metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D10.17D12, D6⋊C4⋊4D5, (C6×D5).7D4, C60⋊5C4⋊4C2, C6.15(D4×D5), D6⋊Dic5⋊7C2, C2.18(D5×D12), C30.39(C2×D4), (C2×C20).17D6, D10⋊C4⋊4S3, C10.15(C2×D12), (C2×C12).13D10, (C2×C60).6C22, C6.54(C4○D20), C30.Q8⋊20C2, (C2×Dic5).31D6, (C22×D5).48D6, C30.117(C4○D4), C6.70(D4⋊2D5), C3⋊3(D10.12D4), (C2×C30).104C23, (C22×S3).11D10, C5⋊2(C23.21D6), C15⋊8(C22.D4), C10.28(D4⋊2S3), C2.15(D20⋊5S3), (C2×Dic3).101D10, (C6×Dic5).60C22, C2.15(C30.C23), (C2×Dic15).84C22, (C10×Dic3).64C22, (C5×D6⋊C4)⋊4C2, (C2×D5×Dic3)⋊7C2, (C2×C4).42(S3×D5), (C2×C15⋊D4).1C2, (C3×D10⋊C4)⋊4C2, (D5×C2×C6).18C22, C22.172(C2×S3×D5), (S3×C2×C10).16C22, (C2×C6).116(C22×D5), (C2×C10).116(C22×S3), SmallGroup(480,490)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D10.17D12
G = < a,b,c,d | a10=b2=c12=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a5b, dcd=a5c-1 >
Subgroups: 812 in 156 conjugacy classes, 48 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, C20, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C5×S3, C3×D5, C30, C22.D4, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C4⋊Dic3, D6⋊C4, D6⋊C4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C5×Dic3, C3×Dic5, Dic15, C60, C6×D5, C6×D5, S3×C10, C2×C30, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C5×C22⋊C4, C2×C4×D5, C2×C5⋊D4, C23.21D6, D5×Dic3, C15⋊D4, C6×Dic5, C10×Dic3, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, D10.12D4, D6⋊Dic5, C30.Q8, C3×D10⋊C4, C5×D6⋊C4, C60⋊5C4, C2×D5×Dic3, C2×C15⋊D4, D10.17D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, D12, C22×S3, C22.D4, C22×D5, C2×D12, D4⋊2S3, S3×D5, C4○D20, D4×D5, D4⋊2D5, C23.21D6, C2×S3×D5, D10.12D4, D20⋊5S3, D5×D12, C30.C23, D10.17D12
►On 240 points
Generators in S240
(1 34 119 147 142 18 73 233 71 50)(2 35 120 148 143 19 74 234 72 51)(3 36 109 149 144 20 75 235 61 52)(4 25 110 150 133 21 76 236 62 53)(5 26 111 151 134 22 77 237 63 54)(6 27 112 152 135 23 78 238 64 55)(7 28 113 153 136 24 79 239 65 56)(8 29 114 154 137 13 80 240 66 57)(9 30 115 155 138 14 81 229 67 58)(10 31 116 156 139 15 82 230 68 59)(11 32 117 145 140 16 83 231 69 60)(12 33 118 146 141 17 84 232 70 49)(37 203 216 92 166 221 103 175 182 130)(38 204 205 93 167 222 104 176 183 131)(39 193 206 94 168 223 105 177 184 132)(40 194 207 95 157 224 106 178 185 121)(41 195 208 96 158 225 107 179 186 122)(42 196 209 85 159 226 108 180 187 123)(43 197 210 86 160 227 97 169 188 124)(44 198 211 87 161 228 98 170 189 125)(45 199 212 88 162 217 99 171 190 126)(46 200 213 89 163 218 100 172 191 127)(47 201 214 90 164 219 101 173 192 128)(48 202 215 91 165 220 102 174 181 129)
(1 136)(2 57)(3 138)(4 59)(5 140)(6 49)(7 142)(8 51)(9 144)(10 53)(11 134)(12 55)(13 143)(14 52)(15 133)(16 54)(17 135)(18 56)(19 137)(20 58)(21 139)(22 60)(23 141)(24 50)(25 68)(26 145)(27 70)(28 147)(29 72)(30 149)(31 62)(32 151)(33 64)(34 153)(35 66)(36 155)(37 169)(38 211)(39 171)(40 213)(41 173)(42 215)(43 175)(44 205)(45 177)(46 207)(47 179)(48 209)(61 81)(63 83)(65 73)(67 75)(69 77)(71 79)(74 154)(76 156)(78 146)(80 148)(82 150)(84 152)(85 129)(86 166)(87 131)(88 