metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.39D4, D12.37D10, C60.156C23, Dic6.38D10, Dic10.38D6, C4○D12.4D5, (C2×C20).95D6, (C2×C30).52D4, C30.84(C2×D4), C60.7C4⋊7C2, (C6×Dic10)⋊1C2, (C2×Dic10)⋊8S3, C15⋊Q16⋊14C2, (C2×C12).95D10, C15⋊8(C8.C22), C20.D6⋊14C2, C3⋊4(D4.9D10), C5⋊4(Q8.11D6), C4.24(C15⋊D4), C12.31(C5⋊D4), C20.82(C3⋊D4), (C2×C60).32C22, C20.95(C22×S3), C12.93(C22×D5), C15⋊3C8.26C22, (C5×D12).43C22, C22.5(C15⋊D4), (C5×Dic6).45C22, (C3×Dic10).45C22, C4.129(C2×S3×D5), (C2×C4).13(S3×D5), C6.78(C2×C5⋊D4), (C5×C4○D12).1C2, C10.79(C2×C3⋊D4), C2.12(C2×C15⋊D4), (C2×C6).55(C5⋊D4), (C2×C10).11(C3⋊D4), SmallGroup(480,385)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.37D10
G = < a,b,c,d | a12=b2=c10=1, d2=a6, bab=a-1, ac=ca, dad-1=a7, cbc-1=a6b, dbd-1=a3b, dcd-1=c-1 >
Subgroups: 476 in 120 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, C2×C10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, C5×S3, C30, C30, C8.C22, C5⋊2C8, Dic10, Dic10, C2×Dic5, C2×C20, C2×C20, C5×D4, C5×Q8, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, C5×Dic3, C3×Dic5, C60, S3×C10, C2×C30, C4.Dic5, D4.D5, C5⋊Q16, C2×Dic10, C5×C4○D4, Q8.11D6, C15⋊3C8, C3×Dic10, C3×Dic10, C6×Dic5, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, C2×C60, D4.9D10, C20.D6, C15⋊Q16, C60.7C4, C6×Dic10, C5×C4○D12, D12.37D10
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, C5⋊D4, C22×D5, C2×C3⋊D4, S3×D5, C2×C5⋊D4, Q8.11D6, C15⋊D4, C2×S3×D5, D4.9D10, C2×C15⋊D4, D12.37D10
►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 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 205)(14 216)(15 215)(16 214)(17 213)(18 212)(19 211)(20 210)(21 209)(22 208)(23 207)(24 206)(25 221)(26 220)(27 219)(28 218)(29 217)(30 228)(31 227)(32 226)(33 225)(34 224)(35 223)(36 222)(49 141)(50 140)(51 139)(52 138)(53 137)(54 136)(55 135)(56 134)(57 133)(58 144)(59 143)(60 142)(61 172)(62 171)(63 170)(64 169)(65 180)(66 179)(67 178)(68 177)(69 176)(70 175)(71 174)(72 173)(73 130)(74 129)(75 128)(76 127)(77 126)(78 125)(79 124)(80 123)(81 122)(82 121)(83 132)(84 131)(85 237)(86 236)(87 235)(88 234)(89 233)(90 232)(91 231)(92 230)(93 229)(94 240)(95 239)(96 238)(97 197)(98 196)(99 195)(100 194)(101 193)(102 204)(103 203)(104 202)(105 201)(106 200)(107 199)(108 198)(109 186)(110 185)(111 184)(112 183)(113 182)(114 181)(115 192)(116 191)(117 190)(118 189)(119 188)(120 187)(145 167)(146 166)(147 165)(148 164)(149 163)(150 162)(151 161)(152 160)(153 159)(154 158)(155 157)(156 168)
(1 146 204 177 127)(2 147 193 178 128)(3 148 194 179 129)(4 149 195 180 130)(5 150 196 169 131)(6 151 197 170 132)(7 152 198 171 121)(8 153 199 172 122)(9 154 200 173 123)(10 155 201 174 124)(11 156 202 175 125)(12 145 203 176 126)(13 235 219 140 119)(14 236 220 141 120)(15 237 221 142 109)(16 238 222 143 110)(17 239 223 144 111)(18 240 224 133 112)(19 229 225 134 113)(20 230 226 135 114)(21 231 227 136 115)(22 232 228 137 116)(23 233 217 138 117)(24 234 218 139 118)(25 54 186 209 85 31 60 192 215 91)(26 55 187 210 86 32 49 181 216 92)(27 56 188 211 87 33 50 182 205 93)(28 57 189 212 88 34 51 183 206 94)(29 58 190 213 89 35 52 184 207 95)(30 59 191 214 90 36 53 185 208 96)(37 164 106 66 80 43 158 100 72 74)(38 165 107 67 81 44 159 101 61 75)(39 166 108 68 82 45 160 102 62 76)(40 167 97 69 83 46 161 103 63 77)(41 168 98 70 84 47 162 104 64 78)(42 157 99 71 73 48 163 105 65 79)
(1 119 7 113)(2 114 8 120)(3 109 9 115)(4 116 10 110)(5 111 11 117)(6 118 12 112)(13 121 19 127)(14 128 20 122)(15 123 21 129)(16 130 22 124)(17 125 23 131)(18 132 24 126)(25 103 31 97)(26 98 32 104)(27 105 33 99)(28 100 34 106)(29 107 35 101)(30 102 36 108)(37 189 43 183)(38 184 44 190)(39 191 45 185)(40 186 46 192)(41 181 47 187)(42 188 48 182)(49 162 55 168)(50 157 56 163)(51 164 57 158)(52 159 58 165)(53 166 59 160)(54 161 60 167)(61 95 67 89)(62 90 68 96)(63 85 69 91)(64 92 70 86)(65 87 71 93)(66 94 72 88)(73 205 79 211)(74 212 80 206)(75 207 81 213)(76 214 82 208)(77 209 83 215)(78 216 84 210)(133 151 139 145)(134 146 140 152)(135 153 141 147)(136 148 142 154)(137 155 143 149)(138 150 144 156)(169 239 175 233)(170 234 176 240)(171 229 177 235)(172 236 178 230)(173 231 179 237)(174 238 180 232)(193 226 199 220)(194 221 200 227)(195 228 201 222)(196 223 202 217)(197 218 203 224)(198 225 204 219)
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,205)(14,216)(15,215)(16,214)(17,213)(18,212)(19,211)(20,210)(21,209)(22,208)(23,207)(24,206)(25,221)(26,220)(27,219)(28,218)(29,217)(30,228)(31,227)(32,226)(33,225)(34,224)(35,223)(36,222)(49,141)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,144)(59,143)(60,142)(61,172)(62,171)(63,170)(64,169)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,132)(84,131)(85,237)(86,236)(87,235)(88,234)(89,233)(90,232)(91,231)(92,230)(93,229)(94,240)(95,239)(96,238)(97,197)(98,196)(99,195)(100,194)(101,193)(102,204)(103,203)(104,202)(105,201)(106,200)(107,199)(108,198)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)(152,160)(153,159)(154,158)(155,157)(156,168), (1,146,204,177,127)(2,147,193,178,128)(3,148,194,179,129)(4,149,195,180,130)(5,150,196,169,131)(6,151,197,170,132)(7,152,198,171,121)(8,153,199,172,122)(9,154,200,173,123)(10,155,201,174,124)(11,156,202,175,125)(12,145,203,176,126)(13,235,219,140,119)(14,236,220,141,120)(15,237,221,142,109)(16,238,222,143,110)(17,239,223,144,111)(18,240,224,133,112)(19,229,225,134,113)(20,230,226,135,114)(21,231,227,136,115)(22,232,228,137,116)(23,233,217,138,117)(24,234,218,139,118)(25,54,186,209,85,31,60,192,215,91)(26,55,187,210,86,32,49,181,216,92)(27,56,188,211,87,33,50,182,205,93)(28,57,189,212,88,34,51,183,206,94)(29,58,190,213,89,35,52,184,207,95)(30,59,191,214,90,36,53,185,208,96)(37,164,106,66,80,43,158,100,72,74)(38,165,107,67,81,44,159,101,61,75)(39,166,108,68,82,45,160,102,62,76)(40,167,97,69,83,46,161,103,63,77)(41,168,98,70,84,47,162,104,64,78)(42,157,99,71,73,48,163,105,65,79), (1,119,7,113)(2,114,8,120)(3,109,9,115)(4,116,10,110)(5,111,11,117)(6,118,12,112)(13,121,19,127)(14,128,20,122)(15,123,21,129)(16,130,22,124)(17,125,23,131)(18,132,24,126)(25,103,31,97)(26,98,32,104)(27,105,33,99)(28,100,34,106)(29,107,35,101)(30,102,36,108)(37,189,43,183)(38,184,44,190)(39,191,45,185)(40,186,46,192)(41,181,47,187)(42,188,48,182)(49,162,55,168)(50,157,56,163)(51,164,57,158)(52,159,58,165)(53,166,59,160)(54,161,60,167)(61,95,67,89)(62,90,68,96)(63,85,69,91)(64,92,70,86)(65,87,71,93)(66,94,72,88)(73,205,79,211)(74,212,80,206)(75,207,81,213)(76,214,82,208)(77,209,83,215)(78,216,84,210)(133,151,139,145)(134,146,140,152)(135,153,141,147)(136,148,142,154)(137,155,143,149)(138,150,144,156)(169,239,175,233)(170,234,176,240)(171,229,177,235)(172,236,178,230)(173,231,179,237)(174,238,180,232)(193,226,199,220)(194,221,200,227)(195,228,201,222)(196,223,202,217)(197,218,203,224)(198,225,204,219)>;
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), (1,45)(2,44)(3,43)(4,42)(5,41)(6,40)(7,39)(8,38)(9,37)(10,48)(11,47)(12,46)(13,205)(14,216)(15,215)(16,214)(17,213)(18,212)(19,211)(20,210)(21,209)(22,208)(23,207)(24,206)(25,221)(26,220)(27,219)(28,218)(29,217)(30,228)(31,227)(32,226)(33,225)(34,224)(35,223)(36,222)(49,141)(50,140)(51,139)(52,138)(53,137)(54,136)(55,135)(56,134)(57,133)(58,144)(59,143)(60,142)(61,172)(62,171)(63,170)(64,169)(65,180)(66,179)(67,178)(68,177)(69,176)(70,175)(71,174)(72,173)(73,130)(74,129)(75,128)(76,127)(77,126)(78,125)(79,124)(80,123)(81,122)(82,121)(83,132)(84,131)(85,237)(86,236)(87,235)(88,234)(89,233)(90,232)(91,231)(92,230)(93,229)(94,240)(95,239)(96,238)(97,197)(98,196)(99,195)(100,194)(101,193)(102,204)(103,203)(104,202)(105,201)(106,200)(107,199)(108,198)(109,186)(110,185)(111,184)(112,183)(113,182)(114,181)(115,192)(116,191)(117,190)(118,189)(119,188)(120,187)(145,167)(146,166)(147,165)(148,164)(149,163)(150,162)(151,161)(152,160)(153,159)(154,158)(155,157)(156,168), (1,146,204,177,127)(2,147,193,178,128)(3,148,194,179,129)(4,149,195,180,130)(5,150,196,169,131)(6,151,197,170,132)(7,152,198,171,121)(8,153,199,172,122)(9,154,200,173,123)(10,155,201,174,124)(11,156,202,175,125)(12,145,203,176,126)(13,235,219,140,119)(14,236,220,141,120)(15,237,221,142,109)(16,238,222,143,110)(17,239,223,144,111)(18,240,224,133,112)(19,229,225,134,113)(20,230,226,135,114)(21,231,227,136,115)(22,232,228,137,116)(23,233,217,138,117)(24,234,218,139,118)(25,54,186,209,85,31,60,192,215,91)(26,55,187,210,86,32,49,181,216,92)(27,56,188,211,87,33,50,182,205,93)(28,57,189,212,88,34,51,183,206,94)(29,58,190,213,89,35,52,184,207,95)(30,59,191,214,90,36,53,185,208,96)(37,164,106,66,80,43,158,100,72,74)(38,165,107,67,81,44,159,101,61,75)(39,166,108,68,82,45,160,102,62,76)(40,167,97,69,83,46,161,103,63,77)(41,168,98,70,84,47,162,104,64,78)(42,157,99,71,73,48,163,105,65,79), (1,119,7,113)(2,114,8,120)(3,109,9,115)(4,116,10,110)(5,111,11,117)(6,118,12,112)(13,121,19,127)(14,128,20,122)(15,123,21,129)(16,130,22,124)(17,125,23,131)(18,132,24,126)(25,103,31,97)(26,98,32,104)(27,105,33,99)(28,100,34,106)(29,107,35,101)(30,102,36,108)(37,189,43,183)(38,184,44,190)(39,191,45,185)(40,186,46,192)(41,181,47,187)(42,188,48,182)(49,162,55,168)(50,157,56,163)(51,164,57,158)(52,159,58,165)(53,166,59,160)(54,161,60,167)(61,95,67,89)(62,90,68,96)(63,85,69,91)(64,92,70,86)(65,87,71,93)(66,94,72,88)(73,205,79,211)(74,212,80,206)(75,207,81,213)(76,214,82,208)(77,209,83,215)(78,216,84,210)(133,151,139,145)(134,146,140,152)(135,153,141,147)(136,148,142,154)(137,155,143,149)(138,150,144,156)(169,239,175,233)(170,234,176,240)(171,229,177,235)(172,236,178,230)(173,231,179,237)(174,238,180,232)(193,226,199,220)(194,221,200,227)(195,228,201,222)(196,223,202,217)(197,218,203,224)(198,225,204,219) );
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)], [(1,45),(2,44),(3,43),(4,42),(5,41),(6,40),(7,39),(8,38),(9,37),(10,48),(11,47),(12,46),(13,205),(14,216),(15,215),(16,214),(17,213),(18,212),(19,211),(20,210),(21,209),(22,208),(23,207),(24,206),(25,221),(26,220),(27,219),(28,218),(29,217),(30,228),(31,227),(32,226),(33,225),(34,224),(35,223),(36,222),(49,141),(50,140),(51,139),(52,138),(53,137),(54,136),(55,135),(56,134),(57,133),(58,144),(59,143),(60,142),(61,172),(62,171),(63,170),(64,169),(65,180),(66,179),(67,178),(68,177),(69,176),(70,175),(71,174),(72,173),(73,130),(74,129),(75,128),(76,127),(77,126),(78,125),(79,124),(80,123),(81,122),(82,121),(83,132),(84,131),(85,237),(86,236),(87,235),(88,234),(89,233),(90,232),(91,231),(92,230),(93,229),(94,240),(95,239),(96,238),(97,197),(98,196),(99,195),(100,194),(101,193),(102,204),(103,203),(104,202),(105,201),(106,200),(107,199),(108,198),(109,186),(110,185),(111,184),(112,183),(113,182),(114,181),(115,192),(116,191),(117,190),(118,189),(119,188),(120,187),(145,167),(146,166),(147,165),(148,164),(149,163),(150,162),(151,161),(152,160),(153,159),(154,158),(155,157),(156,168)], [(1,146,204,177,127),(2,147,193,178,128),(3,148,194,179,129),(4,149,195,180,130),(5,150,196,169,131),(6,151,197,170,132),(7,152,198,171,121),(8,153,199,172,122),(9,154,200,173,123),(10,155,201,174,124),(11,156,202,175,125),(12,145,203,176,126),(13,235,219,140,119),(14,236,220,141,120),(15,237,221,142,109),(16,238,222,143,110),(17,239,223,144,111),(18,240,224,133,112),(19,229,225,134,113),(20,230,226,135,114),(21,231,227,136,115),(22,232,228,137,116),(23,233,217,138,117),(24,234,218,139,118),(25,54,186,209,85,31,60,192,215,91),(26,55,187,210,86,32,49,181,216,92),(27,56,188,211,87,33,50,182,205,93),(28,57,189,212,88,34,51,183,206,94),(29,58,190,213,89,35,52,184,207,95),(30,59,191,214,90,36,53,185,208,96),(37,164,106,66,80,43,158,100,72,74),(38,165,107,67,81,44,159,101,61,75),(39,166,108,68,82,45,160,102,62,76),(40,167,97,69,83,46,161,103,63,77),(41,168,98,70,84,47,162,104,64,78),(42,157,99,71,73,48,163,105,65,79)], [(1,119,7,113),(2,114,8,120),(