direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×D10⋊Dic3, (C2×C6).44D20, (C2×C30).66D4, C6.64(C2×D20), C30⋊4(C22⋊C4), D10⋊6(C2×Dic3), C30.215(C2×D4), C6⋊3(D10⋊C4), (C23×D5).3S3, C23.61(S3×D5), (C2×Dic3)⋊19D10, (C22×D5)⋊5Dic3, (C22×Dic3)⋊1D5, (C22×D5).93D6, (C22×C10).99D6, (C22×C6).82D10, C10⋊2(C6.D4), C30.137(C22×C4), (C2×C30).177C23, C22.16(D5×Dic3), (C2×Dic15)⋊30C22, (C22×Dic15)⋊10C2, (C10×Dic3)⋊23C22, C22.23(C15⋊D4), C22.23(C3⋊D20), (C22×C30).39C22, C10.29(C22×Dic3), (D5×C2×C6)⋊5C4, C6.92(C2×C4×D5), C15⋊8(C2×C22⋊C4), (C6×D5)⋊26(C2×C4), C3⋊4(C2×D10⋊C4), (Dic3×C2×C10)⋊1C2, (C2×C6).55(C4×D5), C2.2(C2×C3⋊D20), C6.89(C2×C5⋊D4), C2.2(C2×C15⋊D4), C5⋊3(C2×C6.D4), C2.16(C2×D5×Dic3), (D5×C22×C6).1C2, C22.77(C2×S3×D5), C10.90(C2×C3⋊D4), (C2×C30).112(C2×C4), (C2×C6).57(C5⋊D4), (D5×C2×C6).110C22, (C2×C10).35(C3⋊D4), (C2×C10).38(C2×Dic3), (C2×C6).189(C22×D5), (C2×C10).189(C22×S3), SmallGroup(480,611)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D10⋊Dic3
G = < a,b,c,d,e | a2=b10=c2=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b5c, ede-1=d-1 >
Subgroups: 1148 in 264 conjugacy classes, 100 normal (30 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, C6, C6, C6, C2×C4, C23, C23, D5, C10, C10, Dic3, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C22×C4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C22×C6, C22×C6, C3×D5, C30, C30, C2×C22⋊C4, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C6.D4, C22×Dic3, C22×Dic3, C23×C6, C5×Dic3, Dic15, C6×D5, C6×D5, C2×C30, C2×C30, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C2×C6.D4, C10×Dic3, C10×Dic3, C2×Dic15, C2×Dic15, D5×C2×C6, D5×C2×C6, C22×C30, C2×D10⋊C4, D10⋊Dic3, Dic3×C2×C10, C22×Dic15, D5×C22×C6, C2×D10⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, Dic3, D6, C22⋊C4, C22×C4, C2×D4, D10, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C4×D5, D20, C5⋊D4, C22×D5, C6.D4, C22×Dic3, C2×C3⋊D4, S3×D5, D10⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C2×C6.D4, D5×Dic3, C15⋊D4, C3⋊D20, C2×S3×D5, C2×D10⋊C4, D10⋊Dic3, C2×D5×Dic3, C2×C15⋊D4, C2×C3⋊D20, C2×D10⋊Dic3
►On 240 points
Generators in S240
(1 112)(2 113)(3 114)(4 115)(5 116)(6 117)(7 118)(8 119)(9 120)(10 111)(11 160)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(21 102)(22 103)(23 104)(24 105)(25 106)(26 107)(27 108)(28 109)(29 110)(30 101)(31 126)(32 127)(33 128)(34 129)(35 130)(36 121)(37 122)(38 123)(39 124)(40 125)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 86)(52 87)(53 88)(54 89)(55 90)(56 81)(57 82)(58 83)(59 84)(60 85)(61 96)(62 97)(63 98)(64 99)(65 100)(66 91)(67 92)(68 93)(69 94)(70 95)(131 226)(132 227)(133 228)(134 229)(135 230)(136 221)(137 222)(138 223)(139 224)(140 225)(141 236)(142 237)(143 238)(144 239)(145 240)(146 231)(147 232)(148 233)(149 234)(150 235)(161 196)(162 197)(163 198)(164 199)(165 200)(166 191)(167 192)(168 193)(169 194)(170 195)(171 206)(172 207)(173 208)(174 209)(175 210)(176 201)(177 202)(178 203)(179 204)(180 205)(181 216)(182 217)(183 218)(184 219)(185 220)(186 211)(187 212)(188 213)(189 214)(190 215)
(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)(2 9)(3 8)(4 7)(5 6)(11 15)(12 14)(16 20)(17 19)(21 26)(22 25)(23 24)(27 30)(28 29)(31 38)(32 37)(33 36)(34 35)(39 40)(41 48)(42 47)(43 46)(44 45)(49 50)(51 52)(53 60)(54 59)(55 58)(56 57)(61 68)(62 67)(63 66)(64 65)(69 70)(71 78)(72 77)(73 76)(74 75)(79 80)(81 82)(83 90)(84 89)(85 88)(86 87)(91 98)(92 97)(93 96)(94 95)(99 100)(101 108)(102 107)(103 106)(104 105)(109 110)(111 112)(113 120)(114 119)(115 118)(116 117)(121 128)(122 127)(123 126)(124 125)(129 130)(131 133)(134 140)(135 139)(136 138)(141 147)(142 146)(143 145)(148 150)(151 153)(154 160)(155 159)(156 158)(161 163)(164 170)(165 169)(166 168)(171 177)(172 176)(173 175)(178 180)(181 183)(184 190)(185 189)(186 188)(191 193)(194 200)(195 199)(196 198)(201 207)(202 206)(203 205)(208 210)(211 213)(214 220)(215 219)(216 218)(221 223)(224 230)(225 229)(226 228)(231 237)(232 236)(233 235)(238 240)
(1 45 40 57 29 70)(2 46 31 58 30 61)(3 47 32 59 21 62)(4 48 33 60 22 63)(5 49 34 51 23 64)(6 50 35 52 24 65)(7 41 36 53 25 66)(8 42 37 54 26 67)(9 43 38 55 27 68)(10 44 39 56 28 69)(11 200 232 215 230 202)(12 191 233 216 221 203)(13 192 234 217 222 204)(14 193 235 218 223 205)(15 194 236 219 224 206)(16 195 237 220 225 207)(17 196 238 211 226 208)(18 197 239 212 227 209)(19 198 240 213 228 210)(20 199 231 214 229 201)(71 126 83 101 96 113)(72 127 84 102 97 114)(73 128 85 103 98 115)(74 129 86 104 99 116)(75 130 87 105 100 117)(76 121 88 106 91 118)(77 122 89 107 92 119)(78 123 90 108 93 120)(79 124 81 109 94 111)(80 125 82 110 95 112)(131 173 156 161 143 186)(132 174 157 162 144 187)(133 175 158 163 145 188)(134 176 159 164 146 189)(135 177 160 165 147 190)(136 178 151 166 148 181)(137 179 152 167 149 182)(138 180 153 168 150 183)(139 171 154 169 141 184)(140 172 155 170 142 185)
(1 177 57 147)(2 178 58 148)(3 179 59 149)(4 180 60 150)(5 171 51 141)(6 172 52 142)(7 173 53 143)(8 174 54 144)(9 175 55 145)(10 176 56 146)(11 125 215 95)(12 126 216 96)(13 127 217 97)(14 128 218 98)(15 129 219 99)(16 130 220 100)(17 121 211 91)(18 122 212 92)(19 123 213 93)(20 124 214 94)(21 167 47 137)(22 168 48 138)(23 169 49 139)(24 170 50 140)(25 161 41 131)(26 162 42 132)(27 163 43 133)(28 164 44 134)(29 165 45 135)(30 166 46 136)(31 181 61 151)(32 182 62 152)(33 183 63 153)(34 184 64 154)(35 185 65 155)(36 186 66 156)(37 187 67 157)(38 188 68 158)(39 189 69 159)(40 190 70 160)(71 221 101 191)(72 222 102 192)(73 223 103 193)(74 224 104 194)(75 225 105 195)(76 226 106 196)(77 227 107 197)(78 228 108 198)(79 229 109 199)(80 230 110 200)(81 231 111 201)(82 232 112 202)(83 233 113 203)(84 234 114 204)(85 235 115 205)(86 236 116 206)(87 237 117 207)(88 238 118 208)(89 239 119 209)(90 240 120 210)
G:=sub<Sym(240)| (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,160)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,101)(31,126)(32,127)(33,128)(34,129)(35,130)(36,121)(37,122)(38,123)(39,124)(40,125)(41,76)(42,77)(43,78)(44,79)(45,80)(46,71)(47,72)(48,73)(49,74)(50,75)(51,86)(52,87)(53,88)(54,89)(55,90)(56,81)(57,82)(58,83)(59,84)(60,85)(61,96)(62,97)(63,98)(64,99)(65,100)(66,91)(67,92)(68,93)(69,94)(70,95)(131,226)(132,227)(133,228)(134,229)(135,230)(136,221)(137,222)(138,223)(139,224)(140,