168)(89 121)(90 158)(91 123)(92 160)(93 125)(94 162)(95 127)(96 164)(97 203)(98 104)(99 193)(100 106)(101 195)(102 108)(103 197)(105 199)(107 201)(109 115)(110 230)(111 117)(112 232)(113 119)(114 234)(116 236)(118 238)(120 240)(122 192)(124 182)(126 184)(128 186)(130 188)(132 190)(157 191)(159 181)(161 183)(163 185)(165 187)(167 189)(170 222)(172 224)(174 226)(176 228)(178 218)(180 220)(194 200)(196 202)(198 204)(206 217)(208 219)(210 221)(212 223)(214 225)(216 227)(229 235)(231 237)(233 239)
(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)
(1 124)(2 159)(3 122)(4 157)(5 132)(6 167)(7 130)(8 165)(9 128)(10 163)(11 126)(12 161)(13 129)(14 164)(15 127)(16 162)(17 125)(18 160)(19 123)(20 158)(21 121)(22 168)(23 131)(24 166)(25 224)(26 39)(27 222)(28 37)(29 220)(30 47)(31 218)(32 45)(33 228)(34 43)(35 226)(36 41)(38 78)(40 76)(42 74)(44 84)(46 82)(48 80)(49 87)(50 188)(51 85)(52 186)(53 95)(54 184)(55 93)(56 182)(57 91)(58 192)(59 89)(60 190)(61 179)(62 207)(63 177)(64 205)(65 175)(66 215)(67 173)(68 213)(69 171)(70 211)(71 169)(72 209)(73 227)(75 225)(77 223)(79 221)(81 219)(83 217)(86 142)(88 140)(90 138)(92 136)(94 134)(96 144)(97 233)(98 118)(99 231)(100 116)(101 229)(102 114)(103 239)(104 112)(105 237)(106 110)(107 235)(108 120)(109 195)(111 193)(113 203)(115 201)(117 199)(119 197)(133 185)(135 183)(137 181)(139 191)(141 189)(143 187)(145 212)(146 170)(147 210)(148 180)(149 208)(150 178)(151 206)(152 176)(153 216)(154 174)(155 214)(156 172)(194 236)(196 234)(198 232)(200 230)(202 240)(204 238)
G:=sub<Sym(240)| (1,34,119,147,142,18,73,233,71,50)(2,35,120,148,143,19,74,234,72,51)(3,36,109,149,144,20,75,235,61,52)(4,25,110,150,133,21,76,236,62,53)(5,26,111,151,134,22,77,237,63,54)(6,27,112,152,135,23,78,238,64,55)(7,28,113,153,136,24,79,239,65,56)(8,29,114,154,137,13,80,240,66,57)(9,30,115,155,138,14,81,229,67,58)(10,31,116,156,139,15,82,230,68,59)(11,32,117,145,140,16,83,231,69,60)(12,33,118,146,141,17,84,232,70,49)(37,203,216,92,166,221,103,175,182,130)(38,204,205,93,167,222,104,176,183,131)(39,193,206,94,168,223,105,177,184,132)(40,194,207,95,157,224,106,178,185,121)(41,195,208,96,158,225,107,179,186,122)(42,196,209,85,159,226,108,180,187,123)(43,197,210,86,160,227,97,169,188,124)(44,198,211,87,161,228,98,170,189,125)(45,199,212,88,162,217,99,171,190,126)(46,200,213,89,163,218,100,172,191,127)(47,201,214,90,164,219,101,173,192,128)(48,202,215,91,165,220,102,174,181,129), (1,136)(2,57)(3,138)(4,59)(5,140)(6,49)(7,142)(8,51)(9,144)(10,53)(11,134)(12,55)(13,143)(14,52)(15,133)(16,54)(17,135)(18,56)(19,137)(20,58)(21,139)(22,60)(23,141)(24,50)(25,68)(26,145)(27,70)(28,147)(29,72)(30,149)(31,62)(32,151)(33,64)(34,153)(35,66)(36,155)(37,169)(38,211)(39,171)(40,213)(41,173)(42,215)(43,175)(44,205)(45,177)(46,207)(47,179)(48,209)(61,81)(63,83)(65,73)(67,75)(69,77)(71,79)(74,154)(76,156)(78,146)(80,148)(82,150)(84,152)(85,129)(86,166)(87,131)(88,168)(89,121)(90,158)(91,123)(92,160)(93,125)(94,162)(95,127)(96,164)(97,203)(98,104)(99,193)(100,106)(101,195)(102,108)(103,197)(105,199)(107,201)(109,115)(110,230)(111,117)(112,232)(113,119)(114,234)(116,236)(118,