metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10⋊1D12, D6⋊C4⋊3D5, (C6×D5)⋊9D4, C6.21(D4×D5), C15⋊5(C4⋊D4), C5⋊2(C12⋊7D4), C30.58(C2×D4), C2.23(D5×D12), C3⋊4(D10⋊D4), (C3×Dic5)⋊13D4, (C2×C20).202D6, C10.22(C2×D12), D30⋊3C4⋊27C2, C30.83(C4○D4), C6.38(C4○D20), (C2×C12).269D10, Dic5⋊4(C3⋊D4), C30.Q8⋊25C2, (C22×D5).90D6, C10.41(C4○D12), (C2×C60).392C22, (C2×C30).138C23, (C2×Dic5).182D6, (C2×Dic3).43D10, (C22×S3).15D10, C2.27(D6.D10), (C6×Dic5).209C22, (C10×Dic3).86C22, (C22×D15).46C22, (C2×Dic15).107C22, (C2×C4×D5)⋊13S3, (D5×C2×C12)⋊21C2, (C5×D6⋊C4)⋊28C2, (C2×C15⋊D4)⋊2C2, (C2×C5⋊D12)⋊3C2, (C2×C3⋊D20)⋊2C2, C2.18(D5×C3⋊D4), (C2×C4).132(S3×D5), C10.38(C2×C3⋊D4), C22.190(C2×S3×D5), (S3×C2×C10).30C22, (D5×C2×C6).106C22, (C2×C6).150(C22×D5), (C2×C10).150(C22×S3), SmallGroup(480,524)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10⋊D12
G = < a,b,c,d | a10=b2=c12=d2=1, bab=cac-1=dad=a-1, cbc-1=a8b, dbd=a3b, dcd=c-1 >
Subgroups: 1148 in 188 conjugacy classes, 52 normal (44 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, Dic5, C20, D10, D10, C2×C10, C2×C10, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C5×S3, C3×D5, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, D6⋊C4, C2×D12, C2×C3⋊D4, C22×C12, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, D30, C2×C30, C10.D4, D10⋊C4, C5×C22⋊C4, C2×C4×D5, C2×D20, C2×C5⋊D4, C12⋊7D4, C15⋊D4, C3⋊D20, C5⋊D12, D5×C12, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C22×D15, D10⋊D4, C30.Q8, C5×D6⋊C4, D30⋊3C4, C2×C15⋊D4, C2×C3⋊D20, C2×C5⋊D12, D5×C2×C12, D10⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C3⋊D4, C22×S3, C4⋊D4, C22×D5, C2×D12, C4○D12, C2×C3⋊D4, S3×D5, C4○D20, D4×D5, C12⋊7D4, C2×S3×D5, D10⋊D4, D6.D10, D5×D12, D5×C3⋊D4, D10⋊D12
►On 240 points
Generators in S240
(1 83 197 163 52 106 180 224 210 192)(2 181 211 225 169 107 53 164 198 84)(3 73 199 165 54 108 170 226 212 182)(4 183 213 227 171 97 55 166 200 74)(5 75 201 167 56 98 172 228 214 184)(6 185 215 217 173 99 57 168 202 76)(7 77 203 157 58 100 174 218 216 186)(8 187 205 219 175 101 59 158 204 78)(9 79 193 159 60 102 176 220 206 188)(10 189 207 221 177 103 49 160 194 80)(11 81 195 161 50 104 178 222 208 190)(12 191 209 223 179 105 51 162 196 82)(13 232 148 34 88 61 131 111 39 143)(14 144 40 112 132 62 89 35 149 233)(15 234 150 36 90 63 121 113 41 133)(16 134 42 114 122 64 91 25 151 235)(17 236 152 26 92 65 123 115 43 135)(18 136 44 116 124 66 93 27 153 237)(19 238 154 28 94 67 125 117 45 137)(20 138 46 118 126 68 95 29 155 239)(21 240 156 30 96 69 127 119 47 139)(22 140 48 120 128 70 85 31 145 229)(23 230 146 32 86 71 129 109 37 141)(24 142 38 110 130 72 87 33 147 231)
