direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊5S3, D20⋊27D6, C30.2C24, C60.110C23, Dic30⋊28C22, Dic15.2C23, (C6×D20)⋊7C2, (C4×S3)⋊14D10, (C2×D20)⋊15S3, C30⋊1(C4○D4), C6⋊3(C4○D20), C6.2(C23×D5), (C2×C20).305D6, C15⋊D4⋊8C22, C10.2(S3×C23), C10⋊2(D4⋊2S3), (C6×D5).1C23, (S3×C20)⋊16C22, (C2×Dic30)⋊24C2, (C2×C12).159D10, (C3×D20)⋊22C22, (D5×Dic3)⋊5C22, D6.22(C22×D5), (C22×D5).67D6, D10.1(C22×S3), (S3×C10).24C23, (C2×C30).221C23, (C2×C60).124C22, C20.160(C22×S3), (C22×S3).80D10, C12.121(C22×D5), (C2×Dic3).189D10, (C5×Dic3).25C23, Dic3.32(C22×D5), (C10×Dic3).206C22, (C2×Dic15).148C22, (S3×C2×C4)⋊3D5, (S3×C2×C20)⋊4C2, C15⋊1(C2×C4○D4), C3⋊3(C2×C4○D20), C5⋊2(C2×D4⋊2S3), C4.109(C2×S3×D5), C2.6(C22×S3×D5), (C2×D5×Dic3)⋊19C2, (C2×C15⋊D4)⋊17C2, C22.93(C2×S3×D5), (C2×C4).115(S3×D5), (D5×C2×C6).58C22, (S3×C2×C10).97C22, (C2×C6).233(C22×D5), (C2×C10).233(C22×S3), SmallGroup(480,1074)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1468 in 328 conjugacy classes, 116 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×6], C3, C4 [×2], C4 [×6], C22, C22 [×12], C5, S3 [×2], C6, C6 [×2], C6 [×4], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], D5 [×4], C10, C10 [×2], C10 [×2], Dic3 [×2], Dic3 [×4], C12 [×2], D6 [×2], D6 [×2], C2×C6, C2×C6 [×8], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×4], C20 [×2], C20 [×2], D10 [×4], D10 [×4], C2×C10, C2×C10 [×4], Dic6 [×4], C4×S3 [×4], C2×Dic3, C2×Dic3 [×10], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C5×S3 [×2], C3×D5 [×4], C30, C30 [×2], C2×C4○D4, Dic10 [×4], C4×D5 [×8], D20 [×4], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20, C2×C20 [×5], C22×D5 [×2], C22×C10, C2×Dic6, S3×C2×C4, D4⋊2S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×2], Dic15 [×4], C60 [×2], C6×D5 [×4], C6×D5 [×4], S3×C10 [×2], S3×C10 [×2], C2×C30, C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×8], C2×C5⋊D4 [×2], C22×C20, C2×D4⋊2S3, D5×Dic3 [×8], C15⋊D4 [×8], C3×D20 [×4], S3×C20 [×4], C10×Dic3, Dic30 [×4], C2×Dic15 [×2], C2×C60, D5×C2×C6 [×2], S3×C2×C10, C2×C4○D20, D20⋊5S3 [×8], C2×D5×Dic3 [×2], C2×C15⋊D4 [×2], C6×D20, S3×C2×C20, C2×Dic30, C2×D20⋊5S3
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], D4⋊2S3 [×2], S3×C23, S3×D5, C4○D20 [×2], C23×D5, C2×D4⋊2S3, C2×S3×D5 [×3], C2×C4○D20, D20⋊5S3 [×2], C22×S3×D5, C2×D20⋊5S3
Generators and relations
G = < a,b,c,d,e | a2=b20=c2=d3=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b10c, ede=d-1 >
►On 240 points
Generators in S240
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 21)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(41 115)(42 116)(43 117)(44 118)(45 119)(46 120)(47 101)(48 102)(49 103)(50 104)(51 105)(52 106)(53 107)(54 108)(55 109)(56 110)(57 111)(58 112)(59 113)(60 114)(61 200)(62 181)(63 182)(64 183)(65 184)(66 185)(67 186)(68 187)(69 188)(70 189)(71 190)(72 191)(73 192)(74 193)(75 194)(76 195)(77 196)(78 197)(79 198)(80 199)(81 235)(82 236)(83 237)(84 238)(85 239)(86 240)(87 221)(88 222)(89 223)(90 224)(91 225)(92 226)(93 227)(94 228)(95 229)(96 230)(97 231)(98 232)(99 233)(100 234)(121 217)(122 218)(123 219)(124 220)(125 201)(126 202)(127 203)(128 204)(129 205)(130 206)(131 207)(132 208)(133 209)(134 210)(135 211)(136 212)(137 213)(138 214)(139 215)(140 216)(141 168)(142 169)(143 170)(144 171)(145 172)(146 173)(147 174)(148 175)(149 176)(150 177)(151 178)(152 179)(153 180)(154 161)(155 162)(156 163)(157 164)(158 165)(159 166)(160 167)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 46)(42 45)(43 44)(47 60)(48 59)(49 58)(50 57)(51 56)(52 55)(53 54)(61 66)(62 65)(63 64)(67 80)(68 79)(69 78)(70 77)(71 76)(72 75)(73 74)(81 96)(82 95)(83 94)(84 93)(85 92)(86 91)(87 90)(88 89)(97 100)(98 99)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)(115 120)(116 119)(117 118)(121 132)(122 131)(123 130)(124 129)(125 128)(126 127)(133 140)(134 139)(135 138)(136 137)(141 148)(142 147)(143 146)(144 145)(149 160)(150 159)(151 158)(152 157)(153 156)(154 155)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 184)(182 183)(185 200)(186 199)(187 198)(188 197)(189 196)(190 195)(191 194)(192 193)(201 204)(202 203)(205 220)(206 219)(207 218)(208 217)(209 216)(210 215)(211 214)(212 213)(221 224)(222 223)(225 240)(226 239)(227 238)(228 237)(229 236)(230 235)(231 234)(232 233)
(1 203 172)(2 204 173)(3 205 174)(4 206 175)(5 207 176)(6 208 177)(7 209 178)(8 210 179)(9 211 180)(10 212 161)(11 213 162)(12 214 163)(13 215 164)(14 216 165)(15 217 166)(16 218 167)(17 219 168)(18 220 169)(19 201 170)(20 202 171)(21 140 158)(22 121 159)(23 122 160)(24 123 141)(25 124 142)(26 125 143)(27 126 144)(28 127 145)(29 128 146)(30 129 147)(31 130 148)(32 131 149)(33 132 150)(34 133 151)(35 134 152)(36 135 153)(37 136 154)(38 137 155)(39 138 156)(40 139 157)(41 61 240)(42 62 221)(43 63 222)(44 64 223)(45 65 224)(46 66 225)(47 67 226)(48 68 227)(49 69 228)(50 70 229)(51 71 230)(52 72 231)(53 73 232)(54 74 233)(55 75 234)(56 76 235)(57 77 236)(58 78 237)(59 79 238)(60 80 239)(81 110 195)(82 111 196)(83 112 197)(84 113 198)(85 114 199)(86 115 200)(87 116 181)(88 117 182)(89 118 183)(90 119 184)(91 120 185)(92 101 186)(93 102 187)(94 103 188)(95 104 189)(96 105 190)(97 106 191)(98 107 192)(99 108 193)(100 109 194)
(1 188)(2 189)(3 190)(4 191)(5 192)(6 193)(7 194)(8 195)(9 196)(10 197)(11 198)(12 199)(13 200)(14 181)(15 182)(16 183)(17 184)(18 185)(19 186)(20 187)(21 62)(22 63)(23 64)(24 65)(25 66)(26 67)(27 68)(28 69)(29 70)(30 71)(31 72)(32 73)(33 74)(34 75)(35 76)(36 77)(37 78)(38 79)(39 80)(40 61)(41 139)(42 140)(43 121)(44 122)(45 123)(46 124)(47 125)(48 126)(49 127)(50 128)(51 129)(52 130)(53 131)(54 132)(55 133)(56 134)(57 135)(58 136)(59 137)(60 138)(81 179)(82 180)(83 161)(84 162)(85 163)(86 164)(87 165)(88 166)(89 167)(90 168)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(97 175)(98 176)(99 177)(100 178)(101 201)(102 202)(103 203)(104 204)(105 205)(106 206)(107 207)(108 208)(109 209)(110 210)(111 211)(112 212)(113 213)(114 214)(115 215)(116 216)(117 217)(118 218)(119 219)(120 220)(141 224)(142 225)(143 226)(144 227)(145 228)(146 229)(147 230)(148 231)(149 232)(150 233)(151 234)(152 235)(153 236)(154 237)(155 238)(156 239)(157 240)(158 221)(159 222)(160 223)
G:=sub<Sym(240)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,200)(62,181)(63,182)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,197)(79,198)(80,199)(81,235)(82,236)(83,237)(84,238)(85,239)(86,240)(87,221)(88,222)(89,223)(90,224)(91,225)(92,226)(93,227)(94,228)(95,229)(96,230)(97,231)(98,232)(99,233)(100,234)(121,217)(122,218)(123,219)(124,220)(125,201)(126,202)(127,203)(128,204)(129,205)(130,206)(131,207)(132,208)(133,209)(134,210)(135,211)(136,212)(137,213)(138,214)(139,215)(140,216)(141,168)(142,169)(143,170)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,161)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,100)(98,99)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,184)(182,183)(185,200)(186,199)(187,198)(188,197)(189,196)(190,195)(191,194)(192,193)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,224)(222,223)(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)(232,233), (1,203,172)(2,204,173)(3,205,174)(4,206,175)(5,207,176)(6,208,177)(7,209,178)(8,210,179)(9,211,180)(10,212,161)(11,213,162)(12,214,163)(13,215,164)(14,216,165)(15,217,166)(16,218,167)(17,219,168)(18,220,169)(19,201,170)(20,202,171)(21,140,158)(22,121,159)(23,122,160)(24,123,141)(25,124,142)(26,125,143)(27,126,144)(28,127,145)(29,128,146)(30,129,147)(31,130,148)(32,131,149)(33,132,150)(34,133,151)(35,134,152)(36,135,153)(37,136,154)(38,137,155)(39,138,156)(40,139,157)(41,61,240)(42,62,221)(43,63,222)(44,64,223)(45,65,224)(46,66,225)(47,67,226)(48,68,227)(49,69,228)(50,70,229)(51,71,230)(52,72,231)(53,73,232)(54,74,233)(55,75,234)(56,76,235)(57,77,236)(58,78,237)(59,79,238)(60,80,239)(81,110,195)(82,111,196)(83,112,197)(84,113,198)(85,114,199)(86,115,200)(87,116,181)(88,117,182)(89,118,183)(90,119,184)(91,120,185)(92,101,186)(93,102,187)(94,103,188)(95,104,189)(96,105,190)(97,106,191)(98,107,192)(99,108,193)(100,109,194), (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,197)(11,198)(12,199)(13,200)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,61)(41,139)(42,140)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(81,179)(82,180)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,201)(102,202)(103,203)(104,204)(105,205)(106,206)(107,207)(108,208)(109,209)(110,210)(111,211)(112,212)(113,213)(114,214)(115,215)(116,216)(117,217)(118,218)(119,219)(120,220)(141,224)(142,225)(143,226)(144,227)(145,228)(146,229)(147,230)(148,231)(149,232)(150,233)(151,234)(152,235)(153,236)(154,237)(155,238)(156,239)(157,240)(158,221)(159,222)(160,223)>;
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,21)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(41,115)(42,116)(43,117)(44,118)(45,119)(46,120)(47,101)(48,102)(49,103)(50,104)(51,105)(52,106)(53,107)(54,108)(55,109)(56,110)(57,111)(58,112)(59,113)(60,114)(61,200)(62,181)(63,182)(64,183)(65,184)(66,185)(67,186)(68,187)(69,188)(70,189)(71,190)(72,191)(73,192)(74,193)(75,194)(76,195)(77,196)(78,197)(79,198)(80,199)(81,235)(82,236)(83,237)(84,238)(85,239)(86,240)(87,221)(88,222)(89,223)(90,224)(91,225)(92,226)(93,227)(94,228)(95,229)(96,230)(97,231)(98,232)(99,233)(100,234)(121,217)(122,218)(123,219)(124,220)(125,201)(126,202)(127,203)(128,204)(129,205)(130,206)(131,207)(132,208)(133,209)(134,210)(135,211)(136,212)(137,213)(138,214)(139,215)(140,216)(141,168)(142,169)(143,170)(144,171)(145,172)(146,173)(147,174)(148,175)(149,176)(150,177)(151,178)(152,179)(153,180)(154,161)(155,162)(156,163)(157,164)(158,165)(159,166)(160,167), (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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(35,40)(36,39)(37,38)(41,46)(42,45)(43,44)(47,60)(48,59)(49,58)(50,57)(51,56)(52,55)(53,54)(61,66)(62,65)(63,64)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)(97,100)(98,99)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)(115,120)(116,119)(117,118)(121,132)(122,131)(123,130)(124,129)(125,128)(126,127)(133,140)(134,139)(135,138)(136,137)(141,148)(142,147)(143,146)(144,145)(149,160)(150,159)(151,158)(152,157)(153,156)(154,155)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,184)(182,183)(185,200)(186,199)(187,198)(188,197)(189,196)(190,195)(191,194)(192,193)(201,204)(202,203)(205,220)(206,219)(207,218)(208,217)(209,216)(210,215)(211,214)(212,213)(221,224)(222,223)(225,240)(226,239)(227,238)(228,237)(229,236)(230,235)(231,234)(232,233), (1,203,172)(2,204,173)(3,205,174)(4,206,175)(5,207,176)(6,208,177)(7,209,178)(8,210,179)(9,211,180)(10,212,161)(11,213,162)(12,214,163)(13,215,164)(14,216,165)(15,217,166)(16,218,167)(17,219,168)(18,220,169)(19,201,170)(20,202,171)(21,140,158)(22,121,159)(23,122,160)(24,123,141)(25,124,142)(26,125,143)(27,126,144)(28,127,145)(29,128,146)(30,129,147)(31,130,148)(32,131,149)(33,132,150)(34,133,151)(35,134,152)(36,135,153)(37,136,154)(38,137,155)(39,138,156)(40,139,157)(41,61,240)(42,62,221)(43,63,222)(44,64,223)(45,65,224)(46,66,225)(47,67,226)(48,68,227)(49,69,228)(50,70,229)(51,71,230)(52,72,231)(53,73,232)(54,74,233)(55,75,234)(56,76,235)(57,77,236)(58,78,237)(59,79,238)(60,80,239)(81,110,195)(82,111,196)(83,112,197)(84,113,198)(85,114,199)(86,115,200)(87,116,181)(88,117,182)(89,118,183)(90,119,184)(91,120,185)(92,101,186)(93,102,187)(94,103,188)(95,104,189)(96,105,190)(97,106,191)(98,107,192)(99,108,193)(100,109,194), (1,188)(2,189)(3,190)(4,191)(5,192)(6,193)(7,194)(8,195)(9,196)(10,197)(11,198)(12,199)(13,200)(14,181)(15,182)(16,183)(17,184)(18,185)(19,186)(20,187)(21,62)(22,63)(23,64)(24,65)(25,66)(26,67)(27,68)(28,69)(29,70)(30,71)(31,72)(32,73)(33,74)(34,75)(35,76)(36,77)(37,78)(38,79)(39,80)(40,61)(41,139)(42,140)(43,121)(44,122)(45,123)(46,124)(47,125)(48,126)(49,127)(50,128)(51,129)(52,130)(53,131)(54,132)(55,133)(56,134)(57,135)(58,136)(59,137)(60,138)(81,179)(82,180)(83,161)(84,162)(85,163)(86,164)(87,165)(88,166)(89,167)(90,168)(91,169)(92,170)(93,171)(94,172)(95,173)(96,174)(97,175)(98,176)(99,177)(100,178)(101,201)(102,202)(103,203)(104,204)(105,205)(106,206)(107,207)(108,208)(109,209)(110,210)(111,211)(112,212)(113,213)(114,214)(115,215)(116,216)(117,217)(118,218)(119,219)(120,220)(141,224)(142,225)(143,226)(144,227)(145,228)(146,229)(147,230)(148,231)(149,232)(150,233)(151,234)(152,235)(153,236)(154,237)(155,238)(156,239)(157,240)(158,221)(159,222)(160,223) );
G=PermutationGroup([(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,21),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(41,115),(42,116),(43,117),(44,118),(45,119),(46,120),(47,101),(48,102),(49,103),(50,104),(51,105),(52,106),(53,107),(54,108),(55,109),(56,110),(57,111),(58,112),(59,113),(60,114),(61,200),(62,181),(63,182),(64,183),(65,184),(66,185),(67,186),(68,187),(69,188),(70,189),(71,190),(72,191),(73,192),(74,193),(75,194),(76,195),(77,196),(78,197),(79,198),(80,199),(81,235),(82,236),(83,237),(84,238),(85,239),(86,240),(87,221),(88,222),(89,223),(90,224),(91,225),(92,226),(93,227),(94,228),(95,229),(96,230),(97,231),(98,232),(99,233),(100,234),(121,217),(122,218),(123,219),(124,220),(125,201),(126,202),(127,203),(128,204),(129,205),(130,206),(131,207),(132,208),(133,209),(134,210),(135,211),(136,212),(137,213),(138,214),(139,215),(140,216),(141,168),(142,169),(143,170),(144,171),(145,172),(146,173),(147,174),(148,175),(149,176),(150,177),(151,178),(152,179),(153,180),(154,161),(155,162),(156,163),(157,164),(158,165),(159,166),(160,167)], [(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)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28),(35,40),(36,39),(37,38),(41,46),(42,45),(43,44),(47,60),(48,59),(49,58),(50,57),(51,56),(52,55),(53,54),(61,66),(62,65),(63,64),(67,80),(68,79),(69,78),(70,77),(71,76),(72,75),(73,74),(81,96),(82,95),(83,94),(84,93),(85,92),(86,91),(87,90),(88,89),(97,100),(98,99),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108),(115,120),(116,119),(117,118),(121,132),(122,131),(123,130),(124,129),(125,128),(126,127),(133,140),(134,139),(135,138),(136,137),(141,148),(142,147),(143,146),(144,145),(149,160),(150,159),(151,158),(152,157),(153,156),(154,155),(161,162),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172),(181,184),(182,183),(185,200),(186,199),(187,198),(188,197),(189,196),(190,195),(191,194),(192,193),(201,204),(202,203),(205,220),(206,219),(207,218),(208,217),(209,216),(210,215),(211,214),(212,213),(221,224),(222,223),(225,240),(226,239),(227,238),(228,237),(229,236),(230,235),(231,234),(232,233)], [(1,203,172),(2,204,173),(3,205,174),(4,206,175),(5,207,176),(6,208,177),(7,209,178),(8,210,179),(9,211,180),(10,212,161),(11,213,162),(12,214,163),(13,215,164),(14,216,165),(15,217,166),(16,218,167),(17,219,168),(18,220,169),(19,201,170),(20,202,171),(21,140,158),(22,121,159),(23,122,160),(24,123,141),(25,124,142),(26,125,143),(27,126,144),(28,127,145),(29,128,146),(30,129,147),(31,130,148),(32,131,149),(33,132,150),(34,133,151),(35,134,152),(36,135,153),(37,136,154),(38,137,155),(39,138,156),(40,139,157),(41,61,240),(42,62,221),(43,63,222),(44,64,223),(45,65,224),(46,66,225),(47,67,226),(48,68,227),(49,69,228),(50,70,229),(51,71,230),(52,72,231),(53,73,232),(54,74,233),(55,75,234),(56,76,235),(57,77,236),(58,78,237),(59,79,238),(60,80,239),(81,110,195),(82,111,196),(83,112,197),(84,113,198),(85,114,199),(86,115,200),(87,116,181),(88,117,182),(89,118,183),(90,119,184),(91,120,185),(92,101,186),(93,102,187),(94,103,188),(95,104,189),(96,105,190),(97,106,191),(98,107,192),(99,108,193),(100,109,194)], [(1,188),(2,189),(3,190),(4,191),(5,192),(6,193),(7,194),(8,195),(9,196),(10,197),(11,198),(12,199),(13,200),(14,181),(15,182),(16,183),(17,184),(18,185),(19,186),(20,187),(21,62),(22,63),(23,64),(24,65),(25,66),(26,67),(27,68),(28,69),(29,70),(30,71),(31,72),(32,73),(33,74),(34,75),(35,76),(36,77),(37,78),(38,79),(39,80),(40,61),(41,139),(42,140),(43,121),(44,122),(45,123),(46,124),(47,125),(48,126),(49,127),(50,128),(51,129),(52,130),(53,131),(54,132),(55,133),(56,134),(57,135),(58,136),(59,137),(60,138),(81,179),(82,180),(83,161),(84,162),(85,163),(86,164),(87,165),(88,166),(89,167),(90,168),(91,169),(92,170),(93,171),(94,172),(95,173),(96,174),(97,175),(98,176),(99,177),(100,178),(101,201),(102,202),(103,203),(104,204),(105,205),(106,206),(107,207),(108,208),(109,209),(110,210),(111,211),(112,212),(113,213),(114,214),(115,215),(116,216),(117,217),(118,218),(119,219),(120,220),(141,224),(142,225),(143,226),(144,227),(145,228),(146,229),(147,230),(148,231),(149,232),(150,233),(151,234),(152,235),(153,236),(154,237),(155,238),(156,239),(157,240),(158,221),(159,222),(160,223)])
Matrix representation ►G ⊆ GL6(𝔽61)
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 23 |
| 0 | 0 | 0 | 0 | 51 | 30 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 43 | 43 | 0 | 0 |
| 0 | 0 | 0 | 0 | 31 | 23 |
| 0 | 0 | 0 | 0 | 6 | 30 |
| 1 | 46 | 0 | 0 | 0 | 0 |
| 49 | 59 | 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 | 1 |
| 54 | 3 | 0 | 0 | 0 | 0 |
| 45 | 7 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 25 | 52 |
| 0 | 0 | 0 | 0 | 49 | 36 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,60,0,0,0,0,1,0,0,0,0,0,0,0,31,51,0,0,0,0,23,30],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,18,43,0,0,0,0,1,43,0,0,0,0,0,0,31,6,0,0,0,0,23,30],[1,49,0,0,0,0,46,59,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[54,45,0,0,0,0,3,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,25,49,0,0,0,0,52,36] >;
78 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 | 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 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | 4 | 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 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D10 | C4○D20 | D4⋊2S3 | S3×D5 | C2×S3×D5 | C2×S3×D5 | D20⋊5S3 |
| kernel | C2×D20⋊5S3 | D20⋊5S3 | C2×D5×Dic3 | C2×C15⋊D4 | C6×D20 | S3×C2×C20 | C2×Dic30 | C2×D20 | S3×C2×C4 | D20 | C2×C20 | C22×D5 | C30 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | C6 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 2 | 4 | 8 | 2 | 2 | 2 | 16 | 2 | 2 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes_5S_3 % in TeX
G:=Group("C2xD20:5S3"); // GroupNames label
G:=SmallGroup(480,1074);
// by ID
G=gap.SmallGroup(480,1074);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^20=c^2=d^3=e^2=1,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=b^10*c,e*d*e=d^-1>;
// generators/relations
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