3,109,9,115),(4,116,10,110),(5,111,11,117),(6,118,12,112),(13,121,19,127),(14,128,20,122),(15,123,21,129),(16,130,22,124),(17,125,23,131),(18,132,24,126),(25,103,31,97),(26,98,32,104),(27,105,33,99),(28,100,34,106),(29,107,35,101),(30,102,36,108),(37,189,43,183),(38,184,44,190),(39,191,45,185),(40,186,46,192),(41,181,47,187),(42,188,48,182),(49,162,55,168),(50,157,56,163),(51,164,57,158),(52,159,58,165),(53,166,59,160),(54,161,60,167),(61,95,67,89),(62,90,68,96),(63,85,69,91),(64,92,70,86),(65,87,71,93),(66,94,72,88),(73,205,79,211),(74,212,80,206),(75,207,81,213),(76,214,82,208),(77,209,83,215),(78,216,84,210),(133,151,139,145),(134,146,140,152),(135,153,141,147),(136,148,142,154),(137,155,143,149),(138,150,144,156),(169,239,175,233),(170,234,176,240),(171,229,177,235),(172,236,178,230),(173,231,179,237),(174,238,180,232),(193,226,199,220),(194,221,200,227),(195,228,201,222),(196,223,202,217),(197,218,203,224),(198,225,204,219)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 2 | 2 | 2 | 12 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 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 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | C3⋊D4 | C3⋊D4 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D5 | Q8.11D6 | C15⋊D4 | C2×S3×D5 | C15⋊D4 | D4.9D10 | D12.37D10 |
| kernel | D12.37D10 | C20.D6 | C15⋊Q16 | C60.7C4 | C6×Dic10 | C5×C4○D12 | C2×Dic10 | C60 | C2×C30 | C4○D12 | Dic10 | C2×C20 | Dic6 | D12 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of D12.37D10 ►in GL6(𝔽241)
| 240 | 0 | 0 | 0 | 0 | 0 |
| 0 | 240 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 226 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 225 | 0 |
| 27 | 138 | 0 | 0 | 0 | 0 |
| 103 | 214 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 16 |
| 0 | 0 | 0 | 0 | 225 | 0 |
| 0 | 0 | 0 | 15 | 0 | 0 |
| 0 | 0 | 226 | 0 | 0 | 0 |
| 1 | 189 | 0 | 0 | 0 | 0 |
| 52 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 240 | 0 |
| 0 | 0 | 0 | 0 | 0 | 240 |
| 85 | 200 | 0 | 0 | 0 | 0 |
| 41 | 156 | 0 | 0 | 0 | 0 |
| 0 | 0 | 172 | 96 | 0 | 0 |
| 0 | 0 | 96 | 69 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 69 |
| 0 | 0 | 0 | 0 | 69 | 145 |
G:=sub<GL(6,GF(241))| [240,0,0,0,0,0,0,240,0,0,0,0,0,0,0,226,0,0,0,0,15,0,0,0,0,0,0,0,0,225,0,0,0,0,16,0],[27,103,0,0,0,0,138,214,0,0,0,0,0,0,0,0,0,226,0,0,0,0,15,0,0,0,0,225,0,0,0,0,16,0,0,0],[1,52,0,0,0,0,189,189,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[85,41,0,0,0,0,200,156,0,0,0,0,0,0,172,96,0,0,0,0,96,69,0,0,0,0,0,0,96,69,0,0,0,0,69,145] >;
D12.37D10 in GAP, Magma, Sage, TeX
D_{12}._{37}D_{10} % in TeX
G:=Group("D12.37D10"); // GroupNames label
G:=SmallGroup(480,385);
// by ID
G=gap.SmallGroup(480,385);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,219,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^10=1,d^2=a^6,b*a*b=a^-1,a*c=c*a,d*a*d^-1=a^7,c*b*c^-1=a^6*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations