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, C6⋊2(C4○D20), C30⋊5(C4○D4), 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, C5⋊D12⋊21C22, C3⋊D20⋊20C22, C15⋊D4⋊21C22, C10.11(S3×C23), (C6×D5).38C23, (C22×D5).96D6, D6.24(C22×D5), (S3×C10).27C23, C20.186(C22×S3), (C2×C60).245C22, (C2×C30).230C23, (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, (C10×Dic3).209C22, (C2×Dic15).234C22, (C22×D15).119C22, (S3×C2×C20)⋊9C2, (D5×C2×C12)⋊9C2, (C2×C4×D5)⋊15S3, (S3×C2×C4)⋊15D5, C3⋊2(C2×C4○D20), C5⋊2(C2×C4○D12), C15⋊5(C2×C4○D4), (C2×C4×D15)⋊30C2, (C2×C15⋊Q8)⋊26C2, C4.159(C2×S3×D5), (C2×C3⋊D20)⋊24C2, (C2×C15⋊D4)⋊24C2, (C2×C5⋊D12)⋊24C2, C2.15(C22×S3×D5), C22.99(C2×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
Generators and relations for C2×D6.D10
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 >
Subgroups: 1564 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C10, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C2×C10, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C30, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C2×Dic6, S3×C2×C4, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, S3×C10, D30, D30, C2×C30, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C4○D20, C2×C5⋊D4, C22×C20, C2×C4○D12, C15⋊D4, C3⋊D20, C5⋊D12, C15⋊Q8, D5×C12, C6×Dic5, S3×C20, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, C2×C4○D20, D6.D10, 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, C22, S3, C23, D5, D6, C4○D4, C24, D10, C22×S3, C2×C4○D4, C22×D5, C4○D12, S3×C23, S3×D5, C4○D20, C23×D5, C2×C4○D12, C2×S3×D5, C2×C4○D20, D6.D10, C22×S3×D5, C2×D6.D10
►On 240 points
Generators in S240
(1 185)(2 186)(3 187)(4 188)(5 189)(6 190)(7 191)(8 192)(9 193)(10 194)(11 195)(12 196)(13 197)(14 198)(15 199)(16 200)(17 181)(18 182)(19 183)(20 184)(21 239)(22 240)(23 221)(24 222)(25 223)(26 224)(27 225)(28 226)(29 227)(30 228)(31 229)(32 230)(33 231)(34 232)(35 233)(36 234)(37 235)(38 236)(39 237)(40 238)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 120)(50 101)(51 102)(52 103)(53 104)(54 105)(55 106)(56 107)(57 108)(58 109)(59 110)(60 111)(61 91)(62 92)(63 93)(64 94)(65 95)(66 96)(67 97)(68 98)(69 99)(70 100)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(121 213)(122 214)(123 215)(124 216)(125 217)(126 218)(127 219)(128 220)(129 201)(130 202)(131 203)(132 204)(133 205)(134 206)(135 207)(136 208)(137 209)(138 210)(139 211)(140 212)(141 164)(142 165)(143 166)(144 167)(145 168)(146 169)(147 170)(148 171)(149 172)(150 173)(151 174)(152 175)(153 176)(154 177)(155 178)(156 179)(157 180)(158 161)(159 162)(160 163)
(1 154 224 11 144 234)(2 155 225 12 145 235)(3 156 226 13 146 236)(4 157 227 14 147 237)(5 158 228 15 148 238)(6 159 229 16 149 239)(7 160 230 17 150 240)(8 141 231 18 151 221)(9 142 232 19 152 222)(10 143 233 20 153 223)(21 190 162 31 200 172)(22 191 163 32 181 173)(23 192 164 33 182 174)(24 193 165 34 183 175)(25 194 166 35 184 176)(26 195 167 36 185 177)(27 196 168 37 186 178)(28 197 169 38 187 179)(29 198 170 39 188 180)(30 199 171 40 189 161)(41 72 213 51 62 203)(42 73 214 52 63 204)(43 74 215 53 64 205)(44 75 216 54 65 206)(45 76 217 55 66 207)(46 77 218 56 67 208)(47 78 219 57 68 209)(48 79 220 58 69 210)(49 80 201 59 70 211)(50 61 202 60 71 212)(81 140 101 91 130 111)(82 121 102 92 131 112)(83 122 103 93 132 113)(84 123 104 94 133 114)(85 124 105 95 134 115)(86 125 106 96 135 116)(87 126 107 97 136 117)(88 127 108 98 137 118)(89 128 109 99 138 119)(90 129 110 100 139 120)
(1 129)(2 130)(3 131)(4 132)(5 133)(6 134)(7 135)(8 136)(9 137)(10 138)(11 139)(12 140)(13 121)(14 122)(15 123)(16 124)(17 125)(18 126)(19 127)(20 128)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 41)(39 42)(40 43)(61 178)(62 179)(63 180)(64 161)(65 162)(66 163)(67 164)(68 165)(69 166)(70 167)(71 168)(72 169)(73 170)(74 171)(75 172)(76 173)(77 174)(78 175)(79 176)(80 177)(81 145)(82 146)(83 147)(84 148)(85 149)(86 150)(87 151)(88 152)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 141)(98 142)(99 143)(100 144)(101 225)(102 226)(103 227)(104 228)(105 229)(106 230)(107 231)(108 232)(109 233)(110 234)(111 235)(112 236)(113 237)(114 238)(115 239)(116 240)(117 221)(118 222)(119 223)(120 224)(181 217)(182 218)(183 219)(184 220)(185 201)(186 202)(187 203)(188 204)(189 205)(190 206)(191 207)(192 208)(193 209)(194 210)(195 211)(196 212)(197 213)(198 214)(199 215)(200 216)
(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 30 31 40)(22 39 32 29)(23 28 33 38)(24 37 34 27)(25 26 35 36)(41 56 51 46)(42 45 52 55)(43 54 53 44)(47 50 57 60)(48 59 58 49)(61 68 71 78)(62 77 72 67)(63 66 73 76)(64 75 74 65)(69 80 79 70)(81 88 91 98)(82 97 92 87)(83 86 93 96)(84 95 94 85)(89 100 99 90)(101 108 111 118)(102 117 112 107)(103 106 113 116)(104 115 114 105)(109 120 119 110)(121 136 131 126)(122 125 132 135)(123 134 133 124)(127 130 137 140)(128 139 138 129)(141 146 151 156)(142 155 152 145)(143 144 153 154)(147 160 157 150)(148 149 158 159)(161 162 171 172)(163 180 173 170)(164 169 174 179)(165 178 175 168)(166 167 176 177)(181 198 191 188)(182 187 192 197)(183 196 193 186)(184 185 194 195)(189 190 199 200)(201 220 211 210)(202 209 212 219)(203 218 213 208)(204 207 214 217)(205 216 215 206)(221 226 231 236)(222 235 232 225)(223 224 233 234)(227 240 237 230)(228 229 238 239)
G:=sub<Sym(240)| (1,185)(2,186)(3,187)(4,188)(5,189)(6,190)(7,191)(8,192)(9,193)(10,194)(11,195)(12,196)(13,197)(14,198)(15,199)(16,200)(17,181)(18,182)(19,183)(20,184)(21,239)(22,240)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(136,208)(137,209)(138,210)(139,211)(140,212)(141,164)(142,165)(143,166)(144,167)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,161)(159,162)(160,163), (1,154,224,11,144,234)(2,155,225,12,145,235)(3,156,226,13,146,236)(4,157,227,14,147,237)(5,158,228,15,148,238)(6,159,229,16,149,239)(7,160,230,17,150,240)(8,141,231,18,151,221)(9,142,232,19,152,222)(10,143,233,20,153,223)(21,190,162,31,200,172)(22,191,163,32,181,173)(23,192,164,33,182,174)(24,193,165,34,183,175)(25,194,166,35,184,176)(26,195,167,36,185,177)(27,196,168,37,186,178)(28,197,169,38,187,179)(29,198,170,39,188,180)(30,199,171,40,189,161)(41,72,213,51,62,203)(42,73,214,52,63,204)(43,74,215,53,64,205)(44,75,216,54,65,206)(45,76,217,55,66,207)(46,77,218,56,67,208)(47,78,219,57,68,209)(48,79,220,58,69,210)(49,80,201,59,70,211)(50,61,202,60,71,212)(81,140