metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊3D4, Dic15⋊6D4, C23.14D30, (C6×D4)⋊4D5, (C2×D4)⋊6D15, (D4×C10)⋊4S3, (D4×C30)⋊4C2, (C2×D60)⋊12C2, C20⋊6(C3⋊D4), C12⋊6(C5⋊D4), C5⋊4(C12⋊3D4), C3⋊4(C20⋊D4), C15⋊8(C4⋊1D4), C4⋊1(C15⋊7D4), (C2×C4).52D30, C6.122(D4×D5), C2.28(D4×D15), (C4×Dic15)⋊6C2, (C2×C20).150D6, C30.327(C2×D4), C10.124(S3×D4), (C2×C12).149D10, (C2×C60).76C22, (C22×C6).67D10, (C22×C10).82D6, (C2×C30).311C23, (C22×C30).23C22, C22.62(C22×D15), (C22×D15).13C22, (C2×Dic15).173C22, (C2×C15⋊7D4)⋊7C2, C6.111(C2×C5⋊D4), C2.16(C2×C15⋊7D4), C10.111(C2×C3⋊D4), (C2×C6).307(C22×D5), (C2×C10).306(C22×S3), SmallGroup(480,905)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊3D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a-1, cbc=b-1 >
Subgroups: 1460 in 216 conjugacy classes, 59 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C2×D4, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, D15, C30, C30, C30, C4⋊1D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4×Dic3, C2×D12, C2×C3⋊D4, C6×D4, Dic15, C60, D30, C2×C30, C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4, D4×C10, C12⋊3D4, D60, C2×Dic15, C15⋊7D4, C2×C60, D4×C15, C22×D15, C22×C30, C20⋊D4, C4×Dic15, C2×D60, C2×C15⋊7D4, D4×C30, C60⋊3D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, D15, C4⋊1D4, C5⋊D4, C22×D5, S3×D4, C2×C3⋊D4, D30, D4×D5, C2×C5⋊D4, C12⋊3D4, C15⋊7D4, C22×D15, C20⋊D4, D4×D15, C2×C15⋊7D4, C60⋊3D4
►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)
(1 127 118 228)(2 156 119 197)(3 125 120 226)(4 154 61 195)(5 123 62 224)(6 152 63 193)(7 121 64 222)(8 150 65 191)(9 179 66 220)(10 148 67 189)(11 177 68 218)(12 146 69 187)(13 175 70 216)(14 144 71 185)(15 173 72 214)(16 142 73 183)(17 171 74 212)(18 140 75 181)(19 169 76 210)(20 138 77 239)(21 167 78 208)(22 136 79 237)(23 165 80 206)(24 134 81 235)(25 163 82 204)(26 132 83 233)(27 161 84 202)(28 130 85 231)(29 159 86 200)(30 128 87 229)(31 157 88 198)(32 126 89 227)(33 155 90 196)(34 124 91 225)(35 153 92 194)(36 122 93 223)(37 151 94 192)(38 180 95 221)(39 149 96 190)(40 178 97 219)(41 147 98 188)(42 176 99 217)(43 145 100 186)(44 174 101 215)(45 143 102 184)(46 172 103 213)(47 141 104 182)(48 170 105 211)(49 139 106 240)(50 168 107 209)(51 137 108 238)(52 166 109 207)(53 135 110 236)(54 164 111 205)(55 133 112 234)(56 162 113 203)(57 131 114 232)(58 160 115 201)(59 129 116 230)(60 158 117 199)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 52)(26 51)(27 50)(28 49)(29 48)(30 47)(31 46)(32 45)(33 44)(34 43)(35 42)(36 41)(37 40)(38 39)(61 70)(62 69)(63 68)(64 67)(65 66)(71 120)(72 119)(73 118)(74 117)(75 116)(76 115)(77 114)(78 113)(79 112)(80 111)(81 110)(82 109)(83 108)(84 107)(85 106)(86 105)(87 104)(88 103)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(121 189)(122 188)(123 187)(124 186)(125 185)(126 184)(127 183)(128 182)(129 181)(130 240)(131 239)(132 238)(133 237)(134 236)(135 235)(136 234)(137 233)(138 232)(139 231)(140 230)(141 229)(142 228)(143 227)(144 226)(145 225)(146 224)(147 223)(148 222)(149 221)(150 220)(151 219)(152 218)(153 217)(154 216)(155 215)(156 214)(157 213)(158 212)(159 211)(160 210)(161 209)(162 208)(163 207)(164 206)(165 205)(166 204)(167 203)(168 202)(169 201)(170 200)(171 199)(172 198)(173 197)(174 196)(175 195)(176 194)(177 193)(178 192)(179 191)(180 190)
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), (1,127,118,228)(2,156,119,197)(3,125,120,226)(4,154,61,195)(5,123,62,224)(6,152,63,193)(7,121,64,222)(8,150,65,191)(9,179,66,220)(10,148,67,189)(11,177,68,218)(12,146,69,187)(13,175,70,216)(14,144,71,185)(15,173,72,214)(16,142,73,183)(17,171,74,212)(18,140,75,181)(19,169,76,210)(20,138,77,239)(21,167,78,208)(22,136,79,237)(23,165,80,206)(24,134,81,235)(25,163,82,204)(26,132,83,233)(27,161,84,202)(28,130,85,231)(29,159,86,200)(30,128,87,229)(31,157,88,198)(32,126,89,227)(33,155,90,196)(34,124,91,225)(35,153,92,194)(36,122,93,223)(37,151,94,192)(38,180,95,221)(39,149,96,190)(40,178,97,219)(41,147,98,188)(42,176,99,217)(43,145,100,186)(44,174,101,215)(45,143,102,184)(46,172,103,213)(47,141,104,182)(48,170,105,211)(49,139,106,240)(50,168,107,209)(51,137,108,238)(52,166,109,207)(53,135,110,236)(54,164,111,205)(55,133,112,234)(56,162,113,203)(57,131,114,232)(58,160,115,201)(59,129,116,230)(60,158,117,199), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(61,70)(62,69)(63,68)(64,67)(65,66)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(127,183)(128,182)(129,181)(130,240)(131,239)(132,238)(133,237)(134,236)(135,235)(136,234)(137,233)(138,232)(139,231)(140,230)(141,229)(142,228)(143,227)(144,226)(145,225)(146,224)(147,223)(148,222)(149,221)(150,220)(151,219)(152,218)(153,217)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190)>;
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), (1,127,118,228)(2,156,119,197)(3,125,120,226)(4,154,61,195)(5,123,62,224)(6,152,63,193)(7,121,64,222)(8,150,65,191)(9,179,66,220)(10,148,67,189)(11,177,68,218)(12,146,69,187)(13,175,70,216)(14,144,71,185)(15,173,72,214)(16,142,73,183)(17,171,74,212)(18,140,75,181)(19,169,76,210)(20,138,77,239)(21,167,78,208)(22,136,79,237)(23,165,80,206)(24,134,81,235)(25,163,82,204)(26,132,83,233)(27,161,84,202)(28,130,85,231)(29,159,86,200)(30,128,87,229)(31,157,88,198)(32,126,89,227)(33,155,90,196)(34,124,91,225)(35,153,92,194)(36,122,93,223)(37,151,94,192)(38,180,95,221)(39,149,96,190)(40,178,97,219)(41,147,98,188)(42,176,99,217)(43,145,100,186)(44,174,101,215)(45,143,102,184)(46,172,103,213)(47,141,104,182)(48,170,105,211)(49,139,106,240)(50,168,107,209)(51,137,108,238)(52,166,109,207)(53,135,110,236)(54,164,111,205)(55,133,112,234)(56,162,113,203)(57,131,114,232)(58,160,115,201)(59,129,116,230)(60,158,117,199), (1,16)(2,15)(3,14)(4,13)(5,12)(6,11)(7,10)(8,9)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)(61,70)(62,69)(63,68)(64,67)(65,66)(71,120)(72,119)(73,118)(74,117)(75,116)(76,115)(77,114)(78,113)(79,112)(80,111)(81,110)(82,109)(83,108)(84,107)(85,106)(86,105)(87,104)(88,103)(89,102)(90,101)(91,100)(92,99)(93,98)(94,97)(95,96)(121,189)(122,188)(123,187)(124,186)(125,185)(126,184)(127,183)(128,182)(129,181)(130,240)(131,239)(132,238)(133,237)(134,236)(135,235)(136,234)(137,233)(138,232)(139,231)(140,230)(141,229)(142,228)(143,227)(144,226)(145,225)(146,224)(147,223)(148,222)(149,221)(150,220)(151,219)(152,218)(153,217)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)(160,210)(161,209)(162,208)(163,207)(164,206)(165,205)(166,204)(167,203)(168,202)(169,201)(170,200)(171,199)(172,198)(173,197)(174,196)(175,195)(176,194)(177,193)(178,192)(179,191)(180,190) );
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)], [(1,127,118,228),(2,156,119,197),(3,125,120,226),(4,154,61,195),(5,123,62,224),(6,152,63,193),(7,121,64,222),(8,150,65,191),(9,179,66,220),(10,148,67,189),(11,177,68,218),(12,146,69,187),(13,175,70,216),(14,144,71,185),(15,173,72,214),(16,142,73,183),(17,171,74,212),(18,140,75,181),(19,169,76,210),(20,138,77,239),(21,167,78,208),(22,136,79,237),(23,165,80,206),(24,134,81,235),(25,163,82,204),(26,132,83,233),(27,161,84,202),(28,130,85,231),(29,159,86,200),(30,128,87,229),(31,157,88,198),(32,126,89,227),(33,155,90,196),(34,124,91,225),(35,153,92,194),(36,122,93,223),(37,151,94,192),(38,180,95,221),(39,149,96,190),(40,178,97,219),(41,147,98,188),(42,176,99,217),(43,145,100,186),(44,174,101,215),(45,143,102,184),(46,172,103,213),(47,141,104,182),(48,170,105,211),(49,139,106,240),(50,168,107,209),(51,137,108,238),(52,166,109,207),(53,135,110,236),(54,164,111,205),(55,133,112,234),(56,162,113,203),(57,131,114,232),(58,160,115,201),(59,129,116,230),(60,158,117,199)], [(1,16),(2,15),(3,14),(4,13),(5,12),(6,11),(7,10),(8,9),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,52),(26,51),(27,50),(28,49),(29,48),(30,47),(31,46),(32,45),(33,44),(34,43),(35,42),(36,41),(37,40),(38,39),(61,70),(62,69),(63,68),(64,67),(65,66),(71,120),(72,119),(73,118),(74,117),(75,116),(76,115),(77,114),(78,113),(79,112),(80,111),(81,110),(82,109),(83,108),(84,107),(85,106),(86,105),(87,104),(88,103),(89,102),(90,101),(91,100),(92,99),(93,98),(94,97),(95,96),(121,189),(122,188),(123,187),(124,186),(125,185),(126,184),(127,183),(128,182),(129,181),(130,240),(131,239),(132,238),(133,237),(134,236),(135,235),(136,234),(137,233),(138,232),(139,231),(140,230),(141,229),(142,228),(143,227),(144,226),(145,225),(146,224),(147,223),(148,222),(149,221),(150,220),(151,219),(152,218),(153,217),(154,216),(155,215),(156,214),(157,213),(158,212),(159,211),(160,210),(161,209),(162,208),(163,207),(164,206),(165,205),(166,204),(167,203),(168,202),(169,201),(170,200),(171,199),(172,198),(173,197),(174,196),(175,195),(176,194),(177,193),(178,192),(179,191),(180,190)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 15A | 15B | 15C | 15D | 20A | 20B | 20C | 20D | 30A | ··· | 30L | 30M | ··· | 30AB | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 60 | 60 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | D15 | C5⋊D4 | D30 | D30 | C15⋊7D4 | S3×D4 | D4×D5 | D4×D15 |
| kernel | C60⋊3D4 | C4×Dic15 | C2×D60 | C2×C15⋊7D4 | D4×C30 | D4×C10 | Dic15 | C60 | C6×D4 | C2×C20 | C22×C10 | C2×C12 | C22×C6 | C20 | C2×D4 | C12 | C2×C4 | C23 | C4 | C10 | C6 | C2 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 4 | 8 | 16 | 2 | 4 | 8 |
Matrix representation of C60⋊3D4 ►in GL4(𝔽61) generated by
| 38 | 14 | 0 | 0 |
| 39 | 16 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 3 | 60 |
| 8 | 22 | 0 | 0 |
| 22 | 53 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 17 | 0 | 0 |
| 42 | 43 | 0 | 0 |
| 0 | 0 | 1 | 40 |
| 0 | 0 | 0 | 60 |
G:=sub<GL(4,GF(61))| [38,39,0,0,14,16,0,0,0,0,1,3,0,0,40,60],[8,22,0,0,22,53,0,0,0,0,1,0,0,0,0,1],[18,42,0,0,17,43,0,0,0,0,1,0,0,0,40,60] >;
C60⋊3D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_3D_4 % in TeX
G:=Group("C60:3D4"); // GroupNames label
G:=SmallGroup(480,905);
// by ID
G=gap.SmallGroup(480,905);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,253,120,254,219,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations