direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×Dic3.D10, C30.44C24, D30.22C23, Dic15.24C23, C3⋊D4⋊14D10, C15⋊Q8⋊17C22, C10⋊4(C4○D12), C30⋊10(C4○D4), C6⋊3(D4⋊2D5), (C2×Dic5)⋊23D6, C6.44(C23×D5), C23.47(S3×D5), C15⋊7D4⋊18C22, C5⋊D12⋊18C22, C10.44(S3×C23), D6.20(C22×D5), (C22×C6).99D10, (C22×C10).84D6, (S3×C10).22C23, (C2×C30).247C23, D30.C2⋊16C22, (C22×Dic5)⋊11S3, (C6×Dic5)⋊29C22, (S3×Dic5)⋊16C22, (C22×S3).62D10, (C2×Dic3).133D10, (C22×C30).85C22, (C5×Dic3).23C23, Dic5.46(C22×S3), Dic3.22(C22×D5), (C3×Dic5).51C23, (C22×D15).76C22, (C2×Dic15).154C22, (C10×Dic3).133C22, C5⋊5(C2×C4○D12), C15⋊19(C2×C4○D4), C3⋊4(C2×D4⋊2D5), (C2×C15⋊Q8)⋊25C2, (C2×C6×Dic5)⋊9C2, (C2×C3⋊D4)⋊14D5, (C2×S3×Dic5)⋊23C2, (C10×C3⋊D4)⋊12C2, (C2×C5⋊D12)⋊21C2, (C2×C15⋊7D4)⋊23C2, C2.45(C22×S3×D5), C22.11(C2×S3×D5), (C2×D30.C2)⋊23C2, (C5×C3⋊D4)⋊13C22, (S3×C2×C10).62C22, (C2×C10).15(C22×S3), (C2×C6).253(C22×D5), SmallGroup(480,1116)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×Dic3.D10
G = < a,b,c,d,e | a2=b6=d10=1, c2=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=dbd-1=ebe-1=b-1, dcd-1=ece-1=b3c, ede-1=d-1 >
Subgroups: 1500 in 328 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C5, S3, C6, C6, C6, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C15, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C2×C10, C2×C10, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, D15, C30, C30, C30, C2×C4○D4, Dic10, C4×D5, C2×Dic5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C22×C10, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, S3×C10, S3×C10, D30, D30, C2×C30, C2×C30, C2×C30, C2×Dic10, C2×C4×D5, D4⋊2D5, C22×Dic5, C22×Dic5, C2×C5⋊D4, D4×C10, C2×C4○D12, S3×Dic5, D30.C2, C5⋊D12, C15⋊Q8, C6×Dic5, C6×Dic5, C10×Dic3, C5×C3⋊D4, C2×Dic15, C15⋊7D4, S3×C2×C10, C22×D15, C22×C30, C2×D4⋊2D5, C2×S3×Dic5, Dic3.D10, C2×D30.C2, C2×C5⋊D12, C2×C15⋊Q8, C2×C6×Dic5, C10×C3⋊D4, C2×C15⋊7D4, C2×Dic3.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, D4⋊2D5, C23×D5, C2×C4○D12, C2×S3×D5, C2×D4⋊2D5, Dic3.D10, C22×S3×D5, C2×Dic3.D10
►On 240 points
Generators in S240
(1 11)(2 12)(3 13)(4 14)(5 15)(6 20)(7 16)(8 17)(9 18)(10 19)(21 36)(22 37)(23 38)(24 39)(25 40)(26 33)(27 34)(28 35)(29 31)(30 32)(41 80)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 164)(52 165)(53 166)(54 167)(55 168)(56 169)(57 170)(58 161)(59 162)(60 163)(61 148)(62 149)(63 150)(64 141)(65 142)(66 143)(67 144)(68 145)(69 146)(70 147)(81 158)(82 159)(83 160)(84 151)(85 152)(86 153)(87 154)(88 155)(89 156)(90 157)(91 120)(92 111)(93 112)(94 113)(95 114)(96 115)(97 116)(98 117)(99 118)(100 119)(101 191)(102 192)(103 193)(104 194)(105 195)(106 196)(107 197)(108 198)(109 199)(110 200)(121 206)(122 207)(123 208)(124 209)(125 210)(126 201)(127 202)(128 203)(129 204)(130 205)(131 221)(132 222)(133 223)(134 224)(135 225)(136 226)(137 227)(138 228)(139 229)(140 230)(171 231)(172 232)(173 233)(174 234)(175 235)(176 236)(177 237)(178 238)(179 239)(180 240)(181 213)(182 214)(183 215)(184 216)(185 217)(186 218)(187 219)(188 220)(189 211)(190 212)
(1 75 235 17 240 80)(2 71 231 18 236 76)(3 77 237 19 232 72)(4 73 233 20 238 78)(5 79 239 16 234 74)(6 178 49 14 44 173)(7 174 45 15 50 179)(8 180 41 11 46 175)(9 176 47 12 42 171)(10 172 43 13 48 177)(21 115 105 31 110 120)(22 111 101 32 106 116)(23 117 107 33 102 112)(24 113 103 34 108 118)(25 119 109 35 104 114)(26 192 93 38 98 197)(27 198 99 39 94 193)(28 194 95 40 100 199)(29 200 91 36 96 195)(30 196 97 37 92 191)(51 89 84 56 62 67)(52 68 63 57 85 90)(53 81 86 58 64 69)(54 70 65 59 87 82)(55 83 88 60 66 61)(121 136 186 181 131 126)(122 127 132 182 187 137)(123 138 188 183 133 128)(124 129 134 184 189 139)(125 140 190 185 135 130)(141 146 166 158 153 161)(142 162 154 159 167 147)(143 148 168 160 155 163)(144 164 156 151 169 149)(145 150 170 152 157 165)(201 206 226 218 213 221)(202 222 214 219 227 207)(203 208 228 220 215 223)(204 224 216 211 229 209)(205 210 230 212 217 225)
(1 135 17 140)(2 131 18 136)(3 137 19 132)(4 133 20 138)(5 139 16 134)(6 228 14 223)(7 224 15 229)(8 230 11 225)(9 226 12 221)(10 222 13 227)(21 165 31 170)(22 161 32 166)(23 167 33 162)(24 163 34 168)(25 169 35 164)(26 59 38 54)(27 55 39 60)(28 51 40 56)(29 57 36 52)(30 53 37 58)(41 205 175 212)(42 213 176 206)(43 207 177 214)(44 215 178 208)(45 209 179 216)(46 217 180 210)(47 201 171 218)(48 219 172 202)(49 203 173 220)(50 211 174 204)(61 94 88 198)(62 199 89 95)(63 96 90 200)(64 191 81 97)(65 98 82 192)(66 193 83 99)(67 100 84 194)(68 195 85 91)(69 92 86 196)(70 197 87 93)(71 181 236 121)(72 122 237 182)(73 183 238 123)(74 124 239 184)(75 185 240 125)(76 126 231 186)(77 187 232 127)(78 128 233 188)(79 189 234 129)(80 130 235 190)(101 158 116 141)(102 142 117 159)(103 160 118 143)(104 144 119 151)(105 152 120 145)(106 146 111 153)(107 154 112 147)(108 148 113 155)(109 156 114 149)(110 150 115 157)
(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 40 17 28)(2 39 18 27)(3 38 19 26)(4 37 20 30)(5 36 16 29)(6 32 14 22)(7 31 15 21)(8 35 11 25)(9 34 12 24)(10 33 13 23)(41 119 175 104)(42 118 176 103)(43 117 177 102)(44 116 178 101)(45 115 179 110)(46 114 180 109)(47 113 171 108)(48 112 172 107)(49 111 173 106)(50 120 174 105)(51 140 56 135)(52 139 57 134)(53 138 58 133)(54 137 59 132)(55 136 60 131)(61 186 88 126)(62 185 89 125)(63 184 90 124)(64 183 81 123)(65 182 82 122)(66 181 83 121)(67 190 84 130)(68 189 85 129)(69 188 86 128)(70 187 87 127)(71 99 236 193)(72 98 237 192)(73 97 238 191)(74 96 239 200)(75 95 240 199)(76 94 231 198)(77 93 232 197)(78 92 233 196)(79 91 234 195)(80 100 235 194)(141 215 158 208)(142 214 159 207)(143 213 160 206)(144 212 151 205)(145 211 152 204)(146 220 153 203)(147 219 154 202)(148 218 155 201)(149 217 156 210)(150 216 157 209)(161 223 166 228)(162 222 167 227)(163 221 168 226)(164 230 169 225)(165 229 170 224)
G:=sub<Sym(240)| (1,11)(2,12)(3,13)(4,14)(5,15)(6,20)(7,16)(8,17)(9,18)(10,19)(21,36)(22,37)(23,38)(24,39)(25,40)(26,33)(27,34)(28,35)(29,31)(30,32)(41,80)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,161)(59,162)(60,163)(61,148)(62,149)(63,150)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(81,158)(82,159)(83,160)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,120)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(121,206)(122,207)(123,208)(124,209)(125,210)(126,201)(127,202)(128,203)(129,204)(130,205)(131,221)(132,222)(133,223)(134,224)(135,225)(136,226)(137,227)(138,228)(139,229)(140,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240)(181,213)(182,214)(183,215)(184,216)(185,217)(186,218)(187,219)(188,220)(189,211)(190,212), (1,75,235,17,240,80)(2,71,231,18,236,76)(3,77,237,19,232,72)(4,73,233,20,238,78)(5,79,239,16,234,74)(6,178,49,14,44,173)(7,174,45,15,50,179)(8,180,41,11,46,175)(9,176,47,12,42,171)(10,172,43,13,48,177)(21,115,105,31,110,120)(22,111,101,32,106,116)(23,117,107,33,102,112)(24,113,103,34,108,118)(25,119,109,35,104,114)(26,192,93,38,98,197)(27,198,99,39,94,193)(28,194,95,40,100,199)(29,200,91,36,96,195)(30,196,97,37,92,191)(51,89,84,56,62,67)(52,68,63,57,85,90)(53,81,86,58,64,69)(54,70,65,59,87,82)(55,83,88,60,66,61)(121,136,186,181,131,126)(122,127,132,182,187,137)(123,138,188,183,133,128)(124,129,134,184,189,139)(125,140,190,185,135,130)(141,146,166,158,153,161)(142,162,154,159,167,147)(143,148,168,160,155,163)(144,164,156,151,169,149)(145,150,170,152,157,165)(201,206,226,218,213,221)(202,222,214,219,227,207)(203,208,228,220,215,223)(204,224,216,211,229,209)(205,210,230,212,217,225), (1,135,17,140)(2,131,18,136)(3,137,19,132)(4,133,20,138)(5,139,16,134)(6,228,14,223)(7,224,15,229)(8,230,11,225)(9,226,12,221)(10,222,13,227)(21,165,31,170)(22,161,32,166)(23,167,33,162)(24,163,34,168)(25,169,35,164)(26,59,38,54)(27,55,39,60)(28,51,40,56)(29,57,36,52)(30,53,37,58)(41,205,175,212)(42,213,176,206)(43,207,177,214)(44,215,178,208)(45,209,179,216)(46,217,180,210)(47,201,171,218)(48,219,172,202)(49,203,173,220)(50,211,174,204)(61,94,88,198)(62,199,89,95)(63,96,90,200)(64,191,81,97)(65,98,82,192)(66,193,83,99)(67,100,84,194)(68,195,85,91)(69,92,86,196)(70,197,87,93)(71,181,236,121)(72,122,237,182)(73,183,238,123)(74,124,239,184)(75,185,240,125)(76,126,231,186)(77,187,232,127)(78,128,233,188)(79,189,234,129)(80,130,235,190)(101,158,116,141)(102,142,117,159)(103,160,118,143)(104,144,119,151)(105,152,120,145)(106,146,111,153)(107,154,112,147)(108,148,113,155)(109,156,114,149)(110,150,115,157), (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), (1,40,17,28)(2,39,18,27)(3,38,19,26)(4,37,20,30)(5,36,16,29)(6,32,14,22)(7,31,15,21)(8,35,11,25)(9,34,12,24)(10,33,13,23)(41,119,175,104)(42,118,176,103)(43,117,177,102)(44,116,178,101)(45,115,179,110)(46,114,180,109)(47,113,171,108)(48,112,172,107)(49,111,173,106)(50,120,174,105)(51,140,56,135)(52,139,57,134)(53,138,58,133)(54,137,59,132)(55,136,60,131)(61,186,88,126)(62,185,89,125)(63,184,90,124)(64,183,81,123)(65,182,82,122)(66,181,83,121)(67,190,84,130)(68,189,85,129)(69,188,86,128)(70,187,87,127)(71,99,236,193)(72,98,237,192)(73,97,238,191)(74,96,239,200)(75,95,240,199)(76,94,231,198)(77,93,232,197)(78,92,233,196)(79,91,234,195)(80,100,235,194)(141,215,158,208)(142,214,159,207)(143,213,160,206)(144,212,151,205)(145,211,152,204)(146,220,153,203)(147,219,154,202)(148,218,155,201)(149,217,156,210)(150,216,157,209)(161,223,166,228)(162,222,167,227)(163,221,168,226)(164,230,169,225)(165,229,170,224)>;
G:=Group( (1,11)(2,12)(3,13)(4,14)(5,15)(6,20)(7,16)(8,17)(9,18)(10,19)(21,36)(22,37)(23,38)(24,39)(25,40)(26,33)(27,34)(28,35)(29,31)(30,32)(41,80)(42,71)(43,72)(44,73)(45,74)(46,75)(47,76)(48,77)(49,78)(50,79)(51,164)(52,165)(53,166)(54,167)(55,168)(56,169)(57,170)(58,161)(59,162)(60,163)(61,148)(62,149)(63,150)(64,141)(65,142)(66,143)(67,144)(68,145)(69,146)(70,147)(81,158)(82,159)(83,160)(84,151)(85,152)(86,153)(87,154)(88,155)(89,156)(90,157)(91,120)(92,111)(93,112)(94,113)(95,114)(96,115)(97,116)(98,117)(99,118)(100,119)(101,191)(102,192)(103,193)(104,194)(105,195)(106,196)(107,197)(108,198)(109,199)(110,200)(121,206)(122,207)(123,208)(124,209)(125,210)(126,201)(127,202)(128,203)(129,204)(130,205)(131,221)(132,222)(133,223)(134,224)(135,225)(136,226)(137,227)(138,228)(139,229)(140,230)(171,231)(172,232)(173,233)(174,234)(175,235)(176,236)(177,237)(178,238)(179,239)(180,240)(181,213)(182,214)(183,215)(184,216)(185,217)(186,218)(187,219)(188,220)(189,211)(190,212), (1,75,235,17,240,80)(2,71,231,18,236,76)(3,77,237,19,232,72)(4,73,233,20,238,78)(5,79,239,16,234,74)(6,178,49,14,44,173)(7,174,45,15,50,179)(8,180,41,11,46,175)(9,176,47,12,42,171)(10,172,43,13,48,177)(21,115,105,31,110,120)(22,111,101,32,106,116)(23,117,107,33,102,112)(24,113,103,34,108,118)(25,119,109,35,104,114)(26,192,93,38,98,197)(27,198,99,39,94,193)(28,194,95,40,100,199)(29,200,91,36,96,195)(30,196,97,37,92,191)(51,89,84,56,62,67)(52,68,63,57,85,90)(53,81,86,58,64,69)(54,70,65,59,87,82)(55,83,88,60,66,61)(121,136,186,181,131,126)(122,127,132,182,187,137)(123,138,188,183,133,128)(124,129,134,184,189,139)(125,140,190,185,135,130)(141,146,166,158,153,161)(142,162,154,159,167,147)(143,148,168,160,155,163)(144,164,156,151,169,149)(145,150,170,152,157,165)(201,206,226,218,213,221)(202,222,214,219,227,207)(203,208,228,220,215,223)(204,224,216,211,229,209)(205,210,230,212,217,225), (1,135,17,140)(2,131,18,136)(3,137,19,132)(4,133,20,138)(5,139,16,134)(6,228,14,223)(7,224,15,229)(8,230,11,225)(9,226,12,221)(10,222,13,227)(21,165,31,170)(22,161,32,166)(23,167,33,162)(24,163,34,168)(25,169,35,164)(26,59,38,54)(27,55,39,60)(28,51,40,56)(29,57,36,52)(30,53,37,58)(41,205,175,212)(42,213,176,206)(43,207,177,214)(44,215,178,208)(45,209,179,216)(46,217,180,210)(47,201,171,218)(48,219,172,202)(49,203,173,220)(50,211,174,204)(61,94,88,198)(62,199,89,95)(63,96,90,200)(64,191,81,97)(65,98,82,192)(66,193,83,99)(67,100,84,194)(68,195,85,91)(69,92,86,196)(70,197,87,93)(71,181,236,121)(72,122,237,182)(73,183,238,123)(74,124,239,184)(75,185,240,125)(76,126,231,186)(77,187,232,127)(78,128,233,188)(79,189,234,129)(80,130,235,190)(101,158,116,141)(102,142,117,159)(103,160,118,143)(104,144,119,151)(105,152,120,145)(106,146,111,153)(107,154,112,147)(108,148,113,155)(109,156,114,149)(110,150,115,157), (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), (1,40,17,28)(2,39,18,27)(3,38,19,26)(4,37,20,30)(5,36,16,29)(6,32,14,22)(7,31,15,21)(8,35,11,25)(9,34,12,24)(10,33,13,23)(41,119,175,104)(42,118,176,103)(43,117,177,102)(44,116,178,101)(45,115,179,110)(46,114,180,109)(47,113,171,108)(48,112,172,107)(49,111,173,106)(50,120,174,105)(51,140,56,135)(52,139,57,134)(53,138,58,133)(54,137,59,132)(55,136,60,131)(61,186,88,126)(62,185,89,125)(63,184,90,124)(64,183,81,123)(65,182,82,122)(66,181,83,121)(67,190,84,130)(68,189,85,129)(69,188,86,128)(70,187,87,127)(71,99,236,193)(72,98,237,192)(73,97,238,191)(74,96,239,200)(75,95,240,199)(76,94,231,198)(77,93,232,197)(78,92,233,196)(79,91,234,195)(80,100,235,194)(141,215,158,208)(142,214,159,207)(143,213,160,206)(144,212,151,205)(145,211,152,204)(146,220,153,203)(147,219,154,202)(148,218,155,201)(149,217,156,210)(150,216,157,209)(161,223,166,228)(162,222,167,227)(163,221,168,226)(164,230,169,225)(165,229,170,224) );
G=PermutationGroup([[(1,11),(2,12),(3,13),(4,14),(5,15),(6,20),(7,16),(8,17),(9,18),(10,19),(21,36),(22,37),(23,38),(24,39),(25,40),(26,33),(27,34),(28,35),(29,31),(30,32),(41,80),(42,71),(43,72),(44,73),(45,74),(46,75),(47,76),(48,77),(49,78),(50,79),(51,164),(52,165),(53,166),(54,167),(55,168),(56,169),(57,170),(58,161),(59,162),(60,163),(61,148),(62,149),(63,150),(64,141),(65,142),(66,143),(67,144),(68,145),(69,146),(70,147),(81,158),(82,159),(83,160),(84,151),(85,152),(86,153),(87,154),(88,155),(89,156),(90,157),(91,120),(92,111),(93,112),(94,113),(95,114),(96,115),(97,116),(98,117),(99,118),(100,119),(101,191),(102,192),(103,193),(104,194),(105,195),(106,196),(107,197),(108,198),(109,199),(110,200),(121,206),(122,207),(123,208),(124,209),(125,210),(126,201),(127,202),(128,203),(129,204),(130,205),(131,221),(132,222),(133,223),(134,224),(135,225),(136,226),(137,227),(138,228),(139,229),(140,230),(171,231),(172,232),(173,233),(174,234),(175,235),(176,236),(177,237),(178,238),(179,239),(180,240),(181,213),(182,214),(183,215),(184,216),(185,217),(186,218),(187,219),(188,220),(189,211),(190,212)], [(1,75,235,17,240,80),(2,71,231,18,236,76),(3,77,237,19,232,72),(4,73,233,20,238,78),(5,79,239,16,234,74),(6,178,49,14,44,173),(7,174,45,15,50,179),(8,180,41,11,46,175),(9,176,47,12,42,171),(10,172,43,13,48,177),(21,115,105,31,110,120),(22,111,101,32,106,116),(23,117,107,33,102,112),(24,113,103,34,108,118),(25,119,109,35,104,114),(26,192,93,38,98,197),(27,198,99,39,94,193),(28,194,95,40,100,199),(29,200,91,36,96,195),(30,196,97,37,92,191),(51,89,84,56,62,67),(52,68,63,57,85,90),(53,81,86,58,64,69),(54,70,65,59,87,82),(55,83,88,60,66,61),(121,136,186,181,131,126),(122,127,132,182,187,137),(123,138,188,183,133,128),(124,129,134,184,189,139),(125,140,190,185,135,130),(141,146,166,158,153,161),(142,162,154,159,167,147),(143,148,168,160,155,163),(144,164,156,151,169,149),(145,150,170,152,157,165),(201,206,226,218,213,221),(202,222,214,219,227,207),(203,208,228,220,215,223),(204,224,216,211,229,209),(205,210,230,212,217,225)], [(1,135,17,140),(2,131,18,136),(3,137,19,132),(4,133,20,138),(5,139,16,134),(6,228,14,223),(7,224,15,229),(8,230,11,225),(9,226,12,221),(10,222,13,227),(21,165,31,170),(22,161,32,166),(23,167,33,162),(24,163,34,168),(25,169,35,164),(26,59,38,54),(27,55,39,60),(28,51,40,56),(29,57,36,52),(30,53,37,58),(41,205,175,212),(42,213,176,206),(43,207,177,214),(44,215,178,208),(45,209,179,216),(46,217,180,210),(47,201,171,218),(48,219,172,202),(49,203,173,220),(50,211,174,204),(61,94,88,198),(62,199,89,95),(63,96,90,200),(64,191,81,97),(65,98,82,192),(66,193,83,99),(67,100,84,194),(68,195,85,91),(69,92,86,196),(70,197,87,93),(71,181,236,121),(72,122,237,182),(73,183,238,123),(74,124,239,184),(75,185,240,125),(76,126,231,186),(77,187,232,127),(78,128,233,188),(79,189,234,129),(80,130,235,190),(101,158,116,141),(102,142,117,159),(103,160,118,143),(104,144,119