238)(120,240)(122,192)(124,182)(126,184)(128,186)(130,188)(132,190)(157,191)(159,181)(161,183)(163,185)(165,187)(167,189)(170,222)(172,224)(174,226)(176,228)(178,218)(180,220)(194,200)(196,202)(198,204)(206,217)(208,219)(210,221)(212,223)(214,225)(216,227)(229,235)(231,237)(233,239), (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), (1,124)(2,159)(3,122)(4,157)(5,132)(6,167)(7,130)(8,165)(9,128)(10,163)(11,126)(12,161)(13,129)(14,164)(15,127)(16,162)(17,125)(18,160)(19,123)(20,158)(21,121)(22,168)(23,131)(24,166)(25,224)(26,39)(27,222)(28,37)(29,220)(30,47)(31,218)(32,45)(33,228)(34,43)(35,226)(36,41)(38,78)(40,76)(42,74)(44,84)(46,82)(48,80)(49,87)(50,188)(51,85)(52,186)(53,95)(54,184)(55,93)(56,182)(57,91)(58,192)(59,89)(60,190)(61,179)(62,207)(63,177)(64,205)(65,175)(66,215)(67,173)(68,213)(69,171)(70,211)(71,169)(72,209)(73,227)(75,225)(77,223)(79,221)(81,219)(83,217)(86,142)(88,140)(90,138)(92,136)(94,134)(96,144)(97,233)(98,118)(99,231)(100,116)(101,229)(102,114)(103,239)(104,112)(105,237)(106,110)(107,235)(108,120)(109,195)(111,193)(113,203)(115,201)(117,199)(119,197)(133,185)(135,183)(137,181)(139,191)(141,189)(143,187)(145,212)(146,170)(147,210)(148,180)(149,208)(150,178)(151,206)(152,176)(153,216)(154,174)(155,214)(156,172)(194,236)(196,234)(198,232)(200,230)(202,240)(204,238)>;
G:=Group( (1,34,119,147,142,18,73,233,71,50)(2,35,120,148,143,19,74,234,72,51)(3,36,109,149,144,20,75,235,61,52)(4,25,110,150,133,21,76,236,62,53)(5,26,111,151,134,22,77,237,63,54)(6,27,112,152,135,23,78,238,64,55)(7,28,113,153,136,24,79,239,65,56)(8,29,114,154,137,13,80,240,66,57)(9,30,115,155,138,14,81,229,67,58)(10,31,116,156,139,15,82,230,68,59)(11,32,117,145,140,16,83,231,69,60)(12,33,118,146,141,17,84,232,70,49)(37,203,216,92,166,221,103,175,182,130)(38,204,205,93,167,222,104,176,183,131)(39,193,206,94,168,223,105,177,184,132)(40,194,207,95,157,224,106,178,185,121)(41,195,208,96,158,225,107,179,186,122)(42,196,209,85,159,226,108,180,187,123)(43,197,210,86,160,227,97,169,188,124)(44,198,211,87,161,228,98,170,189,125)(45,199,212,88,162,217,99,171,190,126)(46,200,213,89,163,218,100,172,191,127)(47,201,214,90,164,219,101,173,192,128)(48,202,215,91,165,220,102,174,181,129), (1,136)(2,57)(3,138)(4,59)(5,140)(6,49)(7,142)(8,51)(9,144)(10,53)(11,134)(12,55)(13,143)(14,52)(15,133)(16,54)(17,135)(18,56)(19,137)(20,58)(21,139)(22,60)(23,141)(24,50)(25,68)(26,145)(27,70)(28,147)(29,72)(30,149)(31,62)(32,151)(33,64)(34,153)(35,66)(36,155)(37,169)(38,211)(39,171)(40,213)(41,173)(42,215)(43,175)(44,205)(45,177)(46,207)(47,179)(48,209)(61,81)(63,83)(65,73)(67,75)(69,77)(71,79)(74,154)(76,156)(78,146)(80,148)(82,150)(84,152)(85,129)(86,166)(87,131)(88,168)(89,121)(90,158)(91,123)(92,160)(93,125)(94,162)(95,127)(96,164)(97,203)(98,104)(99,193)(100,106)(101,195)(102,108)(103,197)(105,199)(107,201)(109,115)(110,230)(111,117)(112,232)(113,119)(114,234)(116,236)(118,238)(120,240)(122,192)(124,182)(126,184)(128,186)(130,188)(132,190)(157,191)(159,181)(161,183)(163,185)(165,187)(167,189)(170,222)(172,224)(174,226)(176,228)(178,218)(180,220)(194,200)(196,202)(198,204)(206,217)(208,219)(210,221)(212,223)(214,225)(216,227)(229,235)(231,237)(233,239), (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), (1,124)(2,159)(3,122)(4,157)(5,132)(6,167)(7,130)(8,165)(9,128)(10,163)(11,126)(12,161)(13,129)(14,164)(15,127)(16,162)(17,125)(18,160)(19,123)(20,158)(21,121)(22,168)(23,131)(24,166)(25,224)(26,39)(27,222)(28,37)(29,220)(30,47)(31,218)(32,45)(33,228)(34,43)(35,226)(36,41)(38,78)(40,76)(42,74)(44,84)(46,82)(48,80)(49,87)(50,188)(51,85)(52,186)(53,95)(54,184)(55,93)(56,182)(57,91)(58,192)(59,89)(60,190)(61,179)(62,207)(63,177)(64,205)(65,175)(66,215)(67,173)(68,213)(69,171)(70,211)(71,169)(72,209)(73,227)(75,225)(77,223)(79,221)(81,219)(83,217)(86,142)(88,140)(90,138)(92,136)(94,134)(96,144)(97,233)(98,118)(99,231)(100,116)(101,229)(102,114)(103,239)(104,112)(105,237)(106,110)(107,235)(108,120)(109,195)(111,193)(113,203)(115,201)(117,199)(119,197)(133,185)(135,183)(137,181)(139,191)(141,189)(143,187)(145,212)(146,170)(147,210)(148,180)(149,208)(150,178)(151,206)(152,176)(153,216)(154,174)(155,214)(156,172)(194,236)(196,234)(198,232)(200,230)(202,240)(204,238) );
G=PermutationGroup([[(1,34,119,147,142,18,73,233,71,50),(2,35,120,148,143,19,74,234,72,51),(3,36,109,149,144,20,75,235,61,52),(4,25,110,150,133,21,76,236,62,53),(5,26,111,151,134,22,77,237,63,54),(6,27,112,152,135,23,78,238,64,55),(7,28,113,153,136,24,79,239,65,56),(8,29,114,154,137,13,80,240,66,57),(9,30,115,155,138,14,81,229,67,58),(10,31,116,156,139,15,82,230,68,59),(11,32,117,145,140,16,83,231,69,60),(12,33,118,146,141,17,84,232,70,49),(37,203,216,92,166,221,103,175,182,130),(38,204,205,93,167,222,104,176,183,131),(39,193,206,94,168,223,105,177,184,132),(40,194,207,95,157,224,106,178,185,121),(41,195,208,96,158,225,107,179,186,122),(42,196,209,85,159,226,108,180,187,123),(43,197,210,86,160,227,97,169,188,124),(44,198,211,87,161,228,98,170,189,125),(45,199,212,88,162,217,99,171,190,126),(46,200,213,89,163,218,100,172,191,127),(47,201,214,90,164,219,101,173,192,128),(48,202,215,91,165,220,102,174,181,129)], [(1,136),(2,57),(3,138),(4,59),(5,140),(6,49),(7,142),(8,51),(9,144),(10,53),(11,134),(12,55),(13,143),(14,52),(15,133),(16,54),(17,135),(18,56),(19,137),(20,58),(21,139),(22,60),(23,141),(24,50),(25,68),(26,145),(27,70),(28,147),(29,72),(30,149),(31,62),(32,151),(33,64),(34,153),(35,66),(36,155),(37,169),(38,211),(39,171),(40,213),(41,173),(42,215),(43,175),(44,205),(45,177),(46,207),(47,179),(48,209),(61,81),(63,83),(65,73),(67,75),(69,77),(71,79),(74,154),(76,156),(78,146),(80,148),(82,150),(84,152),(85,129),(86,166),(87,131),(88,168),(89,121),(90,158),(91,123),(92,160),(93,125),(94,162),(95,127),(96,164),(97,203),(98,104),(99,193),(100,106),(101,195),(102,108),(103,197),(105,199),(107,201),(109,115),(110,230),(111,117),(112,232),(113,119),(114,234),(116,236),(118,238),(120,240),(122,192),(124,182),(126,184),(128,186),(130,188),(132,190),(157,191),(159,181),(161,183),(163,185),(165,187),(167,189),(170,222),(172,224),(174,226),(176,228),(178,218),(180,220),(194,200),(196,202),(198,204),(206,217),(208,219),(210,221),(212,223),(214,225),(216,227),(229,235),(231,237),(233,239)], [(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)], [(1,124),(2,159),(3,122),(4,157),(5,132),(6,167),(7,130),(8,165),(9,128),(10,163),(11,126),(12,161),(13,129),(14,164),(15,127),(16,162),(17,125),(18,160),(19,123),(20,158),(21,121),(22,168),(23,131),(24,166),(25,224),(26,39),(27,222),(28,37),(29,220),(30,47),(31,218),(32,45),(33,228),(34,43),(35,226),(36,41),(38,78),(40,76),(42,74),(44,84),(46,82),(48,80),(49,87),(50,188),(51,85),(52,186),(53,95),(54,184),(55,93),(56,182),(57,91),(58,192),(59,89),(60,190),(61,179),(62,207),(63,177),(64,205),(65,175),(66,215),(67,173),(68,213),(69,171),(70,211),(71,169),(72,209),(73,227),(75,225),(77,223),(79,221),(81,219),(83,217),(86,142),(88,140),(90,138),(92,136),(94,134),(96,144),(97,233),(98,118),(99,231),(100,116),(101,229),(102,114),(103,239),(104,112),(105,237),(106,110),(107,235),(108,120),(109,195),(111,193),(113,203),(115,201),(117,199),(119,197),(133,185),(135,183),(137,181),(139,191),(141,189),(143,187),(145,212),(146,170),(147,210),(148,180),(149,208),(150,178),(151,206),(152,176),(153,216),(154,174),(155,214),(156,172),(194,236),(196,234),(198,232),(200,230),(202,240),(204,238)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 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 | 2 | 4 | 6 | 6 | 20 | 30 | 30 | 60 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D5 | D6 | D6 | D6 | C4○D4 | D10 | D10 | D10 | D12 | C4○D20 | D4⋊2S3 | S3×D5 | D4×D5 | D4⋊2D5 | C2×S3×D5 | D20⋊5S3 | D5×D12 | C30.C23 |
| kernel | D10.17D12 | D6⋊Dic5 | C30.Q8 | C3×D10⋊C4 | C5×D6⋊C4 | C60⋊5C4 | C2×D5×Dic3 | C2×C15⋊D4 | D10⋊C4 | C6×D5 | D6⋊C4 | C2×Dic5 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C22×S3 | D10 | C6 | C10 | C2×C4 | C6 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of D10.17D12 ►in GL8(𝔽61)
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 44 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 60 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 60 |
| 50 | 50 | 0 | 0 | 0 | 0 | 0 | 0 |
| 22 | 11 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 54 | 0 | 0 |
| 0 | 0 | 0 | 0 | 54 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 30 | 12 |
| 0 | 0 | 0 | 0 | 0 | 0 | 42 | 31 |
G:=sub<GL(8,GF(61))| [60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,44,17,0,0,0,0,0,0,44,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,2,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,44,60,0,0,0,0,0,0,44,17,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[60,2,0,0,0,0,0,0,60,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,60,60],[50,22,0,0,0,0,0,0,50,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,47,54,0,0,0,0,0,0,54,14,0,0,0,0,0,0,0,0,30,42,0,0,0,0,0,0,12,31] >;
D10.17D12 in GAP, Magma, Sage, TeX
D_{10}._{17}D_{12} % in TeX
G:=Group("D10.17D12"); // GroupNames label
G:=SmallGroup(480,490);
// by ID
G=gap.SmallGroup(480,490);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,64,254,219,142,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^10=b^2=c^12=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^5*b,d*c*d=a^5*c^-1>;
// generators/relations