,101,91,130,111)(82,121,102,92,131,112)(83,122,103,93,132,113)(84,123,104,94,133,114)(85,124,105,95,134,115)(86,125,106,96,135,116)(87,126,107,97,136,117)(88,127,108,98,137,118)(89,128,109,99,138,119)(90,129,110,100,139,120), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43)(61,178)(62,179)(63,180)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,141)(98,142)(99,143)(100,144)(101,225)(102,226)(103,227)(104,228)(105,229)(106,230)(107,231)(108,232)(109,233)(110,234)(111,235)(112,236)(113,237)(114,238)(115,239)(116,240)(117,221)(118,222)(119,223)(120,224)(181,217)(182,218)(183,219)(184,220)(185,201)(186,202)(187,203)(188,204)(189,205)(190,206)(191,207)(192,208)(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)(200,216), (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)(21,30,31,40)(22,39,32,29)(23,28,33,38)(24,37,34,27)(25,26,35,36)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,88,91,98)(82,97,92,87)(83,86,93,96)(84,95,94,85)(89,100,99,90)(101,108,111,118)(102,117,112,107)(103,106,113,116)(104,115,114,105)(109,120,119,110)(121,136,131,126)(122,125,132,135)(123,134,133,124)(127,130,137,140)(128,139,138,129)(141,146,151,156)(142,155,152,145)(143,144,153,154)(147,160,157,150)(148,149,158,159)(161,162,171,172)(163,180,173,170)(164,169,174,179)(165,178,175,168)(166,167,176,177)(181,198,191,188)(182,187,192,197)(183,196,193,186)(184,185,194,195)(189,190,199,200)(201,220,211,210)(202,209,212,219)(203,218,213,208)(204,207,214,217)(205,216,215,206)(221,226,231,236)(222,235,232,225)(223,224,233,234)(227,240,237,230)(228,229,238,239)>;
G:=Group( (1,185)(2,186)(3,187)(4,188)(5,189)(6,190)(7,191)(8,192)(9,193)(10,194)(11,195)(12,196)(13,197)(14,198)(15,199)(16,200)(17,181)(18,182)(19,183)(20,184)(21,239)(22,240)(23,221)(24,222)(25,223)(26,224)(27,225)(28,226)(29,227)(30,228)(31,229)(32,230)(33,231)(34,232)(35,233)(36,234)(37,235)(38,236)(39,237)(40,238)(41,112)(42,113)(43,114)(44,115)(45,116)(46,117)(47,118)(48,119)(49,120)(50,101)(51,102)(52,103)(53,104)(54,105)(55,106)(56,107)(57,108)(58,109)(59,110)(60,111)(61,91)(62,92)(63,93)(64,94)(65,95)(66,96)(67,97)(68,98)(69,99)(70,100)(71,81)(72,82)(73,83)(74,84)(75,85)(76,86)(77,87)(78,88)(79,89)(80,90)(121,213)(122,214)(123,215)(124,216)(125,217)(126,218)(127,219)(128,220)(129,201)(130,202)(131,203)(132,204)(133,205)(134,206)(135,207)(136,208)(137,209)(138,210)(139,211)(140,212)(141,164)(142,165)(143,166)(144,167)(145,168)(146,169)(147,170)(148,171)(149,172)(150,173)(151,174)(152,175)(153,176)(154,177)(155,178)(156,179)(157,180)(158,161)(159,162)(160,163), (1,154,224,11,144,234)(2,155,225,12,145,235)(3,156,226,13,146,236)(4,157,227,14,147,237)(5,158,228,15,148,238)(6,159,229,16,149,239)(7,160,230,17,150,240)(8,141,231,18,151,221)(9,142,232,19,152,222)(10,143,233,20,153,223)(21,190,162,31,200,172)(22,191,163,32,181,173)(23,192,164,33,182,174)(24,193,165,34,183,175)(25,194,166,35,184,176)(26,195,167,36,185,177)(27,196,168,37,186,178)(28,197,169,38,187,179)(29,198,170,39,188,180)(30,199,171,40,189,161)(41,72,213,51,62,203)(42,73,214,52,63,204)(43,74,215,53,64,205)(44,75,216,54,65,206)(45,76,217,55,66,207)(46,77,218,56,67,208)(47,78,219,57,68,209)(48,79,220,58,69,210)(49,80,201,59,70,211)(50,61,202,60,71,212)(81,140,101,91,130,111)(82,121,102,92,131,112)(83,122,103,93,132,113)(84,123,104,94,133,114)(85,124,105,95,134,115)(86,125,106,96,135,116)(87,126,107,97,136,117)(88,127,108,98,137,118)(89,128,109,99,138,119)(90,129,110,100,139,120), (1,129)(2,130)(3,131)(4,132)(5,133)(6,134)(7,135)(8,136)(9,137)(10,138)(11,139)(12,140)(13,121)(14,122)(15,123)(16,124)(17,125)(18,126)(19,127)(20,128)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43)(61,178)(62,179)(63,180)(64,161)(65,162)(66,163)(67,164)(68,165)(69,166)(70,167)(71,168)(72,169)(73,170)(74,171)(75,172)(76,173)(77,174)(78,175)(79,176)(80,177)(81,145)(82,146)(83,147)(84,148)(85,149)(86,150)(87,151)(88,152)(89,153)(90,154)(91,155)(92,156)(93,157)(94,158)(95,159)(96,160)(97,141)(98,142)(99,143)(100,144)(101,225)(102,226)(103,227)(104,228)(105,229)(106,230)(107,231)(108,232)(109,233)(110,234)(111,235)(112,236)(113,237)(114,238)(115,239)(116,240)(117,221)(118,222)(119,223)(120,224)(181,217)(182,218)(183,219)(184,220)(185,201)(186,202)(187,203)(188,204)(189,205)(190,206)(191,207)(192,208)(193,209)(194,210)(195,211)(196,212)(197,213)(198,214)(199,215)(200,216), (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)(21,30,31,40)(22,39,32,29)(23,28,33,38)(24,37,34,27)(25,26,35,36)(41,56,51,46)(42,45,52,55)(43,54,53,44)(47,50,57,60)(48,59,58,49)(61,68,71,78)(62,77,72,67)(63,66,73,76)(64,75,74,65)(69,80,79,70)(81,88,91,98)(82,97,92,87)(83,86,93,96)(84,95,94,85)(89,100,99,90)(101,108,111,118)(102,117,112,107)(103,106,113,116)(104,115,114,105)(109,120,119,110)(121,136,131,126)(122,125,132,135)(123,134,133,124)(127,130,137,140)(128,139,138,129)(141,146,151,156)(142,155,152,145)(143,144,153,154)(147,160,157,150)(148,149,158,159)(161,162,171,172)(163,180,173,170)(164,169,174,179)(165,178,175,168)(166,167,176,177)(181,198,191,188)(182,187,192,197)(183,196,193,186)(184,185,194,195)(189,190,199,200)(201,220,211,210)(202,209,212,219)(203,218,213,208)(204,207,214,217)(205,216,215,206)(221,226,231,236)(222,235,232,225)(223,224,233,234)(227,240,237,230)(228,229,238,239) );
G=PermutationGroup([[(1,185),(2,186),(3,187),(4,188),(5,189),(6,190),(7,191),(8,192),(9,193),(10,194),(11,195),(12,196),(13,197),(14,198),(15,199),(16,200),(17,181),(18,182),(19,183),(20,184),(21,239),(22,240),(23,221),(24,222),(25,223),(26,224),(27,225),(28,226),(29,227),(30,228),(31,229),(32,230),(33,231),(34,232),(35,233),(36,234),(37,235),(38,236),(39,237),(40,238),(41,112),(42,113),(43,114),(44,115),(45,116),(46,117),(47,118),(48,119),(49,120),(50,101),(51,102),(52,103),(53,104),(54,105),(55,106),(56,107),(57,108),(58,109),(59,110),(60,111),(61,91),(62,92),(63,93),(64,94),(65,95),(66,96),(67,97),(68,98),(69,99),(70,100),(71,81),(72,82),(73,83),(74,84),(75,85),(76,86),(77,87),(78,88),(79,89),(80,90),(121,213),(122,214),(123,215),(124,216),(125,217),(126,218),(127,219),(128,220),(129,201),(130,202),(131,203),(132,204),(133,205),(134,206),(135,207),(136,208),(137,209),(138,210),(139,211),(140,212),(141,164),(142,165),(143,166),(144,167),(145,168),(146,169),(147,170),(148,171),(149,172),(150,173),(151,174),(152,175),(153,176),(154,177),(155,178),(156,179),(157,180),(158,161),(159,162),(160,163)], [(1,154,224,11,144,234),(2,155,225,12,145,235),(3,156,226,13,146,236),(4,157,227,14,147,237),(5,158,228,15,148,238),(6,159,229,16,149,239),(7,160,230,17,150,240),(8,141,231,18,151,221),(9,142,232,19,152,222),(10,143,233,20,153,223),(21,190,162,31,200,172),(22,191,163,32,181,173),(23,192,164,33,182,174),(24,193,165,34,183,175),(25,194,166,35,184,176),(26,195,167,36,185,177),(27,196,168,37,186,178),(28,197,169,38,187,179),(29,198,170,39,188,180),(30,199,171,40,189,161),(41,72,213,51,62,203),(42,73,214,52,63,204),(43,74,215,53,64,205),(44,75,216,54,65,206),(45,76,217,55,66,207),(46,77,218,56,67,208),(47,78,219,57,68,209),(48,79,220,58,69,210),(49,80,201,59,70,211),(50,61,202,60,71,212),(81,140,101,91,130,111),(82,121,102,92,131,112),(83,122,103,93,132,113),(84,123,104,94,133,114),(85,124,105,95,134,115),(86,125,106,96,135,116),(87,126,107,97,136,117),(88,127,108,98,137,118),(89,128,109,99,138,119),(90,129,110,100,139,120)], [(1,129),(2,130),(3,131),(4,132),(5,133),(6,134),(7,135),(8,136),(9,137),(10,138),(11,139),(12,140),(13,121),(14,122),(15,123),(16,124),(17,125),(18,126),(19,127),(20,128),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,41),(39,42),(40,43),(61,178),(62,179),(63,180),(64,161),(65,162),(66,163),(67,164),(68,165),(69,166),(70,167),(71,168),(72,169),(73,170),(74,171),(75,172),(76,173),(77,174),(78,175),(79,176),(80,177),(81,145),(82,146),(83,147),(84,148),(85,149),(86,150),(87,151),(88,152),(89,153),(90,154),(91,155),(92,156),(93,157),(94,158),(95,159),(96,160),(97,141),(98,142),(99,143),(100,144),(101,225),(102,226),(103,227),(104,228),(105,229),(106,230),(107,231),(108,232),(109,233),(110,234),(111,235),(112,236),(113,237),(114,238),(115,239),(116,240),(117,221),(118,222),(119,223),(120,224),(181,217),(182,218),(183,219),(184,220),(185,201),(186,202),(187,203),(188,204),(189,205),(190,206),(191,207),(192,208),(193,209),(194,210),(195,211),(196,212),(197,213),(198,214),(199,215),(200,216)], [(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),(21,30,31,40),(22,39,32,29),(23,28,33,38),(24,37,34,27),(25,26,35,36),(41,56,51,46),(42,45,52,55),(43,54,53,44),(47,50,57,60),(48,59,58,49),(61,68,71,78),(62,77,72,67),(63,66,73,76),(64,75,74,65),(69,80,79,70),(81,88,91,98),(82,97,92,87),(83,86,93,96),(84,95,94,85),(89,100,99,90),(101,108,111,118),(102,117,112,107),(103,106,113,116),(104,115,114,105),(109,120,119,110),(121,136,131,126),(122,125,132,135),(123,134,133,124),(127,130,137,140),(128,139,138,129),(141,146,151,156),(142,155,152,145),(143,144,153,154),(147,160,157,150),(148,149,158,159),(161,162,171,172),(163,180,173,170),(164,169,174,179),(165,178,175,168),(166,167,176,177),(181,198,191,188),(182,187,192,197),(183,196,193,186),(184,185,194,195),(189,190,199,200),(201,220,211,210),(202,209,212,219),(203,218,213,208),(204,207,214,217),(205,216,215,206),(221,226,231,236),(222,235,232,225),(223,224,233,234),(227,240,237,230),(228,229,238,239)]])
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 |
Matrix representation of C2×D6.D10 ►in 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] >;
C2×D6.D10 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