metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊10D4, Dic15⋊10D4, C6.47(D4×D5), (C2×D12)⋊10D5, (C6×D20)⋊10C2, (C2×D20)⋊10S3, C15⋊3(C4⋊1D4), C20⋊3(C3⋊D4), C4⋊1(C15⋊D4), C3⋊2(C20⋊D4), C5⋊2(C12⋊3D4), C12⋊3(C5⋊D4), C10.48(S3×D4), (C10×D12)⋊10C2, (C2×C20).132D6, C30.157(C2×D4), (C4×Dic15)⋊27C2, (C2×C12).133D10, (C22×D5).23D6, C2.23(C20⋊D6), (C2×C60).203C22, (C2×C30).153C23, (C22×S3).21D10, (C2×Dic15).214C22, (C2×C15⋊D4)⋊8C2, C6.88(C2×C5⋊D4), (C2×C4).213(S3×D5), C10.89(C2×C3⋊D4), C2.21(C2×C15⋊D4), (D5×C2×C6).37C22, C22.205(C2×S3×D5), (S3×C2×C10).37C22, (C2×C6).165(C22×D5), (C2×C10).165(C22×S3), SmallGroup(480,539)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60⋊10D4
G = < a,b,c | a60=b4=c2=1, bab-1=a29, cac=a19, cbc=b-1 >
Subgroups: 1244 in 216 conjugacy classes, 56 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C5×S3, C3×D5, 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, C6×D5, S3×C10, C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4, D4×C10, C12⋊3D4, C15⋊D4, C3×D20, C5×D12, C2×Dic15, C2×C60, D5×C2×C6, S3×C2×C10, C20⋊D4, C4×Dic15, C2×C15⋊D4, C6×D20, C10×D12, C60⋊10D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4⋊1D4, C5⋊D4, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, D4×D5, C2×C5⋊D4, C12⋊3D4, C15⋊D4, C2×S3×D5, C20⋊D4, C20⋊D6, C2×C15⋊D4, C60⋊10D4
►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)
(1 169 90 232)(2 138 91 201)(3 167 92 230)(4 136 93 199)(5 165 94 228)(6 134 95 197)(7 163 96 226)(8 132 97 195)(9 161 98 224)(10 130 99 193)(11 159 100 222)(12 128 101 191)(13 157 102 220)(14 126 103 189)(15 155 104 218)(16 124 105 187)(17 153 106 216)(18 122 107 185)(19 151 108 214)(20 180 109 183)(21 149 110 212)(22 178 111 181)(23 147 112 210)(24 176 113 239)(25 145 114 208)(26 174 115 237)(27 143 116 206)(28 172 117 235)(29 141 118 204)(30 170 119 233)(31 139 120 202)(32 168 61 231)(33 137 62 200)(34 166 63 229)(35 135 64 198)(36 164 65 227)(37 133 66 196)(38 162 67 225)(39 131 68 194)(40 160 69 223)(41 129 70 192)(42 158 71 221)(43 127 72 190)(44 156 73 219)(45 125 74 188)(46 154 75 217)(47 123 76 186)(48 152 77 215)(49 121 78 184)(50 150 79 213)(51 179 80 182)(52 148 81 211)(53 177 82 240)(54 146 83 209)(55 175 84 238)(56 144 85 207)(57 173 86 236)(58 142 87 205)(59 171 88 234)(60 140 89 203)
(1 16)(2 35)(3 54)(4 13)(5 32)(6 51)(7 10)(8 29)(9 48)(11 26)(12 45)(14 23)(15 42)(17 20)(18 39)(19 58)(21 36)(22 55)(24 33)(25 52)(27 30)(28 49)(31 46)(34 43)(37 40)(38 59)(41 56)(44 53)(47 50)(57 60)(61 94)(62 113)(63 72)(64 91)(65 110)(66 69)(67 88)(68 107)(70 85)(71 104)(73 82)(74 101)(75 120)(76 79)(77 98)(78 117)(80 95)(81 114)(83 92)(84 111)(86 89)(87 108)(90 105)(93 102)(96 99)(97 118)(100 115)(103 112)(106 109)(116 119)(121 235)(122 194)(123 213)(124 232)(125 191)(126 210)(127 229)(128 188)(129 207)(130 226)(131 185)(132 204)(133 223)(134 182)(135 201)(136 220)(137 239)(138 198)(139 217)(140 236)(141 195)(142 214)(143 233)(144 192)(145 211)(146 230)(147 189)(148 208)(149 227)(150 186)(151 205)(152 224)(153 183)(154 202)(155 221)(156 240)(157 199)(158 218)(159 237)(160 196)(161 215)(162 234)(163 193)(164 212)(165 231)(166 190)(167 209)(168 228)(169 187)(170 206)(171 225)(172 184)(173 203)(174 222)(175 181)(176 200)(177 219)(178 238)(179 197)(180 216)
