metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D60⋊14C4, Dic3⋊5D20, C3⋊1(C4×D20), C12⋊2(C4×D5), C15⋊12(C4×D4), C20⋊11(C4×S3), C60⋊16(C2×C4), C2.4(S3×D20), D30⋊11(C2×C4), C4⋊Dic5⋊17S3, (C4×Dic3)⋊6D5, (C5×Dic3)⋊8D4, C10.18(S3×D4), C30.49(C2×D4), C6.18(C2×D20), C5⋊2(Dic3⋊5D4), C4⋊1(D30.C2), (Dic3×C20)⋊6C2, (C2×D60).15C2, (C2×C20).300D6, D30⋊4C4⋊16C2, C30.69(C4○D4), C6.10(C4○D20), (C2×C12).126D10, C2.4(D60⋊C2), (C2×C30).118C23, (C2×C60).119C22, C30.129(C22×C4), (C2×Dic5).112D6, C10.15(Q8⋊3S3), (C2×Dic3).182D10, (C6×Dic5).71C22, (C22×D15).40C22, (C10×Dic3).182C22, C6.49(C2×C4×D5), C10.81(S3×C2×C4), (C3×C4⋊Dic5)⋊5C2, C22.57(C2×S3×D5), (C2×D30.C2)⋊6C2, (C2×C4).110(S3×D5), C2.13(C2×D30.C2), (C2×C6).130(C22×D5), (C2×C10).130(C22×S3), SmallGroup(480,504)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D60⋊14C4
G = < a,b,c | a60=b2=c4=1, bab=a-1, cac-1=a41, cbc-1=a40b >
Subgroups: 1132 in 188 conjugacy classes, 64 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, Dic3, C12, C12, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C2×C12, C22×S3, D15, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic3, D6⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C5×Dic3, C5×Dic3, C3×Dic5, C60, D30, D30, C2×C30, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×D20, Dic3⋊5D4, D30.C2, C6×Dic5, C10×Dic3, D60, C2×C60, C22×D15, C4×D20, D30⋊4C4, C3×C4⋊Dic5, Dic3×C20, C2×D30.C2, C2×D60, D60⋊14C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D5, D6, C22×C4, C2×D4, C4○D4, D10, C4×S3, C22×S3, C4×D4, C4×D5, D20, C22×D5, S3×C2×C4, S3×D4, Q8⋊3S3, S3×D5, C2×C4×D5, C2×D20, C4○D20, Dic3⋊5D4, D30.C2, C2×S3×D5, C4×D20, D60⋊C2, S3×D20, C2×D30.C2, D60⋊14C4
►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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 60)(17 59)(18 58)(19 57)(20 56)(21 55)(22 54)(23 53)(24 52)(25 51)(26 50)(27 49)(28 48)(29 47)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(61 91)(62 90)(63 89)(64 88)(65 87)(66 86)(67 85)(68 84)(69 83)(70 82)(71 81)(72 80)(73 79)(74 78)(75 77)(92 120)(93 119)(94 118)(95 117)(96 116)(97 115)(98 114)(99 113)(100 112)(101 111)(102 110)(103 109)(104 108)(105 107)(121 137)(122 136)(123 135)(124 134)(125 133)(126 132)(127 131)(128 130)(138 180)(139 179)(140 178)(141 177)(142 176)(143 175)(144 174)(145 173)(146 172)(147 171)(148 170)(149 169)(150 168)(151 167)(152 166)(153 165)(154 164)(155 163)(156 162)(157 161)(158 160)(181 193)(182 192)(183 191)(184 190)(185 189)(186 188)(194 240)(195 239)(196 238)(197 237)(198 236)(199 235)(200 234)(201 233)(202 232)(203 231)(204 230)(205 229)(206 228)(207 227)(208 226)(209 225)(210 224)(211 223)(212 222)(213 221)(214 220)(215 219)(216 218)
(1 99 152 210)(2 80 153 191)(3 61 154 232)(4 102 155 213)(5 83 156 194)(6 64 157 235)(7 105 158 216)(8 86 159 197)(9 67 160 238)(10 108 161 219)(11 89 162 200)(12 70 163 181)(13 111 164 222)(14 92 165 203)(15 73 166 184)(16 114 167 225)(17 95 168 206)(18 76 169 187)(19 117 170 228)(20 98 171 209)(21 79 172 190)(22 120 173 231)(23 101 174 212)(24 82 175 193)(25 63 176 234)(26 104 177 215)(27 85 178 196)(28 66 179 237)(29 107 180 218)(30 88 121 199)(31 69 122 240)(32 110 123 221)(33 91 124 202)(34 72 125 183)(35 113 126 224)(36 94 127 205)(37 75 128 186)(38 116 129 227)(39 97 130 208)(40 78 131 189)(41 119 132 230)(42 100 133 211)(43 81 134 192)(44 62 135 233)(45 103 136 214)(46 84 137 195)(47 65 138 236)(48 106 139 217)(49 87 140 198)(50 68 141 239)(51 109 142 220)(52 90 143 201)(53 71 144 182)(54 112 145 223)(55 93 146 204)(56 74 147 185)(57 115 148 226)(58 96 149 207)(59 77 150 188)(60 118 151 229)
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(181,193)(182,192)(183,191)(184,190)(185,189)(186,188)(194,240)(195,239)(196,238)(197,237)(198,236)(199,235)(200,234)(201,233)(202,232)(203,231)(204,230)(205,229)(206,228)(207,227)(208,226)(209,225)(210,224)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218), (1,99,152,210)(2,80,153,191)(3,61,154,232)(4,102,155,213)(5,83,156,194)(6,64,157,235)(7,105,158,216)(8,86,159,197)(9,67,160,238)(10,108,161,219)(11,89,162,200)(12,70,163,181)(13,111,164,222)(14,92,165,203)(15,73,166,184)(16,114,167,225)(17,95,168,206)(18,76,169,187)(19,117,170,228)(20,98,171,209)(21,79,172,190)(22,120,173,231)(23,101,174,212)(24,82,175,193)(25,63,176,234)(26,104,177,215)(27,85,178,196)(28,66,179,237)(29,107,180,218)(30,88,121,199)(31,69,122,240)(32,110,123,221)(33,91,124,202)(34,72,125,183)(35,113,126,224)(36,94,127,205)(37,75,128,186)(38,116,129,227)(39,97,130,208)(40,78,131,189)(41,119,132,230)(42,100,133,211)(43,81,134,192)(44,62,135,233)(45,103,136,214)(46,84,137,195)(47,65,138,236)(48,106,139,217)(49,87,140,198)(50,68,141,239)(51,109,142,220)(52,90,143,201)(53,71,144,182)(54,112,145,223)(55,93,146,204)(56,74,147,185)(57,115,148,226)(58,96,149,207)(59,77,150,188)(60,118,151,229)>;
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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,60)(17,59)(18,58)(19,57)(20,56)(21,55)(22,54)(23,53)(24,52)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(61,91)(62,90)(63,89)(64,88)(65,87)(66,86)(67,85)(68,84)(69,83)(70,82)(71,81)(72,80)(73,79)(74,78)(75,77)(92,120)(93,119)(94,118)(95,117)(96,116)(97,115)(98,114)(99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(121,137)(122,136)(123,135)(124,134)(125,133)(126,132)(127,131)(128,130)(138,180)(139,179)(140,178)(141,177)(142,176)(143,175)(144,174)(145,173)(146,172)(147,171)(148,170)(149,169)(150,168)(151,167)(152,166)(153,165)(154,164)(155,163)(156,162)(157,161)(158,160)(181,193)(182,192)(183,191)(184,190)(185,189)(186,188)(194,240)(195,239)(196,238)(197,237)(198,236)(199,235)(200,234)(201,233)(202,232)(203,231)(204,230)(205,229)(206,228)(207,227)(208,226)(209,225)(210,224)(211,223)(212,222)(213,221)(214,220)(215,219)(216,218), (1,99,152,210)(2,80,153,191)(3,61,154,232)(4,102,155,213)(5,83,156,194)(6,64,157,235)(7,105,158,216)(8,86,159,197)(9,67,160,238)(10,108,161,219)(11,89,162,200)(12,70,163,181)(13,111,164,222)(14,92,165,203)(15,73,166,184)(16,114,167,225)(17,95,168,206)(18,76,169,187)(19,117,170,228)(20,98,171,209)(21,79,172,190)(22,120,173,231)(23,101,174,212)(24,82,175,193)(25,63,176,234)(26,104,177,215)(27,85,178,196)(28,66,179,237)(29,107,180,218)(30,88,121,199)(31,69,122,240)(32,110,123,221)(33,91,124,202)(34,72,125,183)(35,113,126,224)(36,94,127,205)(37,75,128,186)(38,116,129,227)(39,97,130,208)(40,78,131,189)(41,119,132,230)(42,100,133,211)(43,81,134,192)(44,62,135,233)(45,103,136,214)(46,84,137,195)(47,65,138,236)(48,106,139,217)(49,87,140,198)(50,68,141,239)(51,109,142,220)(52,90,143,201)(53,71,144,182)(54,112,145,223)(55,93,146,204)(56,74,147,185)(57,115,148,226)(58,96,149,207)(59,77,150,188)(60,118,151,229) );
