metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.24D10, C60.14C23, Dic10.8D6, Dic30.4C22, D4⋊S3⋊7D5, C3⋊C8.13D10, (C5×D4).6D6, D4⋊2D5⋊2S3, C15⋊13(C4○D8), C3⋊5(D8⋊3D5), D4.D15⋊6C2, D4.10(S3×D5), (C6×D5).10D4, (C4×D5).45D6, C3⋊Dic20⋊3C2, C6.143(D4×D5), D12⋊5D5⋊2C2, (C3×D4).21D10, C30.176(C2×D4), C20.D6⋊3C2, C5⋊2(Q8.13D6), D10.9(C3⋊D4), C20.14(C22×S3), C15⋊3C8.3C22, (C3×Dic5).68D4, (C5×D12).4C22, (D5×C12).6C22, (D4×C15).8C22, C12.14(C22×D5), Dic5.40(C3⋊D4), (C3×Dic10).4C22, (D5×C3⋊C8)⋊3C2, C4.14(C2×S3×D5), (C5×D4⋊S3)⋊6C2, C2.25(D5×C3⋊D4), (C5×C3⋊C8).3C22, (C3×D4⋊2D5)⋊2C2, C10.46(C2×C3⋊D4), SmallGroup(480,566)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.24D10
G = < a,b,c,d | a12=b2=c10=1, d2=a6, bab=a-1, cac-1=dad-1=a7, cbc-1=a3b, dbd-1=a9b, dcd-1=a6c-1 >
Subgroups: 604 in 124 conjugacy classes, 40 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C6, C8, C2×C4, D4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, Dic5, C20, D10, C2×C10, C3⋊C8, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, C5×S3, C3×D5, C30, C30, C4○D8, C5⋊2C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C5⋊D4, C5×D4, C5×D4, C2×C3⋊C8, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C4○D12, C3×C4○D4, C3×Dic5, C3×Dic5, Dic15, C60, C6×D5, S3×C10, C2×C30, C8×D5, Dic20, D4.D5, C5×D8, D4⋊2D5, D4⋊2D5, Q8.13D6, C5×C3⋊C8, C15⋊3C8, S3×Dic5, C15⋊D4, C3×Dic10, D5×C12, C6×Dic5, C3×C5⋊D4, C5×D12, Dic30, D4×C15, D8⋊3D5, D5×C3⋊C8, C20.D6, C3⋊Dic20, C5×D4⋊S3, D4.D15, D12⋊5D5, C3×D4⋊2D5, D12.24D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4○D8, C22×D5, C2×C3⋊D4, S3×D5, D4×D5, Q8.13D6, C2×S3×D5, D8⋊3D5, D5×C3⋊D4, D12.24D10
►On 240 points
Generators in S240
(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 77)(2 76)(3 75)(4 74)(5 73)(6 84)(7 83)(8 82)(9 81)(10 80)(11 79)(12 78)(13 211)(14 210)(15 209)(16 208)(17 207)(18 206)(19 205)(20 216)(21 215)(22 214)(23 213)(24 212)(25 177)(26 176)(27 175)(28 174)(29 173)(30 172)(31 171)(32 170)(33 169)(34 180)(35 179)(36 178)(37 67)(38 66)(39 65)(40 64)(41 63)(42 62)(43 61)(44 72)(45 71)(46 70)(47 69)(48 68)(49 136)(50 135)(51 134)(52 133)(53 144)(54 143)(55 142)(56 141)(57 140)(58 139)(59 138)(60 137)(85 159)(86 158)(87 157)(88 168)(89 167)(90 166)(91 165)(92 164)(93 163)(94 162)(95 161)(96 160)(97 200)(98 199)(99 198)(100 197)(101 196)(102 195)(103 194)(104 193)(105 204)(106 203)(107 202)(108 201)(109 235)(110 234)(111 233)(112 232)(113 231)(114 230)(115 229)(116 240)(117 239)(118 238)(119 237)(120 236)(121 220)(122 219)(123 218)(124 217)(125 228)(126 227)(127 226)(128 225)(129 224)(130 223)(131 222)(132 221)(145 185)(146 184)(147 183)(148 182)(149 181)(150 192)(151 191)(152 190)(153 189)(154 188)(155 187)(156 186)