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), (1,169,90,232)(2,138,91,201)(3,167,92,230)(4,136,93,199)(5,165,94,228)(6,134,95,197)(7,163,96,226)(8,132,97,195)(9,161,98,224)(10,130,99,193)(11,159,100,222)(12,128,101,191)(13,157,102,220)(14,126,103,189)(15,155,104,218)(16,124,105,187)(17,153,106,216)(18,122,107,185)(19,151,108,214)(20,180,109,183)(21,149,110,212)(22,178,111,181)(23,147,112,210)(24,176,113,239)(25,145,114,208)(26,174,115,237)(27,143,116,206)(28,172,117,235)(29,141,118,204)(30,170,119,233)(31,139,120,202)(32,168,61,231)(33,137,62,200)(34,166,63,229)(35,135,64,198)(36,164,65,227)(37,133,66,196)(38,162,67,225)(39,131,68,194)(40,160,69,223)(41,129,70,192)(42,158,71,221)(43,127,72,190)(44,156,73,219)(45,125,74,188)(46,154,75,217)(47,123,76,186)(48,152,77,215)(49,121,78,184)(50,150,79,213)(51,179,80,182)(52,148,81,211)(53,177,82,240)(54,146,83,209)(55,175,84,238)(56,144,85,207)(57,173,86,236)(58,142,87,205)(59,171,88,234)(60,140,89,203), (1,16)(2,35)(3,54)(4,13)(5,32)(6,51)(7,10)(8,29)(9,48)(11,26)(12,45)(14,23)(15,42)(17,20)(18,39)(19,58)(21,36)(22,55)(24,33)(25,52)(27,30)(28,49)(31,46)(34,43)(37,40)(38,59)(41,56)(44,53)(47,50)(57,60)(61,94)(62,113)(63,72)(64,91)(65,110)(66,69)(67,88)(68,107)(70,85)(71,104)(73,82)(74,101)(75,120)(76,79)(77,98)(78,117)(80,95)(81,114)(83,92)(84,111)(86,89)(87,108)(90,105)(93,102)(96,99)(97,118)(100,115)(103,112)(106,109)(116,119)(121,235)(122,194)(123,213)(124,232)(125,191)(126,210)(127,229)(128,188)(129,207)(130,226)(131,185)(132,204)(133,223)(134,182)(135,201)(136,220)(137,239)(138,198)(139,217)(140,236)(141,195)(142,214)(143,233)(144,192)(145,211)(146,230)(147,189)(148,208)(149,227)(150,186)(151,205)(152,224)(153,183)(154,202)(155,221)(156,240)(157,199)(158,218)(159,237)(160,196)(161,215)(162,234)(163,193)(164,212)(165,231)(166,190)(167,209)(168,228)(169,187)(170,206)(171,225)(172,184)(173,203)(174,222)(175,181)(176,200)(177,219)(178,238)(179,197)(180,216)>;
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), (1,169,90,232)(2,138,91,201)(3,167,92,230)(4,136,93,199)(5,165,94,228)(6,134,95,197)(7,163,96,226)(8,132,97,195)(9,161,98,224)(10,130,99,193)(11,159,100,222)(12,128,101,191)(13,157,102,220)(14,126,103,189)(15,155,104,218)(16,124,105,187)(17,153,106,216)(18,122,107,185)(19,151,108,214)(20,180,109,183)(21,149,110,212)(22,178,111,181)(23,147,112,210)(24,176,113,239)(25,145,114,208)(26,174,115,237)(27,143,116,206)(28,172,117,235)(29,141,118,204)(30,170,119,233)(31,139,120,202)(32,168,61,231)(33,137,62,200)(34,166,63,229)(35,135,64,198)(36,164,65,227)(37,133,66,196)(38,162,67,225)(39,131,68,194)(40,160,69,223)(41,129,70,192)(42,158,71,221)(43,127,72,190)(44,156,73,219)(45,125,74,188)(46,154,75,217)(47,123,76,186)(48,152,77,215)(49,121,78,184)(50,150,79,213)(51,179,80,182)(52,148,81,211)(53,177,82,240)(54,146,83,209)(55,175,84,238)(56,144,85,207)(57,173,86,236)(58,142,87,205)(59,171,88,234)(60,140,89,203), (1,16)(2,35)(3,54)(4,13)(5,32)(6,51)(7,10)(8,29)(9,48)(11,26)(12,45)(14,23)(15,42)(17,20)(18,39)(19,58)(21,36)(22,55)(24,33)(25,52)(27,30)(28,49)(31,46)(34,43)(37,40)(38,59)(41,56)(44,53)(47,50)(57,60)(61,94)(62,113)(63,72)(64,91)(65,110)(66,69)(67,88)(68,107)(70,85)(71,104)(73,82)(74,101)(75,120)(76,79)(77,98)(78,117)(80,95)(81,114)(83,92)(84,111)(86,89)(87,108)(90,105)(93,102)(96,99)(97,118)(100,115)(103,112)(106,109)(116,119)(121,235)(122,194)(123,213)(124,232)(125,191)(126,210)(127,229)(128,188)(129,207)(130,226)(131,185)(132,204)(133,223)(134,182)(135,201)(136,220)(137,239)(138,198)(139,217)(140,236)(141,195)(142,214)(143,233)(144,192)(145,211)(146,230)(147,189)(148,208)(149,227)(150,186)(151,205)(152,224)(153,183)(154,202)(155,221)(156,240)(157,199)(158,218)(159,237)(160,196)(161,215)(162,234)(163,193)(164,212)(165,231)(166,190)(167,209)(168,228)(169,187)(170,206)(171,225)(172,184)(173,203)(174,222)(175,181)(176,200)(177,219)(178,238)(179,197)(180,216) );
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)], [(1,169,90,232),(2,138,91,201),(3,167,92,230),(4,136,93,199),(5,165,94,228),(6,134,95,197),(7,163,96,226),(8,132,97,195),(9,161,98,224),(10,130,99,193),(11,159,100,222),(12,128,101,191),(13,157,102,220),(14,126,103,189),(15,155,104,218),(16,124,105,187),(17,153,106,216),(18,122,107,185),(19,151,108,214),(20,180,109,183),(21,149,110,212),(22,178,111,181),(23,147,112,210),(24,176,113,239),(25,145,114,208),(26,174,115,237),(27,143,116,206),(28,172,117,235),(29,141,118,204),(30,170,119,233),(31,139,120,202),(32,168,61,231),(33,137,62,200),(34,166,63,229),(35,135,64,198),(36,164,65,227),(37,133,66,196),(38,162,67,225),(39,131,68,194),(40,160,69,223),(41,129,70,192),(42,158,71,221),(43,127,72,190),(44,156,73,219),(45,125,74,188),(46,154,75,217),(47,123,76,186),(48,152,77,215),(49,121,78,184),(50,150,79,213),(51,179,80,182),(52,148,81,211),(53,177,82,240),(54,146,83,209),(55,175,84,238),(56,144,85,207),(57,173,86,236),(58,142,87,205),(59,171,88,234),(60,140,89,203)], [(1,16),(2,35),(3,54),(4,13),(5,32),(6,51),(7,10),(8,29),(9,48),(11,26),(12,45),(14,23),(15,42),(17,20),(18,39),(19,58),(21,36),(22,55),(24,33),(25,52),(27,30),(28,49),(31,46),(34,43),(37,40),(38,59),(41,56),(44,53),(47,50),(57,60),(61,94),(62,113),(63,72),(64,91),(65,110),(66,69),(67,88),(68,107),(70,85),(71,104),(73,82),(74,101),(75,120),(76,79),(77,98),(78,117),(80,95),(81,114),(83,92),(84,111),(86,89),(87,108),(90,105),(93,102),(96,99),(97,118),(100,115),(103,112),(106,109),(116,119),(121,235),(122,194),(123,213),(124,232),(125,191),(126,210),(127,229),(128,188),(129,207),(130,226),(131,185),(132,204),(133,223),(134,182),(135,201),(136,220),(137,239),(138,198),(139,217),(140,236),(141,195),(142,214),(143,233),(144,192),(145,211),(146,230),(147,189),(148,208),(149,227),(150,186),(151,205),(152,224),(153,183),(154,202),(155,221),(156,240),(157,199),(158,218),(159,237),(160,196),(161,215),(162,234),(163,193),(164,212),(165,231),(166,190),(167,209),(168,228),(169,187),(170,206),(171,225),(172,184),(173,203),(174,222),(175,181),(176,200),(177,219),(178,238),(179,197),(180,216)]])
60 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 | 20A | 20B | 20C | 20D | 30A | ··· | 30F | 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 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 20 | 20 | 2 | 2 | 2 | 30 | 30 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 12 | ··· | 12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | C5⋊D4 | S3×D4 | S3×D5 | D4×D5 | C15⋊D4 | C2×S3×D5 | C20⋊D6 |
| kernel | C60⋊10D4 | C4×Dic15 | C2×C15⋊D4 | C6×D20 | C10×D12 | C2×D20 | Dic15 | C60 | C2×D12 | C2×C20 | C22×D5 | C2×C12 | C22×S3 | C20 | C12 | C10 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 8 | 2 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C60⋊10D4 ►in GL6(𝔽61)
| 18 | 18 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 15 | 0 | 0 |
| 0 | 0 | 12 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 31 | 17 | 0 | 0 | 0 | 0 |
| 8 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 31 | 48 | 0 | 0 |
| 0 | 0 | 41 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 60 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 43 | 60 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(61))| [18,43,0,0,0,0,18,60,0,0,0,0,0,0,60,12,0,0,0,0,15,2,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[31,8,0,0,0,0,17,30,0,0,0,0,0,0,31,41,0,0,0,0,48,30,0,0,0,0,0,0,0,60,0,0,0,0,1,0],[1,43,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C60⋊10D4 in GAP, Magma, Sage, TeX
C_{60}\rtimes_{10}D_4 % in TeX
G:=Group("C60:10D4"); // GroupNames label
G:=SmallGroup(480,539);
// by ID
G=gap.SmallGroup(480,539);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^29,c*a*c=a^19,c*b*c=b^-1>;
// generators/relations