direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×C4×C12, (C4×C60)⋊20C2, (C4×C20)⋊14C6, C60⋊36(C2×C4), C20⋊8(C2×C12), C15⋊10(C2×C42), (C4×Dic5)⋊17C6, Dic5⋊8(C2×C12), (C12×Dic5)⋊35C2, D10.17(C2×C12), (C2×C12).447D10, C10.15(C22×C12), (C2×C30).329C23, C30.173(C22×C4), (C2×C60).547C22, (C6×Dic5).281C22, C5⋊2(C2×C4×C12), C6.98(C2×C4×D5), C2.1(D5×C2×C12), (C2×C4×D5).19C6, C22.9(D5×C2×C6), (D5×C2×C12).40C2, (C2×C4).96(C6×D5), (C6×D5).66(C2×C4), (C2×C20).112(C2×C6), (C3×Dic5)⋊29(C2×C4), (D5×C2×C6).154C22, (C2×C10).12(C22×C6), (C2×Dic5).58(C2×C6), (C22×D5).43(C2×C6), (C2×C6).325(C22×D5), SmallGroup(480,664)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×C4×C12 |
Generators and relations for D5×C4×C12
G = < a,b,c,d | a4=b12=c5=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 528 in 216 conjugacy classes, 138 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C42, C42, C22×C4, Dic5, C20, D10, C2×C10, C2×C12, C2×C12, C22×C6, C3×D5, C30, C2×C42, C4×D5, C2×Dic5, C2×C20, C22×D5, C4×C12, C4×C12, C22×C12, C3×Dic5, C60, C6×D5, C2×C30, C4×Dic5, C4×C20, C2×C4×D5, C2×C4×C12, D5×C12, C6×Dic5, C2×C60, D5×C2×C6, D5×C42, C12×Dic5, C4×C60, D5×C2×C12, D5×C4×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C23, D5, C12, C2×C6, C42, C22×C4, D10, C2×C12, C22×C6, C3×D5, C2×C42, C4×D5, C22×D5, C4×C12, C22×C12, C6×D5, C2×C4×D5, C2×C4×C12, D5×C12, D5×C2×C6, D5×C42, D5×C2×C12, D5×C4×C12
►On 240 points
Generators in S240
(1 36 112 134)(2 25 113 135)(3 26 114 136)(4 27 115 137)(5 28 116 138)(6 29 117 139)(7 30 118 140)(8 31 119 141)(9 32 120 142)(10 33 109 143)(11 34 110 144)(12 35 111 133)(13 200 86 75)(14 201 87 76)(15 202 88 77)(16 203 89 78)(17 204 90 79)(18 193 91 80)(19 194 92 81)(20 195 93 82)(21 196 94 83)(22 197 95 84)(23 198 96 73)(24 199 85 74)(37 236 52 170)(38 237 53 171)(39 238 54 172)(40 239 55 173)(41 240 56 174)(42 229 57 175)(43 230 58 176)(44 231 59 177)(45 232 60 178)(46 233 49 179)(47 234 50 180)(48 235 51 169)(61 163 189 218)(62 164 190 219)(63 165 191 220)(64 166 192 221)(65 167 181 222)(66 168 182 223)(67 157 183 224)(68 158 184 225)(69 159 185 226)(70 160 186 227)(71 161 187 228)(72 162 188 217)(97 125 208 147)(98 126 209 148)(99 127 210 149)(100 128 211 150)(101 129 212 151)(102 130 213 152)(103 131 214 153)(104 132 215 154)(105 121 216 155)(106 122 205 156)(107 123 206 145)(108 124 207 146)
(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)
(1 165 93 104 234)(2 166 94 105 235)(3 167 95 106 236)(4 168 96 107 237)(5 157 85 108 238)(6 158 86 97 239)(7 159 87 98 240)(8 160 88 99 229)(9 161 89 100 230)(10 162 90 101 231)(11 163 91 102 232)(12 164 92 103 233)(13 208 173 117 225)(14 209 174 118 226)(15 210 175 119 227)(16 211 176 120 228)(17 212 177 109 217)(18 213 178 110 218)(19 214 179 111 219)(20 215 180 112 220)(21 216 169 113 221)(22 205 170 114 222)(23 206 171 115 223)(24 207 172 116 224)(25 192 83 