metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D30.15D4, D12.16D10, C60.47C23, Dic15.47D4, Dic10.16D6, D60.18C22, C5⋊D24⋊8C2, C3⋊C8.22D10, C5⋊Q16⋊5S3, C6.81(D4×D5), C15⋊24(C4○D8), Q8⋊2S3⋊5D5, C10.82(S3×D4), C5⋊2C8.22D6, (C5×Q8).28D6, Q8.24(S3×D5), Q8⋊3D15⋊3C2, D12⋊D5⋊3C2, D15⋊2C8⋊7C2, C5⋊3(D24⋊C2), C15⋊SD16⋊8C2, C30.209(C2×D4), (C3×Q8).11D10, C3⋊4(SD16⋊3D5), C20.47(C22×S3), C12.47(C22×D5), (C4×D15).15C22, (C5×D12).17C22, (Q8×C15).17C22, C2.34(D10⋊D6), (C3×Dic10).17C22, C4.47(C2×S3×D5), (C3×C5⋊Q16)⋊7C2, (C5×Q8⋊2S3)⋊7C2, (C5×C3⋊C8).17C22, (C3×C5⋊2C8).17C22, SmallGroup(480,599)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.D10
G = < a,b,c,d | a12=b2=1, c10=a6, d2=a3, bab=a-1, cac-1=a7, ad=da, cbc-1=dbd-1=a3b, dcd-1=a3c9 >
Subgroups: 780 in 124 conjugacy classes, 38 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, S3, C6, C8, C2×C4, D4, Q8, Q8, D5, C10, C10, Dic3, C12, C12, D6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C3⋊C8, C24, C4×S3, D12, D12, C3×Q8, C3×Q8, C5×S3, D15, C30, C4○D8, C5⋊2C8, C40, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C5×D4, C5×Q8, S3×C8, D24, Q8⋊2S3, Q8⋊2S3, C3×Q16, Q8⋊3S3, C3×Dic5, Dic15, C60, C60, S3×C10, D30, D30, C8×D5, C40⋊C2, D4⋊D5, C5⋊Q16, C5×SD16, D4⋊2D5, Q8⋊2D5, D24⋊C2, C5×C3⋊C8, C3×C5⋊2C8, S3×Dic5, C5⋊D12, C3×Dic10, C5×D12, C4×D15, C4×D15, D60, D60, Q8×C15, SD16⋊3D5, D15⋊2C8, C5⋊D24, C15⋊SD16, C3×C5⋊Q16, C5×Q8⋊2S3, D12⋊D5, Q8⋊3D15, D12.D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C22×S3, C4○D8, C22×D5, S3×D4, S3×D5, D4×D5, D24⋊C2, C2×S3×D5, SD16⋊3D5, D10⋊D6, D12.D10
►On 240 points
Generators in S240
(1 228 169 26 190 130 11 238 179 36 200 140)(2 239 170 37 191 121 12 229 180 27 181 131)(3 230 171 28 192 132 13 240 161 38 182 122)(4 221 172 39 193 123 14 231 162 29 183 133)(5 232 173 30 194 134 15 222 163 40 184 124)(6 223 174 21 195 125 16 233 164 31 185 135)(7 234 175 32 196 136 17 224 165 22 186 126)(8 225 176 23 197 127 18 235 166 33 187 137)(9 236 177 34 198 138 19 226 167 24 188 128)(10 227 178 25 199 129 20 237 168 35 189 139)(41 83 151 120 61 209 51 93 141 110 71 219)(42 94 152 111 62 220 52 84 142 101 72 210)(43 85 153 102 63 211 53 95 143 112 73 201)(44 96 154 113 64 202 54 86 144 103 74 212)(45 87 155 104 65 213 55 97 145 114 75 203)(46 98 156 115 66 204 56 88 146 105 76 214)(47 89 157 106 67 215 57 99 147 116 77 205)(48 100 158 117 68 206 58 90 148 107 78 216)(49 91 159 108 69 217 59 81 149 118 79 207)(50 82 160 119 70 208 60 92 150 109 80 218)
(1 208)(2 141)(3 210)(4 143)(5 212)(6 145)(7 214)(8 147)(9 216)(10 149)(11 218)(12 151)(13 220)(14 153)(15 202)(16 155)(17 204)(18 157)(19 206)(20 159)(21 213)(22 156)(23 215)(24 158)(25 217)(26 160)(27 219)(28 142)(29 201)(30 144)(31 203)(32 146)(33 205)(34 148)(35 207)(36 150)(37 209)(38 152)(39 211)(40 154)(41 180)(42 122)(43 162)(44 124)(45 164)(46 126)(47 166)(48 128)(49 168)(50 130)(51 170)(52 132)(53 172)(54 134)(55 174)(56 136)(57 176)(58 138)(59 178)(60 140)(61 191)(62 240)(63 193)(64 222)(65 195)(66 224)(67 197)(68 226)(69 199)(70 228)(71 181)(72 230)(73 183)(74 232)(75 185)(76 234)(77 187)(78 236)(79 189)(80 238)(81 227)(82 190)(83 229)(84 192)(85 231)(86 194)(87 233)(88 196)(89 235)(90 198)(91 237)(92 200)(93 239)(94 182)(95 221)(96 184)(97 223)(98 186)(99 225)(100 188)(101 171)(102 123)(103 173)(104 125)(105 175)(106 127)(107 177)(108 129)(109 179)(110 131)(111 161)(112 133)(113 163)(114 135)(115 165)(116 137)(117 167)(118 139)(119 169)(120 121)
(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)
(1 108 26 59 11 118 36 49)(2 58 37 107 12 48 27 117)(3 106 28 57 13 116 38 47)(4 56 39 105 14 46 29 115)(5 104 30 55 15 114 40 45)(6 54 21 103 16 44 31 113)(7 102 32 53 17 112 22 43)(8 52 23 101 18 42 33 111)(9 120 34 51 19 110 24 41)(10 50 25 119 20 60 35 109)(61 198 93 226 71 188 83 236)(62 225 84 197 72 235 94 187)(63 196 95 224 73 186 85 234)(64 223 86 195 74 233 96 185)(65 194 97 222 75 184 87 232)(66 221 88 193 76 231 98 183)(67 192 99 240 77 182 89 230)(68 239 90 191 78 229 100 181)(69 190 81 238 79 200 91 228)(70 237 92 189 80 227 82 199)(121 216 180 158 131 206 170 148)(122 157 171 215 132 147 161 205)(123 214 162 156 133 204 172 146)(124 155 173 213 134 145 163 203)(125 212 164 154 135 202 174 144)(126 153 175 211 136 143 165 201)(127 210 166 152 137 220 176 142)(128 151 177 209 138 141 167 219)(129 208 168 150 139 218 178 160)(130 149 179 207 140 159 169 217)
G:=sub<Sym(240)| (1,228,169,26,190,130,11,238,179,36,200,140)(2,239,170,37,191,121,12,229,180,27,181,131)(3,230,171,28,192,132,13,240,161,38,182,122)(4,221,172,39,193,123,14,231,162,29,183,133)(5,232,173,30,194,134,15,222,163,40,184,124)(6,223,174,21,195,125,16,233,164,31,185,135)(7,234,175,32,196,136,17,224,165,22,186,126)(8,225,176,23,197,127,18,235,166,33,187,137)(9,236,177,34,198,138,19,226,167,24,188,128)(10,227,178,25,199,129,20,237,168,35,189,139)(41,83,151,120,61,209,51,93,141,110,71,219)(42,94,152,111,62,220,52,84,142,101,72,210)(43,85,153,102,63,211,53,95,143,112,73,201)(44,96,154,113,64,202,54,86,144,103,74,212)(45,87,155,104,65,213,55,97,145,114,75,203)(46,98,156,115,66,204,56,88,146,105,76,214)(47,89,157,106,67,215,57,99,147,116,77,205)(48,100,158,117,68,206,58,90,148,107,78,216)(49,91,159,108,69,217,59,81,149,118,79,207)(50,82,160,119,70,208,60,92,150,109,80,218), (1,208)(2,141)(3,210)(4,143)(5,212)(6,145)(7,214)(8,147)(9,216)(10,149)(11,218)(12,151)(13,220)(14,153)(15,202)(16,155)(17,204)(18,157)(19,206)(20,159)(21,213)(22,156)(23,215)(24,158)(25,217)(26,160)(27,219)(28,142)(29,201)(30,144)(31,203)(32,146)(33,205)(34,148)(35,207)(36,150)(37,209)(38,152)(39,211)(40,154)(41,180)(42,122)(43,162)(44,124)(45,164)(46,126)(47,166)(48,128)(49,168)(50,130)(51,170)(52,132)(53,172)(54,134)(55,174)(56,136)(57,176)(58,138)(59,178)(60,140)(61,191)(62,240)(63,193)(64,222)(65,195)(66,224)(67,197)(68,226)(69,199)(70,228)(71,181)(72,230)(73,183)(74,232)(75,185)(76,234)(77,187)(78,236)(79,189)(80,238)(81,227)(82,190)(83,229)(84,192)(85,231)(86,194)(87,233)(88,196)(89,235)(90,198)(91,237)(92,200)(93,239)(94,182)(95,221)(96,184)(97,223)(98,186)(99,225)(100,188)(101,171)(102,123)(103,173)(104,125)(105,175)(106,127)(107,177)(108,129)(109,179)(110,131)(111,161)(112,133)(113,163)(114,135)(115,165)(116,137)(117,167)(118,139)(119,169)(120,121), (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), (1,108,26,59,11,118,36,49)(2,58,37,107,12,48,27,117)(3,106,28,57,13,116,38,47)(4,56,39,105,14,46,29,115)(5,104,30,55,15,114,40,45)(6,54,21,103,16,44,31,113)(7,102,32,53,17,112,22,43)(8,52,23,101,18,42,33,111)(9,120,34,51,19,110,24,41)(10,50,25,119,20,60,35,109)(61,198,93,226,71,188,83,236)(62,225,84,197,72,235,94,187)(63,196,95,224,73,186,85,234)(64,223,86,195,74,233,96,185)(65,194,97,222,75,184,87,232)(66,221,88,193,76,231,98,183)(67,192,99,240,77,182,89,230)(68,239,90,191,78,229,100,181)(69,190,81,238,79,200,91,228)(70,237,92,189,80,227,82,199)(121,216,180,158,131,206,170,148)(122,157,171,215,132,147,161,205)(123,214,162,156,133,204,172,146)(124,155,173,213,134,145,163,203)(125,212,164,154,135,202,174,144)(126,153,175,211,136,143,165,201)(127,210,166,152,137,220,176,142)(128,151,177,209,138,141,167,219)(129,208,168,150,139,218,178,160)(130,149,179,207,140,159,169,217)>;
G:=Group( (1,228,169,26,190,130,11,238,179,36,200,140)(2,239,170,37,191,121,12,229,180,27,181,131)(3,230,171,28,192,132,13,240,161,38,182,122)(4,221,172,39,193,123,14,231,162,29,183,133)(5,232,173,30,194,134,15,222,163,40,184,124)(6,223,174,21,195,125,16,233,164,31,185,135)(7,234,175,32,196,136,17,224,165,22,186,126)(8,225,176,23,197,127,18,235,166,33,187,137)(9,236,177,34,198,138,19,226,167,24,188,128)(10,227,178,25,199,129,20,237,168,35,189,139)(41,83,151,120,61,209,51,93,141,110,71,219)(42,94,152,111,62,220,52,84,142,101,72,210)(43,85,153,102,63,211,53,95,143,112,73,201)(44,96,154,113,64,202,54,86,144,103,74,212)(45,87,155,104,65,213,55,97,145,114,75,203)(46,98,156,115,66,204,56,88,146,105,76,214)(47,89,157,106,67,215,57,99,147,116,77,205)(48,100,158,117,68,206,58,90,148,107,78,216)(49,91,159,108,69,217,59,81,149,118,79,207)(50,82,160,119,70,208,60,92,150,109,80,218), (1,208)(2,141)(3,210)(4,143)(5,212)(6,145)(7,214)(8,147)(9,216)(10,149)(11,218)(12,151)(13,220)(14,153)(15,202)(16,155)(17,204)(18,157)(19,206)(20,159)(21,213)(22,156)(23,215)(24,158)(25,217)(26,160)(27,219)(28,142)(29,201)(30,144)(31,203)(32,146)(33,205)(34,148)(35,207)(36,150)(37,209)(38,152)(39,211)(40,154)(41,180)(42,122)(43,162)(44,124)(45,164)(46,126)(47,166)(48,128)(49,168)(50,130)(51,170)(52,132)(53,172)(54,134)(55,174)(56,136)(57,176)(58,138)(59,178)(60,140)(61,191)(62,240)(63,193)(64,222)(65,195)(66,224)(67,197)(68,226)(69,199)(70,228)(71,181)(72,230)(73,183)(74,232)(75,185)(76,234)(77,187)(78,236)(79,189)(80,238)(81,227)(82,190)(83,229)(84,192)(85,231)(86,194)(87,233)(88,196)(89,235)(90,198)(91,237)(92,200)(93,239)(94,182)(95,221)(96,184)(97,223)(98,186)(99,225)(100,188)(101,171)(102,123)(103,173)(104,125)(105,175)(106,127)(107,177)(108,129)(109,179)(110,131)(111,161)(112,133)(113,163)(114,135)(115,165)(116,137)(117,167)(118,139)(119,169)(120,121), (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), (1,108,26,59,11,118,36,49)(2,58,37,107,12,48,27,117)(3,106,28,57,13,116,38,47)(4,56,39,105,14,46,29,115)(5,104,30,55,15,114,40,45)(6,54,21,103,16,44,31,113)(7,102,32,53,17,112,22,43)(8,52,23,101,18,42,33,111)(9,120,34,51,19,110,24,41)(10,50,25,119,20,60,35,109)(61,198,93,226,71,188,83,236)(62,225,84,197,72,235,94,187)(63,196,95,224,73,186,85,234)(64,223,86,195,74,233,96,185)(65,194,97,222,75,184,87,232)(66,221,88,193,76,231,98,183)(67,192,99,240,77,182,89,230)(68,239,90,191,78,229,100,181)(69,190,81,238,79,200,91,228)(70,237,92,189,80,227,82,199)(121,216,180,158,131,206,170,148)(122,157,171,215,132,147,161,205)(123,214,162,156,133,204,172,146)(124,155,173,213,134,145,163,203)(125,212,164,154,135,202,174,144)(126,153,175,211,136,143,165,201)(127,210,166,152,137,220,176,142)(128,151,177,209,138,141,167,219)(129,208,168,150,139,218,178,160)(130,149,179,207,140,159,169,217) );
G=PermutationGroup([[(1,228,169,26,190,130,11,238,179,36,200,140),(2,239,170,37,191,121,12,229,180,27,181,131),(3,230,171,28,192,132,13,240,161,38,182,122),(4,221,172,39,193,123,14,231,162,29,183,133),(5,232,173,30,194,134,15,222,163,40,184,124),(6,223,174,21,195,125,16,233,164,31,185,135),(7,234,175,32,196,136,17,224,165,22,186,126),(8,225,176,23,197,127,18,235,166,33,187,137),(9,236,177,34,198,138,19,226,167,24,188,128),(10,227,178,25,199,129,20,237,168,35,189,139),(41,83,151,120,61,209,51,93,141,110,71,219),(42,94,152,111,62,220,52,84,142,101,72,210),(43,85,153,102,63,211,53,95,143,112,73,201),(44,96,154,113,64,202,54,86,144,103,74,212),(45,87,155,104,65,213,55,97,145,114,75,203),(46,98,156,115,66,204,56,88,146,105,76,214),(47,89,157,106,67,215,57,99,147,116,77,205),(48,100,158,117,68,206,58,90,148,107,78,216),(49,91,159,108,69,217,59,81,149,118,79,207),(50,82,160,119,70,208,60,92,150,109,80,218)], [(1,208),(2,141),(3,210),(4,143),(5,212),(6,145),(7,214),(8,147),(9,216),(10,149),(11,218),(12,151),(13,220),(14,153),(15,202),(16,155),(17,204),(18,157),(19,206),(20,159),(21,213),(22,156),(23,215),(24,158),(25,217),(26,160),(27,219),(28,142),(29,201),(30,144),(31,203),(32,146),(33,205),(34,148),(35,207),(36,150),(37,209),(38,152),(39,211),(40,154),(41,180),(42,122),(43,162),(44,124),(45,164),(46,126),(47,166),(48,128),(49,168),(50,130),(51,170),(52,132),(53,172),(54,134),(55,174),(56,136),(57,176),(58,138),(59,178),(60,140),(61,191),(62,240),(63,193),(64,222),(65,195),(66,224),(67,197),(68,226),(69,199),(70,228),(71,181),(72,230),(73,183),(74,232),(75,185),(76,234),(77,187),(78,236),(79,189),(80,238),(81,227),(82,190),(83,229),(84,192),(85,231),(86,194),(87,233),(88,196),(89,235),(90,198),(91,237),(92,200),(93,239),(94,182),(95,221),(96,184),(97,223),(98,186),(99,225),(100,188),(101,171),(102,123),(103,173),(104,125),(105,175),(106,127),(107,177),(108,129),(109,179),(110,131),(111,161),(112,133),(113,163),(114,135),(115,165),(116,137),(117,167),(118,139),(119,169),(120,121)], [(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)], [(1,108,26,59,11,118,36,49),(2,58,37,107,12,48,27,117),(3,106,28,57,13,116,38,47),(4,56,39,105,14,46,29,115),(5,104,30,55,15,114,40,45),(6,54,21,103,16,44,31,113),(7,102,32,53,17,112,22,43),(8,52,23,101,18,42,33,111),(9,120,34,51,19,110,24,41),(10,50,25,119,20,60,35,109),(61,198,93,226,71,188,83,236),(62,225,84,197,72,235,94,187),(63,196,95,224,73,186,85,234),(64,223,86,195,74,233,96,185),(65,194,97,222,75,184,87,232),(66,221,88,193,76,231,98,183),(67,192,99,240,77,182,89,230),(68,239,90,191,78,229,100,181),(69,190,81,238,79,200,91,228),(70,237,92,189,80,227,82,199),(121,216,180,158,131,206,170,148),(122,157,171,215,132,147,161,205),(123,214,162,156,133,204,172,146),(124,155,173,213,134,145,163,203),(125,212,164,154,135,202,174,144),(126,153,175,211,136,143,165,201),(127,210,166,152,137,220,176,142),(128,151,177,209,138,141,167,219),(129,208,168,150,139,218,178,160),(130,149,179,207,140,159,169,217)]])
45 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6 | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 24A | 24B | 30A | 30B | 40A | 40B | 40C | 40D | 60A | ··· | 60F |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 24 | 24 | 30 | 30 | 40 | 40 | 40 | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 12 | 30 | 60 | 2 | 2 | 4 | 15 | 15 | 20 | 2 | 2 | 2 | 6 | 6 | 10 | 10 | 2 | 2 | 24 | 24 | 4 | 8 | 40 | 4 | 4 | 4 | 4 | 8 | 8 | 20 | 20 | 4 | 4 | 12 | 12 | 12 | 12 | 8 | ··· | 8 |
45 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 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 | C4○D8 | S3×D4 | S3×D5 | D4×D5 | D24⋊C2 | C2×S3×D5 | SD16⋊3D5 | D10⋊D6 | D12.D10 |
| kernel | D12.D10 | D15⋊2C8 | C5⋊D24 | C15⋊SD16 | C3×C5⋊Q16 | C5×Q8⋊2S3 | D12⋊D5 | Q8⋊3D15 | C5⋊Q16 | Dic15 | D30 | Q8⋊2S3 | C5⋊2C8 | Dic10 | C5×Q8 | C3⋊C8 | D12 | C3×Q8 | C15 | C10 | Q8 | 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 | 4 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 |
Matrix representation of D12.D10 ►in GL6(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 131 | 177 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 187 | 0 | 0 |
| 0 | 0 | 67 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 74 | 16 | 0 | 0 | 0 | 0 |
| 185 | 167 | 0 | 0 | 0 | 0 |
| 0 | 0 | 239 | 187 | 0 | 0 |
| 0 | 0 | 67 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 51 | 239 | 0 | 0 | 0 | 0 |
| 96 | 190 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 189 | 189 |
| 0 | 0 | 0 | 0 | 52 | 1 |
| 211 | 0 | 0 | 0 | 0 | 0 |
| 236 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 52 | 52 |
| 0 | 0 | 0 | 0 | 240 | 189 |
G:=sub<GL(6,GF(241))| [64,131,0,0,0,0,0,177,0,0,0,0,0,0,239,67,0,0,0,0,187,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[74,185,0,0,0,0,16,167,0,0,0,0,0,0,239,67,0,0,0,0,187,2,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[51,96,0,0,0,0,239,190,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,189,52,0,0,0,0,189,1],[211,236,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,52,240,0,0,0,0,52,189] >;
D12.D10 in GAP, Magma, Sage, TeX
D_{12}.D_{10} % in TeX
G:=Group("D12.D10"); // GroupNames label
G:=SmallGroup(480,599);
// by ID
G=gap.SmallGroup(480,599);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,254,135,100,675,346,185,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=1,c^10=a^6,d^2=a^3,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^9>;
// generators/relations