225)(141,236)(142,237)(143,238)(144,239)(145,240)(146,231)(147,232)(148,233)(149,234)(150,235)(161,196)(162,197)(163,198)(164,199)(165,200)(166,191)(167,192)(168,193)(169,194)(170,195)(171,206)(172,207)(173,208)(174,209)(175,210)(176,201)(177,202)(178,203)(179,204)(180,205)(181,216)(182,217)(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)(190,215), (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)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,112)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,130)(131,133)(134,140)(135,139)(136,138)(141,147)(142,146)(143,145)(148,150)(151,153)(154,160)(155,159)(156,158)(161,163)(164,170)(165,169)(166,168)(171,177)(172,176)(173,175)(178,180)(181,183)(184,190)(185,189)(186,188)(191,193)(194,200)(195,199)(196,198)(201,207)(202,206)(203,205)(208,210)(211,213)(214,220)(215,219)(216,218)(221,223)(224,230)(225,229)(226,228)(231,237)(232,236)(233,235)(238,240), (1,45,40,57,29,70)(2,46,31,58,30,61)(3,47,32,59,21,62)(4,48,33,60,22,63)(5,49,34,51,23,64)(6,50,35,52,24,65)(7,41,36,53,25,66)(8,42,37,54,26,67)(9,43,38,55,27,68)(10,44,39,56,28,69)(11,200,232,215,230,202)(12,191,233,216,221,203)(13,192,234,217,222,204)(14,193,235,218,223,205)(15,194,236,219,224,206)(16,195,237,220,225,207)(17,196,238,211,226,208)(18,197,239,212,227,209)(19,198,240,213,228,210)(20,199,231,214,229,201)(71,126,83,101,96,113)(72,127,84,102,97,114)(73,128,85,103,98,115)(74,129,86,104,99,116)(75,130,87,105,100,117)(76,121,88,106,91,118)(77,122,89,107,92,119)(78,123,90,108,93,120)(79,124,81,109,94,111)(80,125,82,110,95,112)(131,173,156,161,143,186)(132,174,157,162,144,187)(133,175,158,163,145,188)(134,176,159,164,146,189)(135,177,160,165,147,190)(136,178,151,166,148,181)(137,179,152,167,149,182)(138,180,153,168,150,183)(139,171,154,169,141,184)(140,172,155,170,142,185), (1,177,57,147)(2,178,58,148)(3,179,59,149)(4,180,60,150)(5,171,51,141)(6,172,52,142)(7,173,53,143)(8,174,54,144)(9,175,55,145)(10,176,56,146)(11,125,215,95)(12,126,216,96)(13,127,217,97)(14,128,218,98)(15,129,219,99)(16,130,220,100)(17,121,211,91)(18,122,212,92)(19,123,213,93)(20,124,214,94)(21,167,47,137)(22,168,48,138)(23,169,49,139)(24,170,50,140)(25,161,41,131)(26,162,42,132)(27,163,43,133)(28,164,44,134)(29,165,45,135)(30,166,46,136)(31,181,61,151)(32,182,62,152)(33,183,63,153)(34,184,64,154)(35,185,65,155)(36,186,66,156)(37,187,67,157)(38,188,68,158)(39,189,69,159)(40,190,70,160)(71,221,101,191)(72,222,102,192)(73,223,103,193)(74,224,104,194)(75,225,105,195)(76,226,106,196)(77,227,107,197)(78,228,108,198)(79,229,109,199)(80,230,110,200)(81,231,111,201)(82,232,112,202)(83,233,113,203)(84,234,114,204)(85,235,115,205)(86,236,116,206)(87,237,117,207)(88,238,118,208)(89,239,119,209)(90,240,120,210)>;
