direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D20⋊S3, D20⋊22D6, C30.3C24, Dic6⋊20D10, C60.158C23, D30.32C23, Dic15.34C23, (C6×D20)⋊12C2, (C2×D20)⋊12S3, C30⋊2(C4○D4), C6.3(C23×D5), C6⋊2(Q8⋊2D5), (C2×Dic6)⋊12D5, (C2×C20).162D6, C3⋊D20⋊8C22, C10.3(S3×C23), C10⋊1(D4⋊2S3), (C6×D5).2C23, (C10×Dic6)⋊12C2, (C2×C12).160D10, (C4×D15)⋊21C22, (C3×D20)⋊29C22, (D5×Dic3)⋊6C22, D10.2(C22×S3), (C22×D5).68D6, (C2×C60).207C22, C20.122(C22×S3), (C2×C30).222C23, (C5×Dic6)⋊26C22, C12.122(C22×D5), Dic3.2(C22×D5), (C5×Dic3).2C23, (C2×Dic3).129D10, (C2×Dic15).231C22, (C10×Dic3).127C22, (C22×D15).116C22, C15⋊2(C2×C4○D4), (C2×C4×D15)⋊24C2, C5⋊1(C2×D4⋊2S3), C3⋊2(C2×Q8⋊2D5), C4.131(C2×S3×D5), C2.7(C22×S3×D5), (C2×D5×Dic3)⋊20C2, (C2×C3⋊D20)⋊17C2, C22.94(C2×S3×D5), (C2×C4).217(S3×D5), (D5×C2×C6).59C22, (C2×C6).234(C22×D5), (C2×C10).234(C22×S3), SmallGroup(480,1075)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 1596 in 328 conjugacy classes, 116 normal (24 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 [×6], C10, C10 [×2], Dic3 [×4], Dic3 [×2], C12 [×2], D6 [×4], C2×C6, C2×C6 [×8], C15, C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], C20 [×2], C20 [×4], D10 [×4], D10 [×8], C2×C10, Dic6 [×4], C4×S3 [×4], C2×Dic3 [×2], C2×Dic3 [×9], C3⋊D4 [×8], C2×C12, C3×D4 [×4], C22×S3, C22×C6 [×2], C3×D5 [×4], D15 [×2], C30, C30 [×2], C2×C4○D4, C4×D5 [×12], D20 [×4], D20 [×8], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×4], C22×D5 [×2], C22×D5, C2×Dic6, S3×C2×C4, D4⋊2S3 [×8], C22×Dic3 [×2], C2×C3⋊D4 [×2], C6×D4, C5×Dic3 [×4], Dic15 [×2], C60 [×2], C6×D5 [×4], C6×D5 [×4], D30 [×2], D30 [×2], C2×C30, C2×C4×D5 [×3], C2×D20, C2×D20 [×2], Q8⋊2D5 [×8], Q8×C10, C2×D4⋊2S3, D5×Dic3 [×8], C3⋊D20 [×8], C3×D20 [×4], C5×Dic6 [×4], C10×Dic3 [×2], C4×D15 [×4], C2×Dic15, C2×C60, D5×C2×C6 [×2], C22×D15, C2×Q8⋊2D5, D20⋊S3 [×8], C2×D5×Dic3 [×2], C2×C3⋊D20 [×2], C6×D20, C10×Dic6, C2×C4×D15, C2×D20⋊S3
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, Q8⋊2D5 [×2], C23×D5, C2×D4⋊2S3, C2×S3×D5 [×3], C2×Q8⋊2D5, D20⋊S3 [×2], C22×S3×D5, C2×D20⋊S3
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, ebe=b9, cd=dc, ece=b18c, ede=d-1 >
►On 240 points
Generators in S240
(1 215)(2 216)(3 217)(4 218)(5 219)(6 220)(7 201)(8 202)(9 203)(10 204)(11 205)(12 206)(13 207)(14 208)(15 209)(16 210)(17 211)(18 212)(19 213)(20 214)(21 129)(22 130)(23 131)(24 132)(25 133)(26 134)(27 135)(28 136)(29 137)(30 138)(31 139)(32 140)(33 121)(34 122)(35 123)(36 124)(37 125)(38 126)(39 127)(40 128)(41 224)(42 225)(43 226)(44 227)(45 228)(46 229)(47 230)(48 231)(49 232)(50 233)(51 234)(52 235)(53 236)(54 237)(55 238)(56 239)(57 240)(58 221)(59 222)(60 223)(61 184)(62 185)(63 186)(64 187)(65 188)(66 189)(67 190)(68 191)(69 192)(70 193)(71 194)(72 195)(73 196)(74 197)(75 198)(76 199)(77 200)(78 181)(79 182)(80 183)(81 152)(82 153)(83 154)(84 155)(85 156)(86 157)(87 158)(88 159)(89 160)(90 141)(91 142)(92 143)(93 144)(94 145)(95 146)(96 147)(97 148)(98 149)(99 150)(100 151)(101 168)(102 169)(103 170)(104 171)(105 172)(106 173)(107 174)(108 175)(109 176)(110 177)(111 178)(112 179)(113 180)(114 161)(115 162)(116 163)(117 164)(118 165)(119 166)(120 167)
(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 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 60)(42 59)(43 58)(44 57)(45 56)(46 55)(47 54)(48 53)(49 52)(50 51)(61 72)(62 71)(63 70)(64 69)(65 68)(66 67)(73 80)(74 79)(75 78)(76 77)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 112)(102 111)(103 110)(104 109)(105 108)(106 107)(113 120)(114 119)(115 118)(116 117)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 142)(143 160)(144 159)(145 158)(146 157)(147 156)(148 155)(149 154)(150 153)(151 152)(161 166)(162 165)(163 164)(167 180)(168 179)(169 178)(170 177)(171 176)(172 175)(173 174)(181 198)(182 197)(183 196)(184 195)(185 194)(186 193)(187 192)(188 191)(189 190)(199 200)(201 208)(202 207)(203 206)(204 205)(209 220)(210 219)(211 218)(212 217)(213 216)(214 215)(221 226)(222 225)(223 224)(227 240)(228 239)(229 238)(230 237)(231 236)(232 235)(233 234)
(1 107 152)(2 108 153)(3 109 154)(4 110 155)(5 111 156)(6 112 157)(7 113 158)(8 114 159)(9 115 160)(10 116 141)(11 117 142)(12 118 143)(13 119 144)(14 120 145)(15 101 146)(16 102 147)(17 103 148)(18 104 149)(19 105 150)(20 106 151)(21 56 195)(22 57 196)(23 58 197)(24 59 198)(25 60 199)(26 41 200)(27 42 181)(28 43 182)(29 44 183)(30 45 184)(31 46 185)(32 47 186)(33 48 187)(34 49 188)(35 50 189)(36 51 190)(37 52 191)(38 53 192)(39 54 193)(40 55 194)(61 138 228)(62 139 229)(63 140 230)(64 121 231)(65 122 232)(66 123 233)(67 124 234)(68 125 235)(69 126 236)(70 127 237)(71 128 238)(72 129 239)(73 130 240)(74 131 221)(75 132 222)(76 133 223)(77 134 224)(78 135 225)(79 136 226)(80 137 227)(81 215 174)(82 216 175)(83 217 176)(84 218 177)(85 219 178)(86 220 179)(87 201 180)(88 202 161)(89 203 162)(90 204 163)(91 205 164)(92 206 165)(93 207 166)(94 208 167)(95 209 168)(96 210 169)(97 211 170)(98 212 171)(99 213 172)(100 214 173)
(1 129)(2 138)(3 127)(4 136)(5 125)(6 134)(7 123)(8 132)(9 121)(10 130)(11 139)(12 128)(13 137)(14 126)(15 135)(16 124)(17 133)(18 122)(19 131)(20 140)(21 215)(22 204)(23 213)(24 202)(25 211)(26 220)(27 209)(28 218)(29 207)(30 216)(31 205)(32 214)(33 203)(34 212)(35 201)(36 210)(37 219)(38 208)(39 217)(40 206)(41 86)(42 95)(43 84)(44 93)(45 82)(46 91)(47 100)(48 89)(49 98)(50 87)(51 96)(52 85)(53 94)(54 83)(55 92)(56 81)(57 90)(58 99)(59 88)(60 97)(61 108)(62 117)(63 106)(64 115)(65 104)(66 113)(67 102)(68 111)(69 120)(70 109)(71 118)(72 107)(73 116)(74 105)(75 114)(76 103)(77 112)(78 101)(79 110)(80 119)(141 240)(142 229)(143 238)(144 227)(145 236)(146 225)(147 234)(148 223)(149 232)(150 221)(151 230)(152 239)(153 228)(154 237)(155 226)(156 235)(157 224)(158 233)(159 222)(160 231)(161 198)(162 187)(163 196)(164 185)(165 194)(166 183)(167 192)(168 181)(169 190)(170 199)(171 188)(172 197)(173 186)(174 195)(175 184)(176 193)(177 182)(178 191)(179 200)(180 189)
G:=sub<Sym(240)| (1,215)(2,216)(3,217)(4,218)(5,219)(6,220)(7,201)(8,202)(9,203)(10,204)(11,205)(12,206)(13,207)(14,208)(15,209)(16,210)(17,211)(18,212)(19,213)(20,214)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,224)(42,225)(43,226)(44,227)(45,228)(46,229)(47,230)(48,231)(49,232)(50,233)(51,234)(52,235)(53,236)(54,237)(55,238)(56,239)(57,240)(58,221)(59,222)(60,223)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,181)(79,182)(80,183)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,151)(101,168)(102,169)(103,170)(104,171)(105,172)(106,173)(107,174)(108,175)(109,176)(110,177)(111,178)(112,179)(113,180)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167), (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)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,142)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)(161,166)(162,165)(163,164)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(199,200)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,107,152)(2,108,153)(3,109,154)(4,110,155)(5,111,156)(6,112,157)(7,113,158)(8,114,159)(9,115,160)(10,116,141)(11,117,142)(12,118,143)(13,119,144)(14,120,145)(15,101,146)(16,102,147)(17,103,148)(18,104,149)(19,105,150)(20,106,151)(21,56,195)(22,57,196)(23,58,197)(24,59,198)(25,60,199)(26,41,200)(27,42,181)(28,43,182)(29,44,183)(30,45,184)(31,46,185)(32,47,186)(33,48,187)(34,49,188)(35,50,189)(36,51,190)(37,52,191)(38,53,192)(39,54,193)(40,55,194)(61,138,228)(62,139,229)(63,140,230)(64,121,231)(65,122,232)(66,123,233)(67,124,234)(68,125,235)(69,126,236)(70,127,237)(71,128,238)(72,129,239)(73,130,240)(74,131,221)(75,132,222)(76,133,223)(77,134,224)(78,135,225)(79,136,226)(80,137,227)(81,215,174)(82,216,175)(83,217,176)(84,218,177)(85,219,178)(86,220,179)(87,201,180)(88,202,161)(89,203,162)(90,204,163)(91,205,164)(92,206,165)(93,207,166)(94,208,167)(95,209,168)(96,210,169)(97,211,170)(98,212,171)(99,213,172)(100,214,173), (1,129)(2,138)(3,127)(4,136)(5,125)(6,134)(7,123)(8,132)(9,121)(10,130)(11,139)(12,128)(13,137)(14,126)(15,135)(16,124)(17,133)(18,122)(19,131)(20,140)(21,215)(22,204)(23,213)(24,202)(25,211)(26,220)(27,209)(28,218)(29,207)(30,216)(31,205)(32,214)(33,203)(34,212)(35,201)(36,210)(37,219)(38,208)(39,217)(40,206)(41,86)(42,95)(43,84)(44,93)(45,82)(46,91)(47,100)(48,89)(49,98)(50,87)(51,96)(52,85)(53,94)(54,83)(55,92)(56,81)(57,90)(58,99)(59,88)(60,97)(61,108)(62,117)(63,106)(64,115)(65,104)(66,113)(67,102)(68,111)(69,120)(70,109)(71,118)(72,107)(73,116)(74,105)(75,114)(76,103)(77,112)(78,101)(79,110)(80,119)(141,240)(142,229)(143,238)(144,227)(145,236)(146,225)(147,234)(148,223)(149,232)(150,221)(151,230)(152,239)(153,228)(154,237)(155,226)(156,235)(157,224)(158,233)(159,222)(160,231)(161,198)(162,187)(163,196)(164,185)(165,194)(166,183)(167,192)(168,181)(169,190)(170,199)(171,188)(172,197)(173,186)(174,195)(175,184)(176,193)(177,182)(178,191)(179,200)(180,189)>;
G:=Group( (1,215)(2,216)(3,217)(4,218)(5,219)(6,220)(7,201)(8,202)(9,203)(10,204)(11,205)(12,206)(13,207)(14,208)(15,209)(16,210)(17,211)(18,212)(19,213)(20,214)(21,129)(22,130)(23,131)(24,132)(25,133)(26,134)(27,135)(28,136)(29,137)(30,138)(31,139)(32,140)(33,121)(34,122)(35,123)(36,124)(37,125)(38,126)(39,127)(40,128)(41,224)(42,225)(43,226)(44,227)(45,228)(46,229)(47,230)(48,231)(49,232)(50,233)(51,234)(52,235)(53,236)(54,237)(55,238)(56,239)(57,240)(58,221)(59,222)(60,223)(61,184)(62,185)(63,186)(64,187)(65,188)(66,189)(67,190)(68,191)(69,192)(70,193)(71,194)(72,195)(73,196)(74,197)(75,198)(76,199)(77,200)(78,181