(1 207 38 131 98 85 182 111 170 60)(2 214 39 126 99 92 183 118 171 55)(3 209 40 121 100 87 184 113 172 50)(4 216 41 128 101 94 185 120 173 57)(5 211 42 123 102 89 186 115 174 52)(6 206 43 130 103 96 187 110 175 59)(7 213 44 125 104 91 188 117 176 54)(8 208 45 132 105 86 189 112 177 49)(9 215 46 127 106 93 190 119 178 56)(10 210 47 122 107 88 191 114 179 51)(11 205 48 129 108 95 192 109 180 58)(12 212 37 124 97 90 181 116 169 53)(13 71 218 204 167 153 229 25 133 82)(14 66 219 199 168 148 230 32 134 77)(15 61 220 194 157 155 231 27 135 84)(16 68 221 201 158 150 232 34 136 79)(17 63 222 196 159 145 233 29 137 74)(18 70 223 203 160 152 234 36 138 81)(19 65 224 198 161 147 235 31 139 76)(20 72 225 193 162 154 236 26 140 83)(21 67 226 200 163 149 237 33 141 78)(22 62 227 195 164 156 238 28 142 73)(23 69 228 202 165 151 239 35 143 80)(24 64 217 197 166 146 240 30 144 75)
(1 60 7 54)(2 55 8 49)(3 50 9 56)(4 57 10 51)(5 52 11 58)(6 59 12 53)(13 25 19 31)(14 32 20 26)(15 27 21 33)(16 34 22 28)(17 29 23 35)(18 36 24 30)(37 116 43 110)(38 111 44 117)(39 118 45 112)(40 113 46 119)(41 120 47 114)(42 115 48 109)(61 237 67 231)(62 232 68 238)(63 239 69 233)(64 234 70 240)(65 229 71 235)(66 236 72 230)(73 136 79 142)(74 143 80 137)(75 138 81 144)(76 133 82 139)(77 140 83 134)(78 135 84 141)(85 104 91 98)(86 99 92 105)(87 106 93 100)(88 101 94 107)(89 108 95 102)(90 103 96 97)(121 190 127 184)(122 185 128 191)(123 192 129 186)(124 187 130 181)(125 182 131 188)(126 189 132 183)(145 228 151 222)(146 223 152 217)(147 218 153 224)(148 225 154 219)(149 220 155 226)(150 227 156 221)(157 194 163 200)(158 201 164 195)(159 196 165 202)(160 203 166 197)(161 198 167 204)(162 193 168 199)(169 212 175 206)(170 207 176 213)(171 214 177 208)(172 209 178 215)(173 216 179 210)(174 211 180 205)
G:=sub<Sym(240)| (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,211)(14,210)(15,209)(16,208)(17,207)(18,206)(19,205)(20,216)(21,215)(22,214)(23,213)(24,212)(25,177)(26,176)(27,175)(28,174)(29,173)(30,172)(31,171)(32,170)(33,169)(34,180)(35,179)(36,178)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,72)(45,71)(46,70)(47,69)(48,68)(49,136)(50,135)(51,134)(52,133)(53,144)(54,143)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(85,159)(86,158)(87,157)(88,168)(89,167)(90,166)(91,165)(92,164)(93,163)(94,162)(95,161)(96,160)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,204)(106,203)(107,202)(108,201)(109,235)(110,234)(111,233)(112,232)(113,231)(114,230)(115,229)(116,240)(117,239)(118,238)(119,237)(120,236)(121,220)(122,219)(123,218)(124,217)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(145,185)(146,184)(147,183)(148,182)(149,181)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186), (1,207,38,131,98,85,182,111,170,60)(2,214,39,126,99,92,183,118,171,55)(3,209,40,121,100,87,184,113,172,50)(4,216,41,128,101,94,185,120,173,57)(5,211,42,123,102,89,186,115,174,52)(6,206,43,130,103,96,187,110,175,59)(7,213,44,125,104,91,188,117,176,54)(8,208,45,132,105,86,189,112,177,49)(9,215,46,127,106,93,190,119,178,56)(10,210,47,122,107,88,191,114,179,51)(11,205,48,129,108,95,192,109,180,58)(12,212,37,124,97,90,181,116,169,53)(13,71,218,204,167,153,229,25,133,82)(14,66,219,199,168,148,230,32,134,77)(15,61,220,194,157,155,231,27,135,84)(16,68,221,201,158,150,232,34,136,79)(17,63,222,196,159,145,233,29,137,74)(18,70,223,203,160,152,234,36,138,81)(19,65,224,198,161,147,235,31,139,76)(20,72,225,193,162,154,236,26,140,83)(21,67,226,200,163,149,237,33,141,78)(22,62,227,195,164,156,238,28,142,73)(23,69,228,202,165,151,239,35,143,80)(24,64,217,197,166,146,240,30,144,75), (1,60,7,54)(2,55,8,49)(3,50,9,56)(4,57,10,51)(5,52,11,58)(6,59,12,53)(13,25,19,31)(14,32,20,26)(15,27,21,33)(16,34,22,28)(17,29,23,35)(18,36,24,30)(37,116,43,110)(38,111,44,117)(39,118,45,112)(40,113,46,119)(41,120,47,114)(42,115,48,109)(61,237,67,231)(62,232,68,238)(63,239,69,233)(64,234,70,240)(65,229,71,235)(66,236,72,230)(73,136,79,142)(74,143,80,137)(75,138,81,144)(76,133,82,139)(77,140,83,134)(78,135,84,141)(85,104,91,98)(86,99,92,105)(87,106,93,100)(88,101,94,107)(89,108,95,102)(90,103,96,97)(121,190,127,184)(122,185,128,191)(123,192,129,186)(124,187,130,181)(125,182,131,188)(126,189,132,183)(145,228,151,222)(146,223,152,217)(147,218,153,224)(148,225,154,219)(149,220,155,226)(150,227,156,221)(157,194,163,200)(158,201,164,195)(159,196,165,202)(160,203,166,197)(161,198,167,204)(162,193,168,199)(169,212,175,206)(170,207,176,213)(171,214,177,208)(172,209,178,215)(173,216,179,210)(174,211,180,205)>;
G:=Group( (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), (1,77)(2,76)(3,75)(4,74)(5,73)(6,84)(7,83)(8,82)(9,81)(10,80)(11,79)(12,78)(13,211)(14,210)(15,209)(16,208)(17,207)(18,206)(19,205)(20,216)(21,215)(22,214)(23,213)(24,212)(25,177)(26,176)(27,175)(28,174)(29,173)(30,172)(31,171)(32,170)(33,169)(34,180)(35,179)(36,178)(37,67)(38,66)(39,65)(40,64)(41,63)(42,62)(43,61)(44,72)(45,71)(46,70)(47,69)(48,68)(49,136)(50,135)(51,134)(52,133)(53,144)(54,143)(55,142)(56,141)(57,140)(58,139)(59,138)(60,137)(85,159)(86,158)(87,157)(88,168)(89,167)(90,166)(91,165)(92,164)(93,163)(94,162)(95,161)(96,160)(97,200)(98,199)(99,198)(100,197)(101,196)(102,195)(103,194)(104,193)(105,204)(106,203)(107,202)(108,201)(109,235)(110,234)(111,233)(112,232)(113,231)(114,230)(115,229)(116,240)(117,239)(118,238)(119,237)(120,236)(121,220)(122,219)(123,218)(124,217)(125,228)(126,227)(127,226)(128,225)(129,224)(130,223)(131,222)(132,221)(145,185)(146,184)(147,183)(148,182)(149,181)(150,192)(151,191)(152,190)(153,189)(154,188)(155,187)(156,186), (1,207,38,131,98,85,182,111,170,60)(2,214,39,126,99,92,183,118,171,55)(3,209,40,121,100,87,184,113,172,50)(4,216,41,128,101,94,185,120,173,57)(5,211,42,123,102,89,186,115,174,52)(6,206,43,130,103,96,187,110,175,59)(7,213,44,125,104,91,188,117,176,54)(8,208,45,132,105,86,189,112,177,49)(9,215,46,127,106,93,190,119,178,56)(10,210,47,122,107,88,191,114,179,51)(11,205,48,129,108,95,192,109,180,58)(12,212,37,124,97,90,181,116,169,53)(13,71,218,204,167,153,229,25,133,82)(14,66,219,199,168,148,230,32,134,77)(15,61,220,194,157,155,231,27,135,84)(16,68,221,201,158,150,232,34,136,79)(17,63,222,196,159,145,233,29,137,74)(18,70,223,203,160,152,234,36,138,81)(19,65,224,198,161,147,235,31,139,76)(20,72,225,193,162,154,236,26,140,83)(21,67,226,200,163,149,237,33,141,78)(22,62,227,195,164,156,238,28,142,73)(23,69,228,202,165,151,239,35,143,80)(24,64,217,197,166,146,240,30,144,75), (1,60,7,54)(2,55,8,49)(3,50,9,56)(4,57,10,51)(5,52,11,58)(6,59,12,53)(13,25,19,31)(14,32,20,26)(15,27,21,33)(16,34,22,28)(17,29,23,35)(18,36,24,30)(37,116,43,110)(38,111,44,117)(39,118,45,112)(40,113,46,119)(41,120,47,114)(42,115,48,109)(61,237,67,231)(62,232,68,238)(63,239,69,233)(64,234,70,240)(65,229,71,235)(66,236,72,230)(73,136,79,142)(74,143,80,137)(75,138,81,144)(76,133,82,139)(77,140,83,134)(78,135,84,141)(85,104,91,98)(86,99,92,105)(87,106,93,100)(88,101,94,107)(89,108,95,102)(90,103,96,97)(121,190,127,184)(122,185,128,191)(123,192,129,186)(124,187,130,181)(125,182,131,188)(126,189,132,183)(145,228,151,222)(146,223,152,217)(147,218,153,224)(148,225,154,219)(149,220,155,226)(150,227,156,221)(157,194,163,200)(158,201,164,195)(159,196,165,202)(160,203,166,197)(161,198,167,204)(162,193,168,199)(169,212,175,206)(170,207,176,213)(171,214,177,208)(172,209,178,215)(173,216,179,210)(174,211,180,205) );
G=PermutationGroup([[(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)], [(1,77),(2,76),(3,75),(4,74),(5,73),(6,84),(7,83),(8,82),(9,81),(10,80),(11,79),(12,78),(13,211),(14,210),(15,209),(16,208),(17,207),(18,206),(19,205),(20,216),(21,215),(22,214),(23,213),(24,212),(25,177),(26,176),(27,175),(28,174),(29,173),(30,172),(31,171),(32,170),(33,169),(34,180),(35,179),(36,178),(37,67),(38,66),(39,65),(40,64),(41,63),(42,62),(43,61),(44,72),(45,71),(46,70),(47,69),(48,68),(49,136),(50,135),(51,134),(52,133),(53,144),(54,143),(55,142),(56,141),(57,140),(58,139),(59,138),(60,137),(85,159),(86,158),(87,157),(88,168),(89,167),(90,166),(91,165),(92,164),(93,163),(94,162),(95,161),(96,160),(97,200),(98,199),(99,198),(100,197),(101,196),(102,195),(103,194),(104,193),(105,204),(106,203),(107,202),(108,201),(109,235),(110,234),(111,233),(112,232),(113,231),(114,230),(115,229),(116,240),(117,239),(118,238),(119,237),(120,236),(121,220),(122,219),(123,218),(124,217),(125,228),(126,227),(127,226),(128,225),(129,224),(130,223),(131,222),(132,221),(145,185),(146,184),(147,183),(148,182),(149,181),(150,192),(151,191),(152,190),(153,189),(154,188),(155,187),(156,186)], [(1,207,38,131,98,85,182,111,170,60),(2,214,39,126,99,92,183,118,171,55),(3,209,40,121,100,87,184,113,172,50),(4,216,41,128,101,94,185,120,173,57),(5,211,42,123,102,89,186,115,174,52),(6,206,43,130,103,96,187,110,175,59),(7,213,44,125,104,91,188,117,176,54),(8,208,45,132,105,86,189,112,177,49),(9,215,46,127,106,93,190,119,178,56),(10,210,47,122,107,88,191,114,179,51),(11,205,48,129,108,95,192,109,180,58),(12,212,37,124,97,90,181,116,169,53),(13,71,218,204,167,153,229,25,133,82),(14,66,219,199,168,148,230,32,134,77),(15,61,220,194,157,155,231,27,135,84),(16,68,221,201,158,150,232,34,136,79),(17,63,222,196,159,145,233,29,137,74),(18,70,223,203,160,152,234,36,138,81),(19,65,224,198,161,147,235,31,139,76),(20,72,225,193,162,154,236,26,140,83),(21,67,226,200,163,149,237,33,141,78),(22,62,227,195,164,156,238,28,142,73),(23,69,228,202,165,151,239,35,143,80),(24,64,217,197,166,146,240,30,144,75)], [(1,60,7,54),(2,55,8,49),(3,50,9,56),(4,57,10,51),(5,52,11,58),(6,59,12,53),(13,25,19,31),(14,32,20,26),(15,27,21,33),(16,34,22,28),(17,29,23,35),(18,36,24,30),(37,116,43,110),(38,111,44,117),(39,118,45,112),(40,113,46,119),(41,120,47,114),(42,115,48,109),(61,237,67,231),(62,232,68,238),(63,239,69,233),(64,234,70,240),(65,229,71,235),(66,236,72,230),(73,136,79,142),(74,143,80,137),(75,138,81,144),(76,133,82,139),(77,140,83,134),(78,135,84,141),(85,104,91,98),(86,99,92,105),(87,106,93,100),(88,101,94,107),(89,108,95,102),(90,103,96,97),(121,190,127,184),(122,185,128,191),(123,192,129,186),(124,187,130,181),(125,182,131,188),(126,189,132,183),(145,228,151,222),(146,223,152,217),(147,218,153,224),(148,225,154,219),(149,220,155,226),(150,227,156,221),(157,194,163,200),(158,201,164,195),(159,196,165,202),(160,203,166,197),(161,198,167,204),(162,193,168,199),(169,212,175,206),(170,207,176,213),(171,214,177,208),(172,209,178,215),(173,216,179,210),(174,211,180,205)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 30A | 30B | 30C | 30D | 30E | 30F | 40A | 40B | 40C | 40D | 60A | 60B |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 30 | 30 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | 60 |
| size | 1 | 1 | 4 | 10 | 12 | 2 | 2 | 5 | 5 | 20 | 60 | 2 | 2 | 2 | 4 | 4 | 20 | 6 | 6 | 30 | 30 | 2 | 2 | 8 | 8 | 24 | 24 | 4 | 10 | 10 | 20 | 20 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 8 | 8 |
48 irreducible representations
| dim | 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 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C4○D8 | S3×D5 | D4×D5 | Q8.13D6 | C2×S3×D5 | D8⋊3D5 | D5×C3⋊D4 | D12.24D10 |
| kernel | D12.24D10 | D5×C3⋊C8 | C20.D6 | C3⋊Dic20 | C5×D4⋊S3 | D4.D15 | D12⋊5D5 | C3×D4⋊2D5 | D4⋊2D5 | C3×Dic5 | C6×D5 | D4⋊S3 | Dic10 | C4×D5 | C5×D4 | C3⋊C8 | D12 | C3×D4 | Dic5 | D10 | C15 | D4 | C6 | C5 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12.24D10 ►in GL6(𝔽241)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 160 | 177 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 30 | 217 | 0 | 0 |
| 0 | 0 | 168 | 211 | 0 | 0 |
| 0 | 0 | 0 | 0 | 140 | 193 |
| 0 | 0 | 0 | 0 | 92 | 101 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 51 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 192 | 0 | 0 |
| 0 | 0 | 0 | 240 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 190 | 190 | 0 | 0 | 0 | 0 |
| 240 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 240 | 49 | 0 | 0 |
| 0 | 0 | 118 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,160,0,0,0,0,0,177,0,0,0,0,0,0,0,1,0,0,0,0,240,240],[240,0,0,0,0,0,0,240,0,0,0,0,0,0,30,168,0,0,0,0,217,211,0,0,0,0,0,0,140,92,0,0,0,0,193,101],[190,51,0,0,0,0,190,240,0,0,0,0,0,0,1,0,0,0,0,0,192,240,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[190,240,0,0,0,0,190,51,0,0,0,0,0,0,240,118,0,0,0,0,49,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
D12.24D10 in GAP, Magma, Sage, TeX
D_{12}._{24}D_{10} % in TeX
G:=Group("D12.24D10"); // GroupNames label
G:=SmallGroup(480,566);
// by ID
G=gap.SmallGroup(480,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,422,135,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=1,d^2=a^6,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^7,c*b*c^-1=a^3*b,d*b*d^-1=a^9*b,d*c*d^-1=a^6*c^-1>;
// generators/relations