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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,60),(17,59),(18,58),(19,57),(20,56),(21,55),(22,54),(23,53),(24,52),(25,51),(26,50),(27,49),(28,48),(29,47),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(61,91),(62,90),(63,89),(64,88),(65,87),(66,86),(67,85),(68,84),(69,83),(70,82),(71,81),(72,80),(73,79),(74,78),(75,77),(92,120),(93,119),(94,118),(95,117),(96,116),(97,115),(98,114),(99,113),(100,112),(101,111),(102,110),(103,109),(104,108),(105,107),(121,137),(122,136),(123,135),(124,134),(125,133),(126,132),(127,131),(128,130),(138,180),(139,179),(140,178),(141,177),(142,176),(143,175),(144,174),(145,173),(146,172),(147,171),(148,170),(149,169),(150,168),(151,167),(152,166),(153,165),(154,164),(155,163),(156,162),(157,161),(158,160),(181,193),(182,192),(183,191),(184,190),(185,189),(186,188),(194,240),(195,239),(196,238),(197,237),(198,236),(199,235),(200,234),(201,233),(202,232),(203,231),(204,230),(205,229),(206,228),(207,227),(208,226),(209,225),(210,224),(211,223),(212,222),(213,221),(214,220),(215,219),(216,218)], [(1,99,152,210),(2,80,153,191),(3,61,154,232),(4,102,155,213),(5,83,156,194),(6,64,157,235),(7,105,158,216),(8,86,159,197),(9,67,160,238),(10,108,161,219),(11,89,162,200),(12,70,163,181),(13,111,164,222),(14,92,165,203),(15,73,166,184),(16,114,167,225),(17,95,168,206),(18,76,169,187),(19,117,170,228),(20,98,171,209),(21,79,172,190),(22,120,173,231),(23,101,174,212),(24,82,175,193),(25,63,176,234),(26,104,177,215),(27,85,178,196),(28,66,179,237),(29,107,180,218),(30,88,121,199),(31,69,122,240),(32,110,123,221),(33,91,124,202),(34,72,125,183),(35,113,126,224),(36,94,127,205),(37,75,128,186),(38,116,129,227),(39,97,130,208),(40,78,131,189),(41,119,132,230),(42,100,133,211),(43,81,134,192),(44,62,135,233),(45,103,136,214),(46,84,137,195),(47,65,138,236),(48,106,139,217),(49,87,140,198),(50,68,141,239),(51,109,142,220),(52,90,143,201),(53,71,144,182),(54,112,145,223),(55,93,146,204),(56,74,147,185),(57,115,148,226),(58,96,149,207),(59,77,150,188),(60,118,151,229)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
78 irreducible representations
| dim | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C4×S3 | D20 | C4×D5 | C4○D20 | S3×D4 | Q8⋊3S3 | S3×D5 | D30.C2 | C2×S3×D5 | D60⋊C2 | S3×D20 |
| kernel | D60⋊14C4 | D30⋊4C4 | C3×C4⋊Dic5 | Dic3×C20 | C2×D30.C2 | C2×D60 | D60 | C4⋊Dic5 | C5×Dic3 | C4×Dic3 | C2×Dic5 | C2×C20 | C30 | C2×Dic3 | C2×C12 | C20 | Dic3 | C12 | C6 | C10 | C10 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 8 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of D60⋊14C4 ►in GL5(𝔽61)
| 60 | 0 | 0 | 0 | 0 |
| 0 | 32 | 59 | 0 | 0 |
| 0 | 2 | 59 | 0 | 0 |
| 0 | 0 | 0 | 2 | 46 |
| 0 | 0 | 0 | 49 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 1 | 44 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 59 | 15 |
| 0 | 0 | 0 | 12 | 2 |
| 50 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 49 | 60 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,32,2,0,0,0,59,59,0,0,0,0,0,2,49,0,0,0,46,60],[60,0,0,0,0,0,1,0,0,0,0,44,60,0,0,0,0,0,59,12,0,0,0,15,2],[50,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,49,0,0,0,0,60] >;
D60⋊14C4 in GAP, Magma, Sage, TeX
D_{60}\rtimes_{14}C_4 % in TeX
G:=Group("D60:14C4"); // GroupNames label
G:=SmallGroup(480,504);
// by ID
G=gap.SmallGroup(480,504);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,64,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^2=c^4=1,b*a*b=a^-1,c*a*c^-1=a^41,c*b*c^-1=a^40*b>;
// generators/relations