121 51)(26 181 84 122 52)(27 182 73 123 53)(28 183 74 124 54)(29 184 75 125 55)(30 185 76 126 56)(31 186 77 127 57)(32 187 78 128 58)(33 188 79 129 59)(34 189 80 130 60)(35 190 81 131 49)(36 191 82 132 50)(37 136 65 197 156)(38 137 66 198 145)(39 138 67 199 146)(40 139 68 200 147)(41 140 69 201 148)(42 141 70 202 149)(43 142 71 203 150)(44 143 72 204 151)(45 144 61 193 152)(46 133 62 194 153)(47 134 63 195 154)(48 135 64 196 155)
(1 240)(2 229)(3 230)(4 231)(5 232)(6 233)(7 234)(8 235)(9 236)(10 237)(11 238)(12 239)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 57)(26 58)(27 59)(28 60)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(36 56)(37 142)(38 143)(39 144)(40 133)(41 134)(42 135)(43 136)(44 137)(45 138)(46 139)(47 140)(48 141)(61 146)(62 147)(63 148)(64 149)(65 150)(66 151)(67 152)(68 153)(69 154)(70 155)(71 156)(72 145)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(97 164)(98 165)(99 166)(100 167)(101 168)(102 157)(103 158)(104 159)(105 160)(106 161)(107 162)(108 163)(109 171)(110 172)(111 173)(112 174)(113 175)(114 176)(115 177)(116 178)(117 179)(118 180)(119 169)(120 170)(121 186)(122 187)(123 188)(124 189)(125 190)(126 191)(127 192)(128 181)(129 182)(130 183)(131 184)(132 185)(193 199)(194 200)(195 201)(196 202)(197 203)(198 204)(205 228)(206 217)(207 218)(208 219)(209 220)(210 221)(211 222)(212 223)(213 224)(214 225)(215 226)(216 227)
G:=sub<Sym(240)| (1,36,112,134)(2,25,113,135)(3,26,114,136)(4,27,115,137)(5,28,116,138)(6,29,117,139)(7,30,118,140)(8,31,119,141)(9,32,120,142)(10,33,109,143)(11,34,110,144)(12,35,111,133)(13,200,86,75)(14,201,87,76)(15,202,88,77)(16,203,89,78)(17,204,90,79)(18,193,91,80)(19,194,92,81)(20,195,93,82)(21,196,94,83)(22,197,95,84)(23,198,96,73)(24,199,85,74)(37,236,52,170)(38,237,53,171)(39,238,54,172)(40,239,55,173)(41,240,56,174)(42,229,57,175)(43,230,58,176)(44,231,59,177)(45,232,60,178)(46,233,49,179)(47,234,50,180)(48,235,51,169)(61,163,189,218)(62,164,190,219)(63,165,191,220)(64,166,192,221)(65,167,181,222)(66,168,182,223)(67,157,183,224)(68,158,184,225)(69,159,185,226)(70,160,186,227)(71,161,187,228)(72,162,188,217)(97,125,208,147)(98,126,209,148)(99,127,210,149)(100,128,211,150)(101,129,212,151)(102,130,213,152)(103,131,214,153)(104,132,215,154)(105,121,216,155)(106,122,205,156)(107,123,206,145)(108,124,207,146), (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), (1,165,93,104,234)(2,166,94,105,235)(3,167,95,106,236)(4,168,96,107,237)(5,157,85,108,238)(6,158,86,97,239)(7,159,87,98,240)(8,160,88,99,229)(9,161,89,100,230)(10,162,90,101,231)(11,163,91,102,232)(12,164,92,103,233)(13,208,173,117,225)(14,209,174,118,226)(15,210,175,119,227)(16,211,176,120,228)(17,212,177,109,217)(18,213,178,110,218)(19,214,179,111,219)(20,215,180,112,220)(21,216,169,113,221)(22,205,170,114,222)(23,206,171,115,223)(24,207,172,116,224)(25,192,83,121,51)(26,181,84,122,52)(27,182,73,123,53)(28,183,74,124,54)(29,184,75,125,55)(30,185,76,126,56)(31,186,77,127,57)(32,187,78,128,58)(33,188,79,129,59)(34,189,80,130,60)(35,190,81,131,49)(36,191,82,132,50)(37,136,65,197,156)(38,137,66,198,145)(39,138,67,199,146)(40,139,68,200,147)(41,140,69,201,148)(42,141,70,202,149)(43,142,71,203,150)(44,143,72,204,151)(45,144,61,193,152)(46,133,62,194,153)(47,134,63,195,154)(48,135,64,196,155), (1,240)(2,229)(3,230)(4,231)(5,232)(6,233)(7,234)(8,235)(9,236)(10,237)(11,238)(12,239)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,57)(26,58)(27,59)(28,60)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,142)(38,143)(39,144)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,145)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,164)(98,165)(99,166)(100,167)(101,168)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,171)(110,172)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,169)(120,170)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,181)(129,182)(130,183)(131,184)(132,185)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,228)(206,217)(207,218)(208,219)(209,220)(210,221)(211,222)(212,223)(213,224)(214,225)(215,226)(216,227)>;
G:=Group( (1,36,112,134)(2,25,113,135)(3,26,114,136)(4,27,115,137)(5,28,116,138)(6,29,117,139)(7,30,118,140)(8,31,119,141)(9,32,120,142)(10,33,109,143)(11,34,110,144)(12,35,111,133)(13,200,86,75)(14,201,87,76)(15,202,88,77)(16,203,89,78)(17,204,90,79)(18,193,91,80)(19,194,92,81)(20,195,93,82)(21,196,94,83)(22,197,95,84)(23,198,96,73)(24,199,85,74)(37,236,52,170)(38,237,53,171)(39,238,54,172)(40,239,55,173)(41,240,56,174)(42,229,57,175)(43,230,58,176)(44,231,59,177)(45,232,60,178)(46,233,49,179)(47,234,50,180)(48,235,51,169)(61,163,189,218)(62,164,190,219)(63,165,191,220)(64,166,192,221)(65,167,181,222)(66,168,182,223)(67,157,183,224)(68,158,184,225)(69,159,185,226)(70,160,186,227)(71,161,187,228)(72,162,188,217)(97,125,208,147)(98,126,209,148)(99,127,210,149)(100,128,211,150)(101,129,212,151)(102,130,213,152)(103,131,214,153)(104,132,215,154)(105,121,216,155)(106,122,205,156)(107,123,206,145)(108,124,207,146), (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), (1,165,93,104,234)(2,166,94,105,235)(3,167,95,106,236)(4,168,96,107,237)(5,157,85,108,238)(6,158,86,97,239)(7,159,87,98,240)(8,160,88,99,229)(9,161,89,100,230)(10,162,90,101,231)(11,163,91,102,232)(12,164,92,103,233)(13,208,173,117,225)(14,209,174,118,226)(15,210,175,119,227)(16,211,176,120,228)(17,212,177,109,217)(18,213,178,110,218)(19,214,179,111,219)(20,215,180,112,220)(21,216,169,113,221)(22,205,170,114,222)(23,206,171,115,223)(24,207,172,116,224)(25,192,83,121,51)(26,181,84,122,52)(27,182,73,123,53)(28,183,74,124,54)(29,184,75,125,55)(30,185,76,126,56)(31,186,77,127,57)(32,187,78,128,58)(33,188,79,129,59)(34,189,80,130,60)(35,190,81,131,49)(36,191,82,132,50)(37,136,65,197,156)(38,137,66,198,145)(39,138,67,199,146)(40,139,68,200,147)(41,140,69,201,148)(42,141,70,202,149)(43,142,71,203,150)(44,143,72,204,151)(45,144,61,193,152)(46,133,62,194,153)(47,134,63,195,154)(48,135,64,196,155), (1,240)(2,229)(3,230)(4,231)(5,232)(6,233)(7,234)(8,235)(9,236)(10,237)(11,238)(12,239)(13,19)(14,20)(15,21)(16,22)(17,23)(18,24)(25,57)(26,58)(27,59)(28,60)(29,49)(30,50)(31,51)(32,52)(33,53)(34,54)(35,55)(36,56)(37,142)(38,143)(39,144)(40,133)(41,134)(42,135)(43,136)(44,137)(45,138)(46,139)(47,140)(48,141)(61,146)(62,147)(63,148)(64,149)(65,150)(66,151)(67,152)(68,153)(69,154)(70,155)(71,156)(72,145)(73,79)(74,80)(75,81)(76,82)(77,83)(78,84)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(97,164)(98,165)(99,166)(100,167)(101,168)(102,157)(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,171)(110,172)(111,173)(112,174)(113,175)(114,176)(115,177)(116,178)(117,179)(118,180)(119,169)(120,170)(121,186)(122,187)(123,188)(124,189)(125,190)(126,191)(127,192)(128,181)(129,182)(130,183)(131,184)(132,185)(193,199)(194,200)(195,201)(196,202)(197,203)(198,204)(205,228)(206,217)(207,218)(208,219)(209,220)(210,221)(211,222)(212,223)(213,224)(214,225)(215,226)(216,227) );
G=PermutationGroup([[(1,36,112,134),(2,25,113,135),(3,26,114,136),(4,27,115,137),(5,28,116,138),(6,29,117,139),(7,30,118,140),(8,31,119,141),(9,32,120,142),(10,33,109,143),(11,34,110,144),(12,35,111,133),(13,200,86,75),(14,201,87,76),(15,202,88,77),(16,203,89,78),(17,204,90,79),(18,193,91,80),(19,194,92,81),(20,195,93,82),(21,196,94,83),(22,197,95,84),(23,198,96,73),(24,199,85,74),(37,236,52,170),(38,237,53,171),(39,238,54,172),(40,239,55,173),(41,240,56,174),(42,229,57,175),(43,230,58,176),(44,231,59,177),(45,232,60,178),(46,233,49,179),(47,234,50,180),(48,235,51,169),(61,163,189,218),(62,164,190,219),(63,165,191,220),(64,166,192,221),(65,167,181,222),(66,168,182,223),(67,157,183,224),(68,158,184,225),(69,159,185,226),(70,160,186,227),(71,161,187,228),(72,162,188,217),(97,125,208,147),(98,126,209,148),(99,127,210,149),(100,128,211,150),(101,129,212,151),(102,130,213,152),(103,131,214,153),(104,132,215,154),(105,121,216,155),(106,122,205,156),(107,123,206,145),(108,124,207,146)], [(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)], [(1,165,93,104,234),(2,166,94,105,235),(3,167,95,106,236),(4,168,96,107,237),(5,157,85,108,238),(6,158,86,97,239),(7,159,87,98,240),(8,160,88,99,229),(9,161,89,100,230),(10,162,90,101,231),(11,163,91,102,232),(12,164,92,103,233),(13,208,173,117,225),(14,209,174,118,226),(15,210,175,119,227),(16,211,176,120,228),(17,212,177,109,217),(18,213,178,110,218),(19,214,179,111,219),(20,215,180,112,220),(21,216,169,113,221),(22,205,170,114,222),(23,206,171,115,223),(24,207,172,116,224),(25,192,83,121,51),(26,181,84,122,52),(27,182,73,123,53),(28,183,74,124,54),(29,184,75,125,55),(30,185,76,126,56),(31,186,77,127,57),(32,187,78,128,58),(33,188,79,129,59),(34,189,80,130,60),(35,190,81,131,49),(36,191,82,132,50),(37,136,65,197,156),(38,137,66,198,145),(39,138,67,199,146),(40,139,68,200,147),(41,140,69,201,148),(42,141,70,202,149),(43,142,71,203,150),(44,143,72,204,151),(45,144,61,193,152),(46,133,62,194,153),(47,134,63,195,154),(48,135,64,196,155)], [(1,240),(2,229),(3,230),(4,231),(5,232),(6,233),(7,234),(8,235),(9,236),(10,237),(11,238),(12,239),(13,19),(14,20),(15,21),(16,22),(17,23),(18,24),(25,57),(26,58),(27,59),(28,60),(29,49),(30,50),(31,51),(32,52),(33,53),(34,54),(35,55),(36,56),(37,142),(38,143),(39,144),(40,133),(41,134),(42,135),(43,136),(44,137),(45,138),(46,139),(47,140),(48,141),(61,146),(62,147),(63,148),(64,149),(65,150),(66,151),(67,152),(68,153),(69,154),(70,155),(71,156),(72,145),(73,79),(74,80),(75,81),(76,82),(77,83),(78,84),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(97,164),(98,165),(99,166),(100,167),(101,168),(102,157),(103,158),(104,159),(105,160),(106,161),(107,162),(108,163),(109,171),(110,172),(111,173),(112,174),(113,175),(114,176),(115,177),(116,178),(117,179),(118,180),(119,169),(120,170),(121,186),(122,187),(123,188),(124,189),(125,190),(126,191),(127,192),(128,181),(129,182),(130,183),(131,184),(132,185),(193,199),(194,200),(195,201),(196,202),(197,203),(198,204),(205,228),(206,217),(207,218),(208,219),(209,220),(210,221),(211,222),(212,223),(213,224),(214,225),(215,226),(216,227)]])
192 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | ··· | 4L | 4M | ··· | 4X | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | ··· | 12X | 12Y | ··· | 12AV | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 4 | ··· | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 1 | 1 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | ··· | 2 | 1 | ··· | 1 | 5 | ··· | 5 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | D5 | D10 | C3×D5 | C4×D5 | C6×D5 | D5×C12 |
| kernel | D5×C4×C12 | C12×Dic5 | C4×C60 | D5×C2×C12 | D5×C42 | D5×C12 | C4×Dic5 | C4×C20 | C2×C4×D5 | C4×D5 | C4×C12 | C2×C12 | C42 | C12 | C2×C4 | C4 |
| # reps | 1 | 3 | 1 | 3 | 2 | 24 | 6 | 2 | 6 | 48 | 2 | 6 | 4 | 24 | 12 | 48 |
Matrix representation of D5×C4×C12 ►in GL3(𝔽61) generated by
| 50 | 0 | 0 |
| 0 | 60 | 0 |
| 0 | 0 | 60 |
| 60 | 0 | 0 |
| 0 | 21 | 0 |
| 0 | 0 | 21 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 60 | 43 |
| 60 | 0 | 0 |
| 0 | 0 | 60 |
| 0 | 60 | 0 |
G:=sub<GL(3,GF(61))| [50,0,0,0,60,0,0,0,60],[60,0,0,0,21,0,0,0,21],[1,0,0,0,0,60,0,1,43],[60,0,0,0,0,60,0,60,0] >;
D5×C4×C12 in GAP, Magma, Sage, TeX
D_5\times C_4\times C_{12} % in TeX
G:=Group("D5xC4xC12"); // GroupNames label
G:=SmallGroup(480,664);
// by ID
G=gap.SmallGroup(480,664);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,344,142,18822]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^12=c^5=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations