×C10)⋊8Q8, C15⋊Q8⋊7C22, C15⋊2(C22×Q8), C6⋊1(C2×Dic10), C6.6(C23×D5), (C6×Dic10)⋊7C2, (C4×S3).42D10, (C2×C20).306D6, C10.6(S3×C23), C3⋊1(C22×Dic10), (C2×Dic30)⋊25C2, (C2×C12).163D10, D6.30(C22×D5), (S3×C20).48C22, (S3×C10).25C23, (C2×C30).225C23, (C2×C60).125C22, C20.161(C22×S3), (C2×Dic5).135D6, (C22×S3).89D10, C12.124(C22×D5), Dic5.3(C22×S3), (S3×Dic5).9C22, (C3×Dic5).3C23, (C2×Dic3).168D10, (C3×Dic10)⋊20C22, (C5×Dic3).26C23, Dic3.24(C22×D5), (C6×Dic5).126C22, (C2×Dic15).150C22, (C10×Dic3).207C22, C5⋊2(C2×S3×Q8), (S3×C2×C4).6D5, (S3×C2×C20).6C2, (C2×C15⋊Q8)⋊20C2, (C5×S3)⋊1(C2×Q8), C4.110(C2×S3×D5), (C2×S3×Dic5).9C2, C22.95(C2×S3×D5), C2.10(C22×S3×D5), (C2×C4).116(S3×D5), (S3×C2×C10).98C22, (C2×C6).235(C22×D5), (C2×C10).236(C22×S3), SmallGroup(480,1078)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1276 in 312 conjugacy classes, 132 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×4], C3, C4 [×2], C4 [×10], C22, C22 [×6], C5, S3 [×4], C6, C6 [×2], C2×C4, C2×C4 [×17], Q8 [×16], C23, C10, C10 [×2], C10 [×4], Dic3 [×2], Dic3 [×4], C12 [×2], C12 [×4], D6 [×6], C2×C6, C15, C22×C4 [×3], C2×Q8 [×12], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×6], Dic6 [×12], C4×S3 [×4], C4×S3 [×8], C2×Dic3, C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3, C5×S3 [×4], C30, C30 [×2], C22×Q8, Dic10 [×4], Dic10 [×12], C2×Dic5 [×2], C2×Dic5 [×10], C2×C20, C2×C20 [×5], C22×C10, C2×Dic6 [×3], S3×C2×C4, S3×C2×C4 [×2], S3×Q8 [×8], C6×Q8, C5×Dic3 [×2], C3×Dic5 [×4], Dic15 [×4], C60 [×2], S3×C10 [×6], C2×C30, C2×Dic10, C2×Dic10 [×11], C22×Dic5 [×2], C22×C20, C2×S3×Q8, S3×Dic5 [×8], C15⋊Q8 [×8], C3×Dic10 [×4], C6×Dic5 [×2], S3×C20 [×4], C10×Dic3, Dic30 [×4], C2×Dic15 [×2], C2×C60, S3×C2×C10, C22×Dic10, S3×Dic10 [×8], C2×S3×Dic5 [×2], C2×C15⋊Q8 [×2], C6×Dic10, S3×C2×C20, C2×Dic30, C2×S3×Dic10
Quotients:
C1, C2 [×15], C22 [×35], S3, Q8 [×4], C23 [×15], D5, D6 [×7], C2×Q8 [×6], C24, D10 [×7], C22×S3 [×7], C22×Q8, Dic10 [×4], C22×D5 [×7], S3×Q8 [×2], S3×C23, S3×D5, C2×Dic10 [×6], C23×D5, C2×S3×Q8, C2×S3×D5 [×3], C22×Dic10, S3×Dic10 [×2], C22×S3×D5, C2×S3×Dic10
Generators and relations
G = < a,b,c,d,e | a2=b3=c2=d20=1, e2=d10, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
►On 240 points
Generators in S240
(1 211)(2 212)(3 213)(4 214)(5 215)(6 216)(7 217)(8 218)(9 219)(10 220)(11 201)(12 202)(13 203)(14 204)(15 205)(16 206)(17 207)(18 208)(19 209)(20 210)(21 73)(22 74)(23 75)(24 76)(25 77)(26 78)(27 79)(28 80)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 151)(42 152)(43 153)(44 154)(45 155)(46 156)(47 157)(48 158)(49 159)(50 160)(51 141)(52 142)(53 143)(54 144)(55 145)(56 146)(57 147)(58 148)(59 149)(60 150)(81 195)(82 196)(83 197)(84 198)(85 199)(86 200)(87 181)(88 182)(89 183)(90 184)(91 185)(92 186)(93 187)(94 188)(95 189)(96 190)(97 191)(98 192)(99 193)(100 194)(101 176)(102 177)(103 178)(104 179)(105 180)(106 161)(107 162)(108 163)(109 164)(110 165)(111 166)(112 167)(113 168)(114 169)(115 170)(116 171)(117 172)(118 173)(119 174)(120 175)(121 237)(122 238)(123 239)(124 240)(125 221)(126 222)(127 223)(128 224)(129 225)(130 226)(131 227)(132 228)(133 229)(134 230)(135 231)(136 232)(137 233)(138 234)(139 235)(140 236)
(1 36 161)(2 37 162)(3 38 163)(4 39 164)(5 40 165)(6 21 166)(7 22 167)(8 23 168)(9 24 169)(10 25 170)(11 26 171)(12 27 172)(13 28 173)(14 29 174)(15 30 175)(16 31 176)(17 32 177)(18 33 178)(19 34 179)(20 35 180)(41 232 87)(42 233 88)(43 234 89)(44 235 90)(45 236 91)(46 237 92)(47 238 93)(48 239 94)(49 240 95)(50 221 96)(51 222 97)(52 223 98)(53 224 99)(54 225 100)(55 226 81)(56 227 82)(57 228 83)(58 229 84)(59 230 85)(60 231 86)(61 119 204)(62 120 205)(63 101 206)(64 102 207)(65 103 208)(66 104 209)(67 105 210)(68 106 211)(69 107 212)(70 108 213)(71 109 214)(72 110 215)(73 111 216)(74 112 217)(75 113 218)(76 114 219)(77 115 220)(78 116 201)(79 117 202)(80 118 203)(121 186 156)(122 187 157)(123 188 158)(124 189 159)(125 190 160)(126 191 141)(127 192 142)(128 193 143)(129 194 144)(130 195 145)(131 196 146)(132 197 147)(133 198 148)(134 199 149)(135 200 150)(136 181 151)(137 182 152)(138 183 153)(139 184 154)(140 185 155)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 176)(22 177)(23 178)(24 179)(25 180)(26 161)(27 162)(28 163)(29 164)(30 165)(31 166)(32 167)(33 168)(34 169)(35 170)(36 171)(37 172)(38 173)(39 174)(40 175)(41 222)(42 223)(43 224)(44 225)(45 226)(46 227)(47 228)(48 229)(49 230)(50 231)(51 232)(52 233)(53 234)(54 235)(55 236)(56 237)(57 238)(58 239)(59 240)(60 221)(61 109)(62 110)(63 111)(64 112)(65 113)(66 114)(67 115)(68 116)(69 117)(70 118)(71 119)(72 120)(73 101)(74 102)(75 103)(76 104)(77 105)(78 106)(79 107)(80 108)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(121 146)(122 147)(123 148)(124 149)(125 150)(126 151)(127 152)(128 153)(129 154)(130 155)(131 156)(132 157)(133 158)(134 159)(135 160)(136 141)(137 142)(138 143)(139 144)(140 145)(181 191)(182 192)(183 193)(184 194)(185 195)(186 196)(187 197)(188 198)(189 199)(190 200)(201 211)(202 212)(203 213)(204 214)(205 215)(206 216)(207 217)(208 218)(209 219)(210 220)
(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 94 11 84)(2 93 12 83)(3 92 13 82)(4 91 14 81)(5 90 15 100)(6 89 16 99)(7 88 17 98)(8 87 18 97)(9 86 19 96)(10 85 20 95)(21 43 31 53)(22 42 32 52)(23 41 33 51)(24 60 34 50)(25 59 35 49)(26 58 36 48)(27 57 37 47)(28 56 38 46)(29 55 39 45)(30 54 40 44)(61 145 71 155)(62 144 72 154)(63 143 73 153)(64 142 74 152)(65 141 75 151)(66 160 76 150)(67 159 77 149)(68 158 78 148)(69 157 79 147)(70 156 80 146)(101 128 111 138)(102 127 112 137)(103 126 113 136)(104 125 114 135)(105 124 115 134)(106 123 116 133)(107 122 117 132)(108 121 118 131)(109 140 119 130)(110 139 120 129)(161 239 171 229)(162 238 172 228)(163 237 173 227)(164 236 174 226)(165 235 175 225)(166 234 176 224)(167 233 177 223)(168 232 178 222)(169 231 179 221)(170 230 180 240)(181 208 191 218)(182 207 192 217)(183 206 193 216)(184 205 194 215)(185 204 195 214)(186 203 196 213)(187 202 197 212)(188 201 198 211)(189 220 199 210)(190 219 200 209)
G:=sub<Sym(240)| (1,211)(2,212)(3,213)(4,214)(5,215)(6,216)(7,217)(8,218)(9,219)(10,220)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,159)(50,160)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,181)(88,182)(89,183)(90,184)(91,185)(92,186)(93,187)(94,188)(95,189)(96,190)(97,191)(98,192)(99,193)(100,194)(101,176)(102,177)(103,178)(104,179)(105,180)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,237)(122,238)(123,239)(124,240)(125,221)(126,222)(127,223)(128,224)(129,225)(130,226)(131,227)(132,228)(133,229)(134,230)(135,231)(136,232)(137,233)(138,234)(139,235)(140,236), (1,36,161)(2,37,162)(3,38,163)(4,39,164)(5,40,165)(6,21,166)(7,22,167)(8,23,168)(9,24,169)(10,25,170)(11,26,171)(12,27,172)(13,28,173)(14,29,174)(15,30,175)(16,31,176)(17,32,177)(18,33,178)(19,34,179)(20,35,180)(41,232,87)(42,233,88)(43,234,89)(44,235,90)(45,236,91)(46,237,92)(47,238,93)(48,239,94)(49,240,95)(50,221,96)(51,222,97)(52,223,98)(53,224,99)(54,225,100)(55,226,81)(56,227,82)(57,228,83)(58,229,84)(59,230,85)(60,231,86)(61,119,204)(62,120,205)(63,101,206)(64,102,207)(65,103,208)(66,104,209)(67,105,210)(68,106,211)(69,107,212)(70,108,213)(71,109,214)(72,110,215)(73,111,216)(74,112,217)(75,113,218)(76,114,219)(77,115,220)(78,116,201)(79,117,202)(80,118,203)(121,186,156)(122,187,157)(123,188,158)(124,189,159)(125,190,160)(126,191,141)(127,192,142)(128,193,143)(129,194,144)(130,195,145)(131,196,146)(132,197,147)(133,198,148)(134,199,149)(135,200,150)(136,181,151)(137,182,152)(138,183,153)(139,184,154)(140,185,155), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,176)(22,177)(23,178)(24,179)(25,180)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,169)(35,170)(36,171)(37,172)(38,173)(39,174)(40,175)(41,222)(42,223)(43,224)(44,225)(45,226)(46,227)(47,228)(48,229)(49,230)(50,231)(51,232)(52,233)(53,234)(54,235)(55,236)(56,237)(57,238)(58,239)(59,240)(60,221)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,152)(128,153)(129,154)(130,155)(131,156)(132,157)(133,158)(134,159)(135,160)(136,141)(137,142)(138,143)(139,144)(140,145)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220), (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), (1,94,11,84)(2,93,12,83)(3,92,13,82)(4,91,14,81)(5,90,15,100)(6,89,16,99)(7,88,17,98)(8,87,18,97)(9,86,19,96)(10,85,20,95)(21,43,31,53)(22,42,32,52)(23,41,33,51)(24,60,34,50)(25,59,35,49)(26,58,36,48)(27,57,37,47)(28,56,38,46)(29,55,39,45)(30,54,40,44)(61,145,71,155)(62,144,72,154)(63,143,73,153)(64,142,74,152)(65,141,75,151)(66,160,76,150)(67,159,77,149)(68,158,78,148)(69,157,79,147)(70,156,80,146)(101,128,111,138)(102,127,112,137)(103,126,113,136)(104,125,114,135)(105,124,115,134)(106,123,116,133)(107,122,117,132)(108,121,118,131)(109,140,119,130)(110,139,120,129)(161,239,171,229)(162,238,172,228)(163,237,173,227)(164,236,174,226)(165,235,175,225)(166,234,176,224)(167,233,177,223)(168,232,178,222)(169,231,179,221)(170,230,180,240)(181,208,191,218)(182,207,192,217)(183,206,193,216)(184,205,194,215)(185,204,195,214)(186,203,196,213)(187,202,197,212)(188,201,198,211)(189,220,199,210)(190,219,200,209)>;
G:=Group( (1,211)(2,212)(3,213)(4,214)(5,215)(6,216)(7,217)(8,218)(9,219)(10,220)(11,201)(12,202)(13,203)(14,204)(15,205)(16,206)(17,207)(18,208)(19,209)(20,210)(21,73)(22,74)(23,75)(24,76)(25,77)(26,78)(27,79)(28,80)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,151)(42,152)(43,153)(44,154)(45,155)(46,156)(47,157)(48,158)(49,159)(50,160)(51,141)(52,142)(53,143)(54,144)(55,145)(56,146)(57,147)(58,148)(59,149)(60,150)(81,195)(82,196)(83,197)(84,198)(85,199)(86,200)(87,181)(88,182)(89,183)(90,184)(91,185)(92,186)(93,187)(94,188)(95,189)(96,190)(97,191)(98,192)(99,193)(100,194)(101,176)(102,177)(103,178)(104,179)(105,180)(106,161)(107,162)(108,163)(109,164)(110,165)(111,166)(112,167)(113,168)(114,169)(115,170)(116,171)(117,172)(118,173)(119,174)(120,175)(121,237)(122,238)(123,239)(124,240)(125,221)(126,222)(127,223)(128,224)(129,225)(130,226)(131,227)(132,228)(133,229)(134,230)(135,231)(136,232)(137,233)(138,234)(139,235)(140,236), (1,36,161)(2,37,162)(3,38,163)(4,39,164)(5,40,165)(6,21,166)(7,22,167)(8,23,168)(9,24,169)(10,25,170)(11,26,171)(12,27,172)(13,28,173)(14,29,174)(15,30,175)(16,31,176)(17,32,177)(18,33,178)(19,34,179)(20,35,180)(41,232,87)(42,233,88)(43,234,89)(44,235,90)(45,236,91)(46,237,92)(47,238,93)(48,239,94)(49,240,95)(50,221,96)(51,222,97)(52,223,98)(53,224,99)(54,225,100)(55,226,81)(56,227,82)(57,228,83)(58,229,84)(59,230,85)(60,231,86)(61,119,204)(62,120,205)(63,101,206)(64,102,207)(65,103,208)(66,104,209)(67,105,210)(68,106,211)(69,107,212)(70,108,213)(71,109,214)(72,110,215)(73,111,216)(74,112,217)(75,113,218)(76,114,219)(77,115,220)(78,116,201)(79,117,202)(80,118,203)(121,186,156)(122,187,157)(123,188,158)(124,189,159)(125,190,160)(126,191,141)(127,192,142)(128,193,143)(129,194,144)(130,195,145)(131,196,146)(132,197,147)(133,198,148)(134,199,149)(135,200,150)(136,181,151)(137,182,152)(138,183,153)(139,184,154)(140,185,155), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,176)(22,177)(23,178)(24,179)(25,180)(26,161)(27,162)(28,163)(29,164)(30,165)(31,166)(32,167)(33,168)(34,169)(35,170)(36,171)(37,172)(38,173)(39,174)(40,175)(41,222)(42,223)(43,224)(44,225)(45,226)(46,227)(47,228)(48,229)(49,230)(50,231)(51,232)(52,233)(53,234)(54,235)(55,236)(56,237)(57,238)(58,239)(59,240)(60,221)(61,109)(62,110)(63,111)(64,112)(65,113)(66,114)(67,115)(68,116)(69,117)(70,118)(71,119)(72,120)(73,101)(74,102)(75,103)(76,104)(77,105)(78,106)(79,107)(80,108)(81,91)(82,92)(83,93)(84,94)(85,95)(86,96)(87,97)(88,98)(89,99)(90,100)(121,146)(122,147)(123,148)(124,149)(125,150)(126,151)(127,152)(128,153)(129,154)(130,155)(131,156)(132,157)(133,158)(134,159)(135,160)(136,141)(137,142)(138,143)(139,144)(140,145)(181,191)(182,192)(183,193)(184,194)(185,195)(186,196)(187,197)(188,198)(189,199)(190,200)(201,211)(202,212)(203,213)(204,214)(205,215)(206,216)(207,217)(208,218)(209,219)(210,220), (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), (1,94,11,84)(2,93,12,83)(3,92,13,82)(4,91,14,81)(5,90,15,100)(6,89,16,99)(7,88,17,98)(8,87,18,97)(9,86,19,96)(10,85,20,95)(21,43,31,53)(22,42,32,52)(23,41,33,51)(24,60,34,50)(25,59,35,49)(26,58,36,48)(27,57,37,47)(28,56,38,46)(29,55,39,45)(30,54,40,44)(61,145,71,155)(62,144,72,154)(63,143,73,153)(64,142,74,152)(65,141,75,151)(66,160,76,150)(67,159,77,149)(68,158,78,148)(69,157,79,147)(70,156,80,146)(101,128,111,138)(102,127,112,137)(103,126,113,136)(104,125,114,135)(105,124,115,134)(106,123,116,133)(107,122,117,132)(108,121,118,131)(109,140,119,130)(110,139,120,129)(161,239,171,229)(162,238,172,228)(163,237,173,227)(164,236,174,226)(165,235,175,225)(166,234,176,224)(167,233,177,223)(168,232,178,222)(169,231,179,221)(170,230,180,240)(181,208,191,218)(182,207,192,217)(183,206,193,216)(184,205,194,215)(185,204,195,214)(186,203,196,213)(187,202,197,212)(188,201,198,211)(189,220,199,210)(190,219,200,209) );
G=PermutationGroup([(1,211),(2,212),(3,213),(4,214),(5,215),(6,216),(7,217),(8,218),(9,219),(10,220),(11,201),(12,202),(13,203),(14,204),(15,205),(16,206),(17,207),(18,208),(19,209),(20,210),(21,73),(22,74),(23,75),(24,76),(25,77),(26,78),(27,79),(28,80),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,151),(42,152),(43,153),(44,154),(45,155),(46,156),(47,157),(48,158),(49,159),(50,160),(51,141),(52,142),(53,143),(54,144),(55,145),(56,146),(57,147),(58,148),(59,149),(60,150),(81,195),(82,196),(83,197),(84,198),(85,199),(86,200),(87,181),(88,182),(89,183),(90,184),(91,185),(92,186),(93,187),(94,188),(95,189),(96,190),(97,191),(98,192),(99,193),(100,194),(101,176),(102,177),(103,178),(104,179),(105,180),(106,161),(107,162),(108,163),(109,164),(110,165),(111,166),(112,167),(113,168),(114,169),(115,170),(116,171),(117,172),(118,173),(119,174),(120,175),(121,237),(122,238),(123,239),(124,240),(125,221),(126,222),(127,223),(128,224),(129,225),(130,226),(131,227),(132,228),(133,229),(134,230),(135,231),(136,232),(137,233),(138,234),(139,235),(140,236)], [(1,36,161),(2,37,162),(3,38,163),(4,39,164),(5,40,165),(6,21,166),(7,22,167),(8,23,168),(9,24,169),(10,25,170),(11,26,171),(12,27,172),(13,28,173),(14,29,174),(15,30,175),(16,31,176),(17,32,177),(18,33,178),(19,34,179),(20,35,180),(41,232,87),(42,233,88),(43,234,89),(44,235,90),(45,236,91),(46,237,92),(47,238,93),(48,239,94),(49,240,95),(50,221,96),(51,222,97),(52,223,98),(53,224,99),(54,225,100),(55,226,81),(56,227,82),(57,228,83),(58,229,84),(59,230,85),(60,231,86),(61,119,204),(62,120,205),(63,101,206),(64,102,207),(65,103,208),(66,104,209),(67,105,210),(68,106,211),(69,107,212),(70,108,213),(71,109,214),(72,110,215),(73,111,216),(74,112,217),(75,113,218),(76,114,219),(77,115,220),(78,116,201),(79,117,202),(80,118,203),(121,186,156),(122,187,157),(123,188,158),(124,189,159),(125,190,160),(126,191,141),(127,192,142),(128,193,143),(129,194,144),(130,195,145),(131,196,146),(132,197,147),(133,198,148),(134,199,149),(135,200,150),(136,181,151),(137,182,152),(138,183,153),(139,184,154),(140,185,155)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,176),(22,177),(23,178),(24,179),(25,180),(26,161),(27,162),(28,163),(29,164),(30,165),(31,166),(32,167),(33,168),(34,169),(35,170),(36,171),(37,172),(38,173),(39,174),(40,175),(41,222),(42,223),(43,224),(44,225),(45,226),(46,227),(47,228),(48,229),(49,230),(50,231),(51,232),(52,233),(53,234),(54,235),(55,236),(56,237),(57,238),(58,239),(59,240),(60,221),(61,109),(62,110),(63,111),(64,112),(65,113),(66,114),(67,115),(68,116),(69,117),(70,118),(71,119),(72,120),(73,101),(74,102),(75,103),(76,104),(77,105),(78,106),(79,107),(80,108),(81,91),(82,92),(83,93),(84,94),(85,95),(86,96),(87,97),(88,98),(89,99),(90,100),(121,146),(122,147),(123,148),(124,149),(125,150),(126,151),(127,152),(128,153),(129,154),(130,155),(131,156),(132,157),(133,158),(134,159),(135,160),(136,141),(137,142),(138,143),(139,144),(140,145),(181,191),(182,192),(183,193),(184,194),(185,195),(186,196),(187,197),(188,198),(189,199),(190,200),(201,211),(202,212),(203,213),(204,214),(205,215),(206,216),(207,217),(208,218),(209,219),(210,220)], [(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)], [(1,94,11,84),(2,93,12,83),(3,92,13,82),(4,91,14,81),(5,90,15,100),(6,89,16,99),(7,88,17,98),(8,87,18,97),(9,86,19,96),(10,85,20,95),(21,43,31,53),(22,42,32,52),(23,41,33,51),(24,60,34,50),(25,59,35,49),(26,58,36,48),(27,57,37,47),(28,56,38,46),(29,55,39,45),(30,54,40,44),(61,145,71,155),(62,144,72,154),(63,143,73,153),(64,142,74,152),(65,141,75,151),(66,160,76,150),(67,159,77,149),(68,158,78,148),(69,157,79,147),(70,156,80,146),(101,128,111,138),(102,127,112,137),(103,126,113,136),(104,125,114,135),(105,124,115,134),(106,123,116,133),(107,122,117,132),(108,121,118,131),(109,140,119,130),(110,139,120,129),(161,239,171,229),(162,238,172,228),(163,237,173,227),(164,236,174,226),(165,235,175,225),(166,234,176,224),(167,233,177,223),(168,232,178,222),(169,231,179,221),(170,230,180,240),(181,208,191,218),(182,207,192,217),(183,206,193,216),(184,205,194,215),(185,204,195,214),(186,203,196,213),(187,202,197,212),(188,201,198,211),(189,220,199,210),(190,219,200,209)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 50 | 0 | 0 | 0 | 0 | 0 |
| 11 | 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 |
| 0 | 0 | 45 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 59 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 38 | 0 | 0 |
| 0 | 0 | 42 | 28 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,60,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[50,11,0,0,0,0,0,11,0,0,0,0,0,0,17,45,0,0,0,0,60,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,1,0,0,0,0,59,1,0,0,0,0,0,0,33,42,0,0,0,0,38,28,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20P | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 10 | 10 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | + | + | - | - | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | Q8 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | D10 | Dic10 | S3×Q8 | S3×D5 | C2×S3×D5 | C2×S3×D5 | S3×Dic10 |
| kernel | C2×S3×Dic10 | S3×Dic10 | C2×S3×Dic5 | C2×C15⋊Q8 | C6×Dic10 | S3×C2×C20 | C2×Dic30 | C2×Dic10 | S3×C10 | S3×C2×C4 | Dic10 | C2×Dic5 | C2×C20 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 4 | 2 | 4 | 2 | 1 | 8 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times S_3\times Dic_{10} % in TeX
G:=Group("C2xS3xDic10"); // GroupNames label
G:=SmallGroup(480,1078);
// by ID
G=gap.SmallGroup(480,1078);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^3=c^2=d^20=1,e^2=d^10,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,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations
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