G:=Group( (1,112)(2,113)(3,114)(4,115)(5,116)(6,117)(7,118)(8,119)(9,120)(10,111)(11,160)(12,151)(13,152)(14,153)(15,154)(16,155)(17,156)(18,157)(19,158)(20,159)(21,102)(22,103)(23,104)(24,105)(25,106)(26,107)(27,108)(28,109)(29,110)(30,101)(31,126)(32,127)(33,128)(34,129)(35,130)(36,121)(37,122)(38,123)(39,124)(40,125)(41,76)(42,77)(43,78)(44,79)(45,80)(46,71)(47,72)(48,73)(49,74)(50,75)(51,86)(52,87)(53,88)(54,89)(55,90)(56,81)(57,82)(58,83)(59,84)(60,85)(61,96)(62,97)(63,98)(64,99)(65,100)(66,91)(67,92)(68,93)(69,94)(70,95)(131,226)(132,227)(133,228)(134,229)(135,230)(136,221)(137,222)(138,223)(139,224)(140,225)(141,236)(142,237)(143,238)(144,239)(145,240)(146,231)(147,232)(148,233)(149,234)(150,235)(161,196)(162,197)(163,198)(164,199)(165,200)(166,191)(167,192)(168,193)(169,194)(170,195)(171,206)(172,207)(173,208)(174,209)(175,210)(176,201)(177,202)(178,203)(179,204)(180,205)(181,216)(182,217)(183,218)(184,219)(185,220)(186,211)(187,212)(188,213)(189,214)(190,215), (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)(2,9)(3,8)(4,7)(5,6)(11,15)(12,14)(16,20)(17,19)(21,26)(22,25)(23,24)(27,30)(28,29)(31,38)(32,37)(33,36)(34,35)(39,40)(41,48)(42,47)(43,46)(44,45)(49,50)(51,52)(53,60)(54,59)(55,58)(56,57)(61,68)(62,67)(63,66)(64,65)(69,70)(71,78)(72,77)(73,76)(74,75)(79,80)(81,82)(83,90)(84,89)(85,88)(86,87)(91,98)(92,97)(93,96)(94,95)(99,100)(101,108)(102,107)(103,106)(104,105)(109,110)(111,112)(113,120)(114,119)(115,118)(116,117)(121,128)(122,127)(123,126)(124,125)(129,130)(131,133)(134,140)(135,139)(136,138)(141,147)(142,146)(143,145)(148,150)(151,153)(154,160)(155,159)(156,158)(161,163)(164,170)(165,169)(166,168)(171,177)(172,176)(173,175)(178,180)(181,183)(184,190)(185,189)(186,188)(191,193)(194,200)(195,199)(196,198)(201,207)(202,206)(203,205)(208,210)(211,213)(214,220)(215,219)(216,218)(221,223)(224,230)(225,229)(226,228)(231,237)(232,236)(233,235)(238,240), (1,45,40,57,29,70)(2,46,31,58,30,61)(3,47,32,59,21,62)(4,48,33,60,22,63)(5,49,34,51,23,64)(6,50,35,52,24,65)(7,41,36,53,25,66)(8,42,37,54,26,67)(9,43,38,55,27,68)(10,44,39,56,28,69)(11,200,232,215,230,202)(12,191,233,216,221,203)(13,192,234,217,222,204)(14,193,235,218,223,205)(15,194,236,219,224,206)(16,195,237,220,225,207)(17,196,238,211,226,208)(18,197,239,212,227,209)(19,198,240,213,228,210)(20,199,231,214,229,201)(71,126,83,101,96,113)(72,127,84,102,97,114)(73,128,85,103,98,115)(74,129,86,104,99,116)(75,130,87,105,100,117)(76,121,88,106,91,118)(77,122,89,107,92,119)(78,123,90,108,93,120)(79,124,81,109,94,111)(80,125,82,110,95,112)(131,173,156,161,143,186)(132,174,157,162,144,187)(133,175,158,163,145,188)(134,176,159,164,146,189)(135,177,160,165,147,190)(136,178,151,166,148,181)(137,179,152,167,149,182)(138,180,153,168,150,183)(139,171,154,169,141,184)(140,172,155,170,142,185), (1,177,57,147)(2,178,58,148)(3,179,59,149)(4,180,60,150)(5,171,51,141)(6,172,52,142)(7,173,53,143)(8,174,54,144)(9,175,55,145)(10,176,56,146)(11,125,215,95)(12,126,216,96)(13,127,217,97)(14,128,218,98)(15,129,219,99)(16,130,220,100)(17,121,211,91)(18,122,212,92)(19,123,213,93)(20,124,214,94)(21,167,47,137)(22,168,48,138)(23,169,49,139)(24,170,50,140)(25,161,41,131)(26,162,42,132)(27,163,43,133)(28,164,44,134)(29,165,45,135)(30,166,46,136)(31,181,61,151)(32,182,62,152)(33,183,63,153)(34,184,64,154)(35,185,65,155)(36,186,66,156)(37,187,67,157)(38,188,68,158)(39,189,69,159)(40,190,70,160)(71,221,101,191)(72,222,102,192)(73,223,103,193)(74,224,104,194)(75,225,105,195)(76,226,106,196)(77,227,107,197)(78,228,108,198)(79,229,109,199)(80,230,110,200)(81,231,111,201)(82,232,112,202)(83,233,113,203)(84,234,114,204)(85,235,115,205)(86,236,116,206)(87,237,117,207)(88,238,118,208)(89,239,119,209)(90,240,120,210) );
G=PermutationGroup([[(1,112),(2,113),(3,114),(4,115),(5,116),(6,117),(7,118),(8,119),(9,120),(10,111),(11,160),(12,151),(13,152),(14,153),(15,154),(16,155),(17,156),(18,157),(19,158),(20,159),(21,102),(22,103),(23,104),(24,105),(25,106),(26,107),(27,108),(28,109),(29,110),(30,101),(31,126),(32,127),(33,128),(34,129),(35,130),(36,121),(37,122),(38,123),(39,124),(40,125),(41,76),(42,77),(43,78),(44,79),(45,80),(46,71),(47,72),(48,73),(49,74),(50,75),(51,86),(52,87),(53,88),(54,89),(55,90),(56,81),(57,82),(58,83),(59,84),(60,85),(61,96),(62,97),(63,98),(64,99),(65,100),(66,91),(67,92),(68,93),(69,94),(70,95),(131,226),(132,227),(133,228),(134,229),(135,230),(136,221),(137,222),(138,223),(139,224),(140,225),(141,236),(142,237),(143,238),(144,239),(145,240),(146,231),(147,232),(148,233),(149,234),(150,235),(161,196),(162,197),(163,198),(164,199),(165,200),(166,191),(167,192),(168,193),(169,194),(170,195),(171,206),(172,207),(173,208),(174,209),(175,210),(176,201),(177,202),(178,203),(179,204),(180,205),(181,216),(182,217),(183,218),(184,219),(185,220),(186,211),(187,212),(188,213),(189,214),(190,215)], [(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),(2,9),(3,8),(4,7),(5,6),(11,15),(12,14),(16,20),(17,19),(21,26),(22,25),(23,24),(27,30),(28,29),(31,38),(32,37),(33,36),(34,35),(39,40),(41,48),(42,47),(43,46),(44,45),(49,50),(51,52),(53,60),(54,59),(55,58),(56,57),(61,68),(62,67),(63,66),(64,65),(69,70),(71,78),(72,77),(73,76),(74,75),(79,80),(81,82),(83,90),(84,89),(85,88),(86,87),(91,98),(92,97),(93,96),(94,95),(99,100),(101,108),(102,107),(103,106),(104,105),(109,110),(111,112),(113,120),(114,119),(115,118),(116,117),(121,128),(122,127),(123,126),(124,125),(129,130),(131,133),(134,140),(135,139),(136,138),(141,147),(142,146),(143,145),(148,150),(151,153),(154,160),(155,159),(156,158),(161,163),(164,170),(165,169),(166,168),(171,177),(172,176),(173,175),(178,180),(181,183),(184,190),(185,189),(186,188),(191,193),(194,200),(195,199),(196,198),(201,207),(202,206),(203,205),(208,210),(211,213),(214,220),(215,219),(216,218),(221,223),(224,230),(225,229),(226,228),(231,237),(232,236),(233,235),(238,240)], [(1,45,40,57,29,70),(2,46,31,58,30,61),(3,47,32,59,21,62),(4,48,33,60,22,63),(5,49,34,51,23,64),(6,50,35,52,24,65),(7,41,36,53,25,66),(8,42,37,54,26,67),(9,43,38,55,27,68),(10,44,39,56,28,69),(11,200,232,215,230,202),(12,191,233,216,221,203),(13,192,234,217,222,204),(14,193,235,218,223,205),(15,194,236,219,224,206),(16,195,237,220,225,207),(17,196,238,211,226,208),(18,197,239,212,227,209),(19,198,240,213,228,210),(20,199,231,214,229,201),(71,126,83,101,96,113),(72,127,84,102,97,114),(73,128,85,103,98,115),(74,129,86,104,99,116),(75,130,87,105,100,117),(76,121,88,106,91,118),(77,122,89,107,92,119),(78,123,90,108,93,120),(79,124,81,109,94,111),(80,125,82,110,95,112),(131,173,156,161,143,186),(132,174,157,162,144,187),(133,175,158,163,145,188),(134,176,159,164,146,189),(135,177,160,165,147,190),(136,178,151,166,148,181),(137,179,152,167,149,182),(138,180,153,168,150,183),(139,171,154,169,141,184),(140,172,155,170,142,185)], [(1,177,57,147),(2,178,58,148),(3,179,59,149),(4,180,60,150),(5,171,51,141),(6,172,52,142),(7,173,53,143),(8,174,54,144),(9,175,55,145),(10,176,56,146),(11,125,215,95),(12,126,216,96),(13,127,217,97),(14,128,218,98),(15,129,219,99),(16,130,220,100),(17,121,211,91),(18,122,212,92),(19,123,213,93),(20,124,214,94),(21,167,47,137),(22,168,48,138),(23,169,49,139),(24,170,50,140),(25,161,41,131),(26,162,42,132),(27,163,43,133),(28,164,44,134),(29,165,45,135),(30,166,46,136),(31,181,61,151),(32,182,62,152),(33,183,63,153),(34,184,64,154),(35,185,65,155),(36,186,66,156),(37,187,67,157),(38,188,68,158),(39,189,69,159),(40,190,70,160),(71,221,101,191),(72,222,102,192),(73,223,103,193),(74,224,104,194),(75,225,105,195),(76,226,106,196),(77,227,107,197),(78,228,108,198),(79,229,109,199),(80,230,110,200),(81,231,111,201),(82,232,112,202),(83,233,113,203),(84,234,114,204),(85,235,115,205),(86,236,116,206),(87,237,117,207),(88,238,118,208),(89,239,119,209),(90,240,120,210)]])
84 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6G | 6H | ··· | 6O | 10A | ··· | 10N | 15A | 15B | 20A | ··· | 20P | 30A | ··· | 30N |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 |
| size | 1 | 1 | ··· | 1 | 10 | 10 | 10 | 10 | 2 | 6 | 6 | 6 | 6 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | ··· | 2 | 4 | 4 | 6 | ··· | 6 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | + | - | - | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | Dic3 | D6 | D6 | D10 | D10 | C3⋊D4 | C4×D5 | D20 | C5⋊D4 | S3×D5 | D5×Dic3 | C15⋊D4 | C3⋊D20 | C2×S3×D5 |
| kernel | C2×D10⋊Dic3 | D10⋊Dic3 | Dic3×C2×C10 | C22×Dic15 | D5×C22×C6 | D5×C2×C6 | C23×D5 | C2×C30 | C22×Dic3 | C22×D5 | C22×D5 | C22×C10 | C2×Dic3 | C22×C6 | C2×C10 | C2×C6 | C2×C6 | C2×C6 | C23 | C22 | C22 | C22 | C22 |
| # reps | 1 | 4 | 1 | 1 | 1 | 8 | 1 | 4 | 2 | 4 | 2 | 1 | 4 | 2 | 8 | 8 | 8 | 8 | 2 | 4 | 4 | 4 | 2 |
Matrix representation of C2×D10⋊Dic3 ►in GL6(𝔽61)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 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 |
| 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 | 43 |
| 0 | 0 | 0 | 0 | 18 | 18 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 43 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 13 | 0 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 47 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 0 |
| 0 | 0 | 0 | 0 | 0 | 50 |
G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,18,0,0,0,0,43,18],[60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,43,1],[13,0,0,0,0,0,0,47,0,0,0,0,0,0,47,0,0,0,0,0,0,13,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,50,0,0,0,0,0,0,50] >;
C2×D10⋊Dic3 in GAP, Magma, Sage, TeX
C_2\times D_{10}\rtimes {\rm Dic}_3 % in TeX
G:=Group("C2xD10:Dic3"); // GroupNames label
G:=SmallGroup(480,611);
// by ID
G=gap.SmallGroup(480,611);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,64,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^10=c^2=d^6=1,e^2=d^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^5*c,e*d*e^-1=d^-1>;
// generators/relations