)(79,182)(80,183)(81,152)(82,153)(83,154)(84,155)(85,156)(86,157)(87,158)(88,159)(89,160)(90,141)(91,142)(92,143)(93,144)(94,145)(95,146)(96,147)(97,148)(98,149)(99,150)(100,151)(101,168)(102,169)(103,170)(104,171)(105,172)(106,173)(107,174)(108,175)(109,176)(110,177)(111,178)(112,179)(113,180)(114,161)(115,162)(116,163)(117,164)(118,165)(119,166)(120,167), (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)(21,30)(22,29)(23,28)(24,27)(25,26)(31,40)(32,39)(33,38)(34,37)(35,36)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)(73,80)(74,79)(75,78)(76,77)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,112)(102,111)(103,110)(104,109)(105,108)(106,107)(113,120)(114,119)(115,118)(116,117)(121,126)(122,125)(123,124)(127,140)(128,139)(129,138)(130,137)(131,136)(132,135)(133,134)(141,142)(143,160)(144,159)(145,158)(146,157)(147,156)(148,155)(149,154)(150,153)(151,152)(161,166)(162,165)(163,164)(167,180)(168,179)(169,178)(170,177)(171,176)(172,175)(173,174)(181,198)(182,197)(183,196)(184,195)(185,194)(186,193)(187,192)(188,191)(189,190)(199,200)(201,208)(202,207)(203,206)(204,205)(209,220)(210,219)(211,218)(212,217)(213,216)(214,215)(221,226)(222,225)(223,224)(227,240)(228,239)(229,238)(230,237)(231,236)(232,235)(233,234), (1,107,152)(2,108,153)(3,109,154)(4,110,155)(5,111,156)(6,112,157)(7,113,158)(8,114,159)(9,115,160)(10,116,141)(11,117,142)(12,118,143)(13,119,144)(14,120,145)(15,101,146)(16,102,147)(17,103,148)(18,104,149)(19,105,150)(20,106,151)(21,56,195)(22,57,196)(23,58,197)(24,59,198)(25,60,199)(26,41,200)(27,42,181)(28,43,182)(29,44,183)(30,45,184)(31,46,185)(32,47,186)(33,48,187)(34,49,188)(35,50,189)(36,51,190)(37,52,191)(38,53,192)(39,54,193)(40,55,194)(61,138,228)(62,139,229)(63,140,230)(64,121,231)(65,122,232)(66,123,233)(67,124,234)(68,125,235)(69,126,236)(70,127,237)(71,128,238)(72,129,239)(73,130,240)(74,131,221)(75,132,222)(76,133,223)(77,134,224)(78,135,225)(79,136,226)(80,137,227)(81,215,174)(82,216,175)(83,217,176)(84,218,177)(85,219,178)(86,220,179)(87,201,180)(88,202,161)(89,203,162)(90,204,163)(91,205,164)(92,206,165)(93,207,166)(94,208,167)(95,209,168)(96,210,169)(97,211,170)(98,212,171)(99,213,172)(100,214,173), (1,129)(2,138)(3,127)(4,136)(5,125)(6,134)(7,123)(8,132)(9,121)(10,130)(11,139)(12,128)(13,137)(14,126)(15,135)(16,124)(17,133)(18,122)(19,131)(20,140)(21,215)(22,204)(23,213)(24,202)(25,211)(26,220)(27,209)(28,218)(29,207)(30,216)(31,205)(32,214)(33,203)(34,212)(35,201)(36,210)(37,219)(38,208)(39,217)(40,206)(41,86)(42,95)(43,84)(44,93)(45,82)(46,91)(47,100)(48,89)(49,98)(50,87)(51,96)(52,85)(53,94)(54,83)(55,92)(56,81)(57,90)(58,99)(59,88)(60,97)(61,108)(62,117)(63,106)(64,115)(65,104)(66,113)(67,102)(68,111)(69,120)(70,109)(71,118)(72,107)(73,116)(74,105)(75,114)(76,103)(77,112)(78,101)(79,110)(80,119)(141,240)(142,229)(143,238)(144,227)(145,236)(146,225)(147,234)(148,223)(149,232)(150,221)(151,230)(152,239)(153,228)(154,237)(155,226)(156,235)(157,224)(158,233)(159,222)(160,231)(161,198)(162,187)(163,196)(164,185)(165,194)(166,183)(167,192)(168,181)(169,190)(170,199)(171,188)(172,197)(173,186)(174,195)(175,184)(176,193)(177,182)(178,191)(179,200)(180,189) );
G=PermutationGroup([(1,215),(2,216),(3,217),(4,218),(5,219),(6,220),(7,201),(8,202),(9,203),(10,204),(11,205),(12,206),(13,207),(14,208),(15,209),(16,210),(17,211),(18,212),(19,213),(20,214),(21,129),(22,130),(23,131),(24,132),(25,133),(26,134),(27,135),(28,136),(29,137),(30,138),(31,139),(32,140),(33,121),(34,122),(35,123),(36,124),(37,125),(38,126),(39,127),(40,128),(41,224),(42,225),(43,226),(44,227),(45,228),(46,229),(47,230),(48,231),(49,232),(50,233),(51,234),(52,235),(53,236),(54,237),(55,238),(56,239),(57,240),(58,221),(59,222),(60,223),(61,184),(62,185),(63,186),(64,187),(65,188),(66,189),(67,190),(68,191),(69,192),(70,193),(71,194),(72,195),(73,196),(74,197),(75,198),(76,199),(77,200),(78,181),(79,182),(80,183),(81,152),(82,153),(83,154),(84,155),(85,156),(86,157),(87,158),(88,159),(89,160),(90,141),(91,142),(92,143),(93,144),(94,145),(95,146),(96,147),(97,148),(98,149),(99,150),(100,151),(101,168),(102,169),(103,170),(104,171),(105,172),(106,173),(107,174),(108,175),(109,176),(110,177),(111,178),(112,179),(113,180),(114,161),(115,162),(116,163),(117,164),(118,165),(119,166),(120,167)], [(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),(21,30),(22,29),(23,28),(24,27),(25,26),(31,40),(32,39),(33,38),(34,37),(35,36),(41,60),(42,59),(43,58),(44,57),(45,56),(46,55),(47,54),(48,53),(49,52),(50,51),(61,72),(62,71),(63,70),(64,69),(65,68),(66,67),(73,80),(74,79),(75,78),(76,77),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,112),(102,111),(103,110),(104,109),(105,108),(106,107),(113,120),(114,119),(115,118),(116,117),(121,126),(122,125),(123,124),(127,140),(128,139),(129,138),(130,137),(131,136),(132,135),(133,134),(141,142),(143,160),(144,159),(145,158),(146,157),(147,156),(148,155),(149,154),(150,153),(151,152),(161,166),(162,165),(163,164),(167,180),(168,179),(169,178),(170,177),(171,176),(172,175),(173,174),(181,198),(182,197),(183,196),(184,195),(185,194),(186,193),(187,192),(188,191),(189,190),(199,200),(201,208),(202,207),(203,206),(204,205),(209,220),(210,219),(211,218),(212,217),(213,216),(214,215),(221,226),(222,225),(223,224),(227,240),(228,239),(229,238),(230,237),(231,236),(232,235),(233,234)], [(1,107,152),(2,108,153),(3,109,154),(4,110,155),(5,111,156),(6,112,157),(7,113,158),(8,114,159),(9,115,160),(10,116,141),(11,117,142),(12,118,143),(13,119,144),(14,120,145),(15,101,146),(16,102,147),(17,103,148),(18,104,149),(19,105,150),(20,106,151),(21,56,195),(22,57,196),(23,58,197),(24,59,198),(25,60,199),(26,41,200),(27,42,181),(28,43,182),(29,44,183),(30,45,184),(31,46,185),(32,47,186),(33,48,187),(34,49,188),(35,50,189),(36,51,190),(37,52,191),(38,53,192),(39,54,193),(40,55,194),(61,138,228),(62,139,229),(63,140,230),(64,121,231),(65,122,232),(66,123,233),(67,124,234),(68,125,235),(69,126,236),(70,127,237),(71,128,238),(72,129,239),(73,130,240),(74,131,221),(75,132,222),(76,133,223),(77,134,224),(78,135,225),(79,136,226),(80,137,227),(81,215,174),(82,216,175),(83,217,176),(84,218,177),(85,219,178),(86,220,179),(87,201,180),(88,202,161),(89,203,162),(90,204,163),(91,205,164),(92,206,165),(93,207,166),(94,208,167),(95,209,168),(96,210,169),(97,211,170),(98,212,171),(99,213,172),(100,214,173)], [(1,129),(2,138),(3,127),(4,136),(5,125),(6,134),(7,123),(8,132),(9,121),(10,130),(11,139),(12,128),(13,137),(14,126),(15,135),(16,124),(17,133),(18,122),(19,131),(20,140),(21,215),(22,204),(23,213),(24,202),(25,211),(26,220),(27,209),(28,218),(29,207),(30,216),(31,205),(32,214),(33,203),(34,212),(35,201),(36,210),(37,219),(38,208),(39,217),(40,206),(41,86),(42,95),(43,84),(44,93),(45,82),(46,91),(47,100),(48,89),(49,98),(50,87),(51,96),(52,85),(53,94),(54,83),(55,92),(56,81),(57,90),(58,99),(59,88),(60,97),(61,108),(62,117),(63,106),(64,115),(65,104),(66,113),(67,102),(68,111),(69,120),(70,109),(71,118),(72,107),(73,116),(74,105),(75,114),(76,103),(77,112),(78,101),(79,110),(80,119),(141,240),(142,229),(143,238),(144,227),(145,236),(146,225),(147,234),(148,223),(149,232),(150,221),(151,230),(152,239),(153,228),(154,237),(155,226),(156,235),(157,224),(158,233),(159,222),(160,231),(161,198),(162,187),(163,196),(164,185),(165,194),(166,183),(167,192),(168,181),(169,190),(170,199),(171,188),(172,197),(173,186),(174,195),(175,184),(176,193),(177,182),(178,191),(179,200),(180,189)])
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 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 60 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 46 |
| 0 | 0 | 0 | 0 | 9 | 16 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 1 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 45 | 46 |
| 0 | 0 | 0 | 0 | 17 | 16 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 60 | 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 |
| 39 | 48 | 0 | 0 | 0 | 0 |
| 9 | 22 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 18 | 0 | 0 |
| 0 | 0 | 60 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 43 |
| 0 | 0 | 0 | 0 | 23 | 7 |
G:=sub<GL(6,GF(61))| [60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,1,18,0,0,0,0,0,0,45,9,0,0,0,0,46,16],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,43,43,0,0,0,0,1,18,0,0,0,0,0,0,45,17,0,0,0,0,46,16],[0,60,0,0,0,0,1,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[39,9,0,0,0,0,48,22,0,0,0,0,0,0,43,60,0,0,0,0,18,18,0,0,0,0,0,0,54,23,0,0,0,0,43,7] >;
66 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 | 12A | 12B | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 15 | 15 | 15 | 15 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D4⋊2S3 | S3×D5 | Q8⋊2D5 | C2×S3×D5 | C2×S3×D5 | D20⋊S3 |
| kernel | C2×D20⋊S3 | D20⋊S3 | C2×D5×Dic3 | C2×C3⋊D20 | C6×D20 | C10×Dic6 | C2×C4×D15 | C2×D20 | C2×Dic6 | D20 | C2×C20 | C22×D5 | C30 | Dic6 | C2×Dic3 | C2×C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 8 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 4 | 1 | 2 | 4 | 8 | 4 | 2 | 2 | 2 | 4 | 4 | 2 | 8 |
In GAP, Magma, Sage, TeX
C_2\times D_{20}\rtimes S_3 % in TeX
G:=Group("C2xD20:S3"); // GroupNames label
G:=SmallGroup(480,1075);
// by ID
G=gap.SmallGroup(480,1075);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,120,675,346,80,1356,18822]);
// 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,e*b*e=b^9,c*d=d*c,e*c*e=b^18*c,e*d*e=d^-1>;
// generators/relations
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