(1 58)(2 175)(3 60)(4 177)(5 50)(6 179)(7 52)(8 169)(9 54)(10 171)(11 56)(12 173)(13 28)(14 68)(15 30)(16 70)(17 32)(18 72)(19 34)(20 62)(21 36)(22 64)(23 26)(24 66)(25 145)(27 147)(29 149)(31 151)(33 153)(35 155)(37 65)(38 116)(39 67)(40 118)(41 69)(42 120)(43 71)(44 110)(45 61)(46 112)(47 63)(48 114)(49 200)(51 202)(53 204)(55 194)(57 196)(59 198)(73 159)(74 103)(75 161)(76 105)(77 163)(78 107)(79 165)(80 97)(81 167)(82 99)(83 157)(84 101)(85 235)(86 135)(87 237)(88 137)(89 239)(90 139)(91 229)(92 141)(93 231)(94 143)(95 233)(96 133)(98 190)(100 192)(102 182)(104 184)(106 186)(108 188)(109 123)(111 125)(113 127)(115 129)(117 131)(119 121)(122 140)(124 142)(126 144)(128 134)(130 136)(132 138)(146 236)(148 238)(150 240)(152 230)(154 232)(156 234)(158 164)(160 166)(162 168)(170 206)(172 208)(174 210)(176 212)(178 214)(180 216)(181 219)(183 221)(185 223)(187 225)(189 227)(191 217)(193 199)(195 201)(197 203)(205 211)(207 213)(209 215)(218 224)(220 226)(222 228)
(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)
(1 150)(2 149)(3 148)(4 147)(5 146)(6 145)(7 156)(8 155)(9 154)(10 153)(11 152)(12 151)(13 199)(14 198)(15 197)(16 196)(17 195)(18 194)(19 193)(20 204)(21 203)(22 202)(23 201)(24 200)(25 191)(26 190)(27 189)(28 188)(29 187)(30 186)(31 185)(32 184)(33 183)(34 182)(35 181)(36 192)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 60)(46 59)(47 58)(48 57)(61 226)(62 225)(63 224)(64 223)(65 222)(66 221)(67 220)(68 219)(69 218)(70 217)(71 228)(72 227)(73 232)(74 231)(75 230)(76 229)(77 240)(78 239)(79 238)(80 237)(81 236)(82 235)(83 234)(84 233)(85 215)(86 214)(87 213)(88 212)(89 211)(90 210)(91 209)(92 208)(93 207)(94 206)(95 205)(96 216)(97 110)(98 109)(99 120)(100 119)(101 118)(102 117)(103 116)(104 115)(105 114)(106 113)(107 112)(108 111)(121 180)(122 179)(123 178)(124 177)(125 176)(126 175)(127 174)(128 173)(129 172)(130 171)(131 170)(132 169)(133 163)(134 162)(135 161)(136 160)(137 159)(138 158)(139 157)(140 168)(141 167)(142 166)(143 165)(144 164)
G:=sub<Sym(240)| (1,83,197,163,52,106,180,224,210,192)(2,181,211,225,169,107,53,164,198,84)(3,73,199,165,54,108,170,226,212,182)(4,183,213,227,171,97,55,166,200,74)(5,75,201,167,56,98,172,228,214,184)(6,185,215,217,173,99,57,168,202,76)(7,77,203,157,58,100,174,218,216,186)(8,187,205,219,175,101,59,158,204,78)(9,79,193,159,60,102,176,220,206,188)(10,189,207,221,177,103,49,160,194,80)(11,81,195,161,50,104,178,222,208,190)(12,191,209,223,179,105,51,162,196,82)(13,232,148,34,88,61,131,111,39,143)(14,144,40,112,132,62,89,35,149,233)(15,234,150,36,90,63,121,113,41,133)(16,134,42,114,122,64,91,25,151,235)(17,236,152,26,92,65,123,115,43,135)(18,136,44,116,124,66,93,27,153,237)(19,238,154,28,94,67,125,117,45,137)(20,138,46,118,126,68,95,29,155,239)(21,240,156,30,96,69,127,119,47,139)(22,140,48,120,128,70,85,31,145,229)(23,230,146,32,86,71,129,109,37,141)(24,142,38,110,130,72,87,33,147,231), (1,58)(2,175)(3,60)(4,177)(5,50)(6,179)(7,52)(8,169)(9,54)(10,171)(11,56)(12,173)(13,28)(14,68)(15,30)(16,70)(17,32)(18,72)(19,34)(20,62)(21,36)(22,64)(23,26)(24,66)(25,145)(27,147)(29,149)(31,151)(33,153)(35,155)(37,65)(38,116)(39,67)(40,118)(41,69)(42,120)(43,71)(44,110)(45,61)(46,112)(47,63)(48,114)(49,200)(51,202)(53,204)(55,194)(57,196)(59,198)(73,159)(74,103)(75,161)(76,105)(77,163)(78,107)(79,165)(80,97)(81,167)(82,99)(83,157)(84,101)(85,235)(86,135)(87,237)(88,137)(89,239)(90,139)(91,229)(92,141)(93,231)(94,143)(95,233)(96,133)(98,190)(100,192)(102,182)(104,184)(106,186)(108,188)(109,123)(111,125)(113,127)(115,129)(117,131)(119,121)(122,140)(124,142)(126,144)(128,134)(130,136)(132,138)(146,236)(148,238)(150,240)(152,230)(154,232)(156,234)(158,164)(160,166)(162,168)(170,206)(172,208)(174,210)(176,212)(178,214)(180,216)(181,219)(183,221)(185,223)(187,225)(189,227)(191,217)(193,199)(195,201)(197,203)(205,211)(207,213)(209,215)(218,224)(220,226)(222,228), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,156)(8,155)(9,154)(10,153)(11,152)(12,151)(13,199)(14,198)(15,197)(16,196)(17,195)(18,194)(19,193)(20,204)(21,203)(22,202)(23,201)(24,200)(25,191)(26,190)(27,189)(28,188)(29,187)(30,186)(31,185)(32,184)(33,183)(34,182)(35,181)(36,192)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(61,226)(62,225)(63,224)(64,223)(65,222)(66,221)(67,220)(68,219)(69,218)(70,217)(71,228)(72,227)(73,232)(74,231)(75,230)(76,229)(77,240)(78,239)(79,238)(80,237)(81,236)(82,235)(83,234)(84,233)(85,215)(86,214)(87,213)(88,212)(89,211)(90,210)(91,209)(92,208)(93,207)(94,206)(95,205)(96,216)(97,110)(98,109)(99,120)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,168)(141,167)(142,166)(143,165)(144,164)>;
G:=Group( (1,83,197,163,52,106,180,224,210,192)(2,181,211,225,169,107,53,164,198,84)(3,73,199,165,54,108,170,226,212,182)(4,183,213,227,171,97,55,166,200,74)(5,75,201,167,56,98,172,228,214,184)(6,185,215,217,173,99,57,168,202,76)(7,77,203,157,58,100,174,218,216,186)(8,187,205,219,175,101,59,158,204,78)(9,79,193,159,60,102,176,220,206,188)(10,189,207,221,177,103,49,160,194,80)(11,81,195,161,50,104,178,222,208,190)(12,191,209,223,179,105,51,162,196,82)(13,232,148,34,88,61,131,111,39,143)(14,144,40,112,132,62,89,35,149,233)(15,234,150,36,90,63,121,113,41,133)(16,134,42,114,122,64,91,25,151,235)(17,236,152,26,92,65,123,115,43,135)(18,136,44,116,124,66,93,27,153,237)(19,238,154,28,94,67,125,117,45,137)(20,138,46,118,126,68,95,29,155,239)(21,240,156,30,96,69,127,119,47,139)(22,140,48,120,128,70,85,31,145,229)(23,230,146,32,86,71,129,109,37,141)(24,142,38,110,130,72,87,33,147,231), (1,58)(2,175)(3,60)(4,177)(5,50)(6,179)(7,52)(8,169)(9,54)(10,171)(11,56)(12,173)(13,28)(14,68)(15,30)(16,70)(17,32)(18,72)(19,34)(20,62)(21,36)(22,64)(23,26)(24,66)(25,145)(27,147)(29,149)(31,151)(33,153)(35,155)(37,65)(38,116)(39,67)(40,118)(41,69)(42,120)(43,71)(44,110)(45,61)(46,112)(47,63)(48,114)(49,200)(51,202)(53,204)(55,194)(57,196)(59,198)(73,159)(74,103)(75,161)(76,105)(77,163)(78,107)(79,165)(80,97)(81,167)(82,99)(83,157)(84,101)(85,235)(86,135)(87,237)(88,137)(89,239)(90,139)(91,229)(92,141)(93,231)(94,143)(95,233)(96,133)(98,190)(100,192)(102,182)(104,184)(106,186)(108,188)(109,123)(111,125)(113,127)(115,129)(117,131)(119,121)(122,140)(124,142)(126,144)(128,134)(130,136)(132,138)(146,236)(148,238)(150,240)(152,230)(154,232)(156,234)(158,164)(160,166)(162,168)(170,206)(172,208)(174,210)(176,212)(178,214)(180,216)(181,219)(183,221)(185,223)(187,225)(189,227)(191,217)(193,199)(195,201)(197,203)(205,211)(207,213)(209,215)(218,224)(220,226)(222,228), (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), (1,150)(2,149)(3,148)(4,147)(5,146)(6,145)(7,156)(8,155)(9,154)(10,153)(11,152)(12,151)(13,199)(14,198)(15,197)(16,196)(17,195)(18,194)(19,193)(20,204)(21,203)(22,202)(23,201)(24,200)(25,191)(26,190)(27,189)(28,188)(29,187)(30,186)(31,185)(32,184)(33,183)(34,182)(35,181)(36,192)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,60)(46,59)(47,58)(48,57)(61,226)(62,225)(63,224)(64,223)(65,222)(66,221)(67,220)(68,219)(69,218)(70,217)(71,228)(72,227)(73,232)(74,231)(75,230)(76,229)(77,240)(78,239)(79,238)(80,237)(81,236)(82,235)(83,234)(84,233)(85,215)(86,214)(87,213)(88,212)(89,211)(90,210)(91,209)(92,208)(93,207)(94,206)(95,205)(96,216)(97,110)(98,109)(99,120)(100,119)(101,118)(102,117)(103,116)(104,115)(105,114)(106,113)(107,112)(108,111)(121,180)(122,179)(123,178)(124,177)(125,176)(126,175)(127,174)(128,173)(129,172)(130,171)(131,170)(132,169)(133,163)(134,162)(135,161)(136,160)(137,159)(138,158)(139,157)(140,168)(141,167)(142,166)(143,165)(144,164) );
G=PermutationGroup([[(1,83,197,163,52,106,180,224,210,192),(2,181,211,225,169,107,53,164,198,84),(3,73,199,165,54,108,170,226,212,182),(4,183,213,227,171,97,55,166,200,74),(5,75,201,167,56,98,172,228,214,184),(6,185,215,217,173,99,57,168,202,76),(7,77,203,157,58,100,174,218,216,186),(8,187,205,219,175,101,59,158,204,78),(9,79,193,159,60,102,176,220,206,188),(10,189,207,221,177,103,49,160,194,80),(11,81,195,161,50,104,178,222,208,190),(12,191,209,223,179,105,51,162,196,82),(13,232,148,34,88,61,131,111,39,143),(14,144,40,112,132,62,89,35,149,233),(15,234,150,36,90,63,121,113,41,133),(16,134,42,114,122,64,91,25,151,235),(17,236,152,26,92,65,123,115,43,135),(18,136,44,116,124,66,93,27,153,237),(19,238,154,28,94,67,125,117,45,137),(20,138,46,118,126,68,95,29,155,239),(21,240,156,30,96,69,127,119,47,139),(22,140,48,120,128,70,85,31,145,229),(23,230,146,32,86,71,129,109,37,141),(24,142,38,110,130,72,87,33,147,231)], [(1,58),(2,175),(3,60),(4,177),(5,50),(6,179),(7,52),(8,169),(9,54),(10,171),(11,56),(12,173),(13,28),(14,68),(15,30),(16,70),(17,32),(18,72),(19,34),(20,62),(21,36),(22,64),(23,26),(24,66),(25,145),(27,147),(29,149),(31,151),(33,153),(35,155),(37,65),(38,116),(39,67),(40,118),(41,69),(42,120),(43,71),(44,110),(45,61),(46,112),(47,63),(48,114),(49,200),(51,202),(53,204),(55,194),(57,196),(59,198),(73,159),(74,103),(75,161),(76,105),(77,163),(78,107),(79,165),(80,97),(81,167),(82,99),(83,157),(84,101),(85,235),(86,135),(87,237),(88,137),(89,239),(90,139),(91,229),(92,141),(93,231),(94,143),(95,233),(96,133),(98,190),(100,192),(102,182),(104,184),(106,186),(108,188),(109,123),(111,125),(113,127),(115,129),(117,131),(119,121),(122,140),(124,142),(126,144),(128,134),(130,136),(132,138),(146,236),(148,238),(150,240),(152,230),(154,232),(156,234),(158,164),(160,166),(162,168),(170,206),(172,208),(174,210),(176,212),(178,214),(180,216),(181,219),(183,221),(185,223),(187,225),(189,227),(191,217),(193,199),(195,201),(197,203),(205,211),(207,213),(209,215),(218,224),(220,226),(222,228)], [(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)], [(1,150),(2,149),(3,148),(4,147),(5,146),(6,145),(7,156),(8,155),(9,154),(10,153),(11,152),(12,151),(13,199),(14,198),(15,197),(16,196),(17,195),(18,194),(19,193),(20,204),(21,203),(22,202),(23,201),(24,200),(25,191),(26,190),(27,189),(28,188),(29,187),(30,186),(31,185),(32,184),(33,183),(34,182),(35,181),(36,192),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,60),(46,59),(47,58),(48,57),(61,226),(62,225),(63,224),(64,223),(65,222),(66,221),(67,220),(68,219),(69,218),(70,217),(71,228),(72,227),(73,232),(74,231),(75,230),(76,229),(77,240),(78,239),(79,238),(80,237),(81,236),(82,235),(83,234),(84,233),(85,215),(86,214),(87,213),(88,212),(89,211),(90,210),(91,209),(92,208),(93,207),(94,206),(95,205),(96,216),(97,110),(98,109),(99,120),(100,119),(101,118),(102,117),(103,116),(104,115),(105,114),(106,113),(107,112),(108,111),(121,180),(122,179),(123,178),(124,177),(125,176),(126,175),(127,174),(128,173),(129,172),(130,171),(131,170),(132,169),(133,163),(134,162),(135,161),(136,160),(137,159),(138,158),(139,157),(140,168),(141,167),(142,166),(143,165),(144,164)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 12 | 60 | 2 | 2 | 2 | 10 | 10 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | C3⋊D4 | D12 | C4○D12 | C4○D20 | S3×D5 | D4×D5 | C2×S3×D5 | D6.D10 | D5×D12 | D5×C3⋊D4 |
| kernel | D10⋊D12 | C30.Q8 | C5×D6⋊C4 | D30⋊3C4 | C2×C15⋊D4 | C2×C3⋊D20 | C2×C5⋊D12 | D5×C2×C12 | C2×C4×D5 | C3×Dic5 | C6×D5 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | Dic5 | D10 | C10 | C6 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 4 | 2 | 4 | 4 | 4 |
Matrix representation of D10⋊D12 ►in GL6(𝔽61)
| 18 | 18 | 0 | 0 | 0 | 0 |
| 43 | 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 | 18 | 0 | 0 | 0 | 0 |
| 60 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 53 | 60 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 |
| 0 | 0 | 0 | 0 | 47 | 13 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 32 | 0 | 0 |
| 0 | 0 | 21 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 | 49 |
| 0 | 0 | 0 | 0 | 14 | 48 |
G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60],[18,60,0,0,0,0,18,43,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,1,53,0,0,0,0,0,60],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,21,0,0,0,0,0,0,32,0,0,0,0,0,0,47,47,0,0,0,0,0,13],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,0,21,0,0,0,0,32,0,0,0,0,0,0,0,13,14,0,0,0,0,49,48] >;
D10⋊D12 in GAP, Magma, Sage, TeX
D_{10}\rtimes D_{12} % in TeX
G:=Group("D10:D12"); // GroupNames label
G:=SmallGroup(480,524);
// by ID
G=gap.SmallGroup(480,524);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^8*b,d*b*d=a^3*b,d*c*d=c^-1>;
// generators/relations