,151),(105,152,120,145),(106,146,111,153),(107,154,112,147),(108,148,113,155),(109,156,114,149),(110,150,115,157)], [(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)], [(1,40,17,28),(2,39,18,27),(3,38,19,26),(4,37,20,30),(5,36,16,29),(6,32,14,22),(7,31,15,21),(8,35,11,25),(9,34,12,24),(10,33,13,23),(41,119,175,104),(42,118,176,103),(43,117,177,102),(44,116,178,101),(45,115,179,110),(46,114,180,109),(47,113,171,108),(48,112,172,107),(49,111,173,106),(50,120,174,105),(51,140,56,135),(52,139,57,134),(53,138,58,133),(54,137,59,132),(55,136,60,131),(61,186,88,126),(62,185,89,125),(63,184,90,124),(64,183,81,123),(65,182,82,122),(66,181,83,121),(67,190,84,130),(68,189,85,129),(69,188,86,128),(70,187,87,127),(71,99,236,193),(72,98,237,192),(73,97,238,191),(74,96,239,200),(75,95,240,199),(76,94,231,198),(77,93,232,197),(78,92,233,196),(79,91,234,195),(80,100,235,194),(141,215,158,208),(142,214,159,207),(143,213,160,206),(144,212,151,205),(145,211,152,204),(146,220,153,203),(147,219,154,202),(148,218,155,201),(149,217,156,210),(150,216,157,209),(161,223,166,228),(162,222,167,227),(163,221,168,226),(164,230,169,225),(165,229,170,224)]])
72 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 | ··· | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 12A | ··· | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 30A | ··· | 30N |
| 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 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 30 | 30 | 2 | 5 | 5 | 5 | 5 | 6 | 6 | 10 | 10 | 30 | 30 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | C4○D4 | D10 | D10 | D10 | D10 | C4○D12 | S3×D5 | D4⋊2D5 | C2×S3×D5 | Dic3.D10 |
| kernel | C2×Dic3.D10 | C2×S3×Dic5 | Dic3.D10 | C2×D30.C2 | C2×C5⋊D12 | C2×C15⋊Q8 | C2×C6×Dic5 | C10×C3⋊D4 | C2×C15⋊7D4 | C22×Dic5 | C2×C3⋊D4 | C2×Dic5 | C22×C10 | C30 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C10 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 6 | 1 | 4 | 2 | 8 | 2 | 2 | 8 | 2 | 4 | 6 | 8 |
Matrix representation of C2×Dic3.D10 ►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 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 60 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 52 | 43 | 0 | 0 | 0 | 0 |
| 52 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 60 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 17 | 43 |
| 0 | 0 | 0 | 0 | 17 | 0 |
| 11 | 0 | 0 | 0 | 0 | 0 |
| 50 | 50 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 60 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
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],[1,60,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,52,0,0,0,0,43,9,0,0,0,0,0,0,60,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,60,0,0,0,0,0,60,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,17,17,0,0,0,0,43,0],[11,50,0,0,0,0,0,50,0,0,0,0,0,0,1,60,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,1,1] >;
C2×Dic3.D10 in GAP, Magma, Sage, TeX
C_2\times {\rm Dic}_3.D_{10} % in TeX
G:=Group("C2xDic3.D10"); // GroupNames label
G:=SmallGroup(480,1116);
// by ID
G=gap.SmallGroup(480,1116);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=d^10=1,c^2=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations