metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.205D4, C22⋊3Dic30, C23.25D30, (C2×C30)⋊9Q8, C60⋊5C4⋊14C2, (C2×C6)⋊8Dic10, (C2×C4).84D30, C30.71(C2×Q8), (C2×Dic30)⋊9C2, (C2×C10)⋊11Dic6, C30.375(C2×D4), (C2×C20).393D6, (C22×C60).9C2, C15⋊34(C22⋊Q8), C30.4Q8⋊2C2, C2.9(C2×Dic30), (C22×C4).7D15, (C22×C12).6D5, (C2×C12).382D10, C5⋊5(C12.48D4), C4.23(C15⋊7D4), C3⋊5(C20.48D4), (C22×C20).10S3, C10.39(C2×Dic6), C6.39(C2×Dic10), C6.103(C4○D20), C30.175(C4○D4), C12.102(C5⋊D4), C20.102(C3⋊D4), (C2×C30).298C23, (C2×C60).464C22, C30.38D4.4C2, (C22×C10).133D6, (C22×C6).115D10, C10.103(C4○D12), C2.17(D60⋊11C2), C22.54(C22×D15), (C22×C30).138C22, (C2×Dic15).13C22, C6.98(C2×C5⋊D4), C2.5(C2×C15⋊7D4), C10.98(C2×C3⋊D4), (C2×C6).294(C22×D5), (C2×C10).293(C22×S3), SmallGroup(480,889)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.205D4
G = < a,b,c | a60=b4=1, c2=a30, bab-1=cac-1=a-1, cbc-1=a30b-1 >
Subgroups: 660 in 148 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C5, C6, C6, C2×C4, C2×C4, Q8, C23, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C30, C30, C22⋊Q8, Dic10, C2×Dic5, C2×C20, C2×C20, C22×C10, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×Dic6, C22×C12, Dic15, C60, C60, C2×C30, C2×C30, C2×C30, C10.D4, C4⋊Dic5, C23.D5, C2×Dic10, C22×C20, C12.48D4, Dic30, C2×Dic15, C2×C60, C2×C60, C22×C30, C20.48D4, C30.4Q8, C60⋊5C4, C30.38D4, C2×Dic30, C22×C60, C60.205D4
Quotients: C1, C2, C22, S3, D4, Q8, C23, D5, D6, C2×D4, C2×Q8, C4○D4, D10, Dic6, C3⋊D4, C22×S3, D15, C22⋊Q8, Dic10, C5⋊D4, C22×D5, C2×Dic6, C4○D12, C2×C3⋊D4, D30, C2×Dic10, C4○D20, C2×C5⋊D4, C12.48D4, Dic30, C15⋊7D4, C22×D15, C20.48D4, C2×Dic30, D60⋊11C2, C2×C15⋊7D4, C60.205D4
►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)
(1 210 160 81)(2 209 161 80)(3 208 162 79)(4 207 163 78)(5 206 164 77)(6 205 165 76)(7 204 166 75)(8 203 167 74)(9 202 168 73)(10 201 169 72)(11 200 170 71)(12 199 171 70)(13 198 172 69)(14 197 173 68)(15 196 174 67)(16 195 175 66)(17 194 176 65)(18 193 177 64)(19 192 178 63)(20 191 179 62)(21 190 180 61)(22 189 121 120)(23 188 122 119)(24 187 123 118)(25 186 124 117)(26 185 125 116)(27 184 126 115)(28 183 127 114)(29 182 128 113)(30 181 129 112)(31 240 130 111)(32 239 131 110)(33 238 132 109)(34 237 133 108)(35 236 134 107)(36 235 135 106)(37 234 136 105)(38 233 137 104)(39 232 138 103)(40 231 139 102)(41 230 140 101)(42 229 141 100)(43 228 142 99)(44 227 143 98)(45 226 144 97)(46 225 145 96)(47 224 146 95)(48 223 147 94)(49 222 148 93)(50 221 149 92)(51 220 150 91)(52 219 151 90)(53 218 152 89)(54 217 153 88)(55 216 154 87)(56 215 155 86)(57 214 156 85)(58 213 157 84)(59 212 158 83)(60 211 159 82)
(1 111 31 81)(2 110 32 80)(3 109 33 79)(4 108 34 78)(5 107 35 77)(6 106 36 76)(7 105 37 75)(8 104 38 74)(9 103 39 73)(10 102 40 72)(11 101 41 71)(12 100 42 70)(13 99 43 69)(14 98 44 68)(15 97 45 67)(16 96 46 66)(17 95 47 65)(18 94 48 64)(19 93 49 63)(20 92 50 62)(21 91 51 61)(22 90 52 120)(23 89 53 119)(24 88 54 118)(25 87 55 117)(26 86 56 116)(27 85 57 115)(28 84 58 114)(29 83 59 113)(30 82 60 112)(121 219 151 189)(122 218 152 188)(123 217 153 187)(124 216 154 186)(125 215 155 185)(126 214 156 184)(127 213 157 183)(128 212 158 182)(129 211 159 181)(130 210 160 240)(131 209 161 239)(132 208 162 238)(133 207 163 237)(134 206 164 236)(135 205 165 235)(136 204 166 234)(137 203 167 233)(138 202 168 232)(139 201 169 231)(140 200 170 230)(141 199 171 229)(142 198 172 228)(143 197 173 227)(144 196 174 226)(145 195 175 225)(146 194 176 224)(147 193 177 223)(148 192 178 222)(149 191 179 221)(150 190 180 220)
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), (1,210,160,81)(2,209,161,80)(3,208,162,79)(4,207,163,78)(5,206,164,77)(6,205,165,76)(7,204,166,75)(8,203,167,74)(9,202,168,73)(10,201,169,72)(11,200,170,71)(12,199,171,70)(13,198,172,69)(14,197,173,68)(15,196,174,67)(16,195,175,66)(17,194,176,65)(18,193,177,64)(19,192,178,63)(20,191,179,62)(21,190,180,61)(22,189,121,120)(23,188,122,119)(24,187,123,118)(25,186,124,117)(26,185,125,116)(27,184,126,115)(28,183,127,114)(29,182,128,113)(30,181,129,112)(31,240,130,111)(32,239,131,110)(33,238,132,109)(34,237,133,108)(35,236,134,107)(36,235,135,106)(37,234,136,105)(38,233,137,104)(39,232,138,103)(40,231,139,102)(41,230,140,101)(42,229,141,100)(43,228,142,99)(44,227,143,98)(45,226,144,97)(46,225,145,96)(47,224,146,95)(48,223,147,94)(49,222,148,93)(50,221,149,92)(51,220,150,91)(52,219,151,90)(53,218,152,89)(54,217,153,88)(55,216,154,87)(56,215,155,86)(57,214,156,85)(58,213,157,84)(59,212,158,83)(60,211,159,82), (1,111,31,81)(2,110,32,80)(3,109,33,79)(4,108,34,78)(5,107,35,77)(6,106,36,76)(7,105,37,75)(8,104,38,74)(9,103,39,73)(10,102,40,72)(11,101,41,71)(12,100,42,70)(13,99,43,69)(14,98,44,68)(15,97,45,67)(16,96,46,66)(17,95,47,65)(18,94,48,64)(19,93,49,63)(20,92,50,62)(21,91,51,61)(22,90,52,120)(23,89,53,119)(24,88,54,118)(25,87,55,117)(26,86,56,116)(27,85,57,115)(28,84,58,114)(29,83,59,113)(30,82,60,112)(121,219,151,189)(122,218,152,188)(123,217,153,187)(124,216,154,186)(125,215,155,185)(126,214,156,184)(127,213,157,183)(128,212,158,182)(129,211,159,181)(130,210,160,240)(131,209,161,239)(132,208,162,238)(133,207,163,237)(134,206,164,236)(135,205,165,235)(136,204,166,234)(137,203,167,233)(138,202,168,232)(139,201,169,231)(140,200,170,230)(141,199,171,229)(142,198,172,228)(143,197,173,227)(144,196,174,226)(145,195,175,225)(146,194,176,224)(147,193,177,223)(148,192,178,222)(149,191,179,221)(150,190,180,220)>;
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), (1,210,160,81)(2,209,161,80)(3,208,162,79)(4,207,163,78)(5,206,164,77)(6,205,165,76)(7,204,166,75)(8,203,167,74)(9,202,168,73)(10,201,169,72)(11,200,170,71)(12,199,171,70)(13,198,172,69)(14,197,173,68)(15,196,174,67)(16,195,175,66)(17,194,176,65)(18,193,177,64)(19,192,178,63)(20,191,179,62)(21,190,180,61)(22,189,121,120)(23,188,122,119)(24,187,123,118)(25,186,124,117)(26,185,125,116)(27,184,126,115)(28,183,127,114)(29,182,128,113)(30,181,129,112)(31,240,130,111)(32,239,131,110)(33,238,132,109)(34,237,133,108)(35,236,134,107)(36,235,135,106)(37,234,136,105)(38,233,137,104)(39,232,138,103)(40,231,139,102)(41,230,140,101)(42,229,141,100)(43,228,142,99)(44,227,143,98)(45,226,144,97)(46,225,145,96)(47,224,146,95)(48,223,147,94)(49,222,148,93)(50,221,149,92)(51,220,150,91)(52,219,151,90)(53,218,152,89)(54,217,153,88)(55,216,154,87)(56,215,155,86)(57,214,156,85)(58,213,157,84)(59,212,158,83)(60,211,159,82), (1,111,31,81)(2,110,32,80)(3,109,33,79)(4,108,34,78)(5,107,35,77)(6,106,36,76)(7,105,37,75)(8,104,38,74)(9,103,39,73)(10,102,40,72)(11,101,41,71)(12,100,42,70)(13,99,43,69)(14,98,44,68)(15,97,45,67)(16,96,46,66)(17,95,47,65)(18,94,48,64)(19,93,49,63)(20,92,50,62)(21,91,51,61)(22,90,52,120)(23,89,53,119)(24,88,54,118)(25,87,55,117)(26,86,56,116)(27,85,57,115)(28,84,58,114)(29,83,59,113)(30,82,60,112)(121,219,151,189)(122,218,152,188)(123,217,153,187)(124,216,154,186)(125,215,155,185)(126,214,156,184)(127,213,157,183)(128,212,158,182)(129,211,159,181)(130,210,160,240)(131,209,161,239)(132,208,162,238)(133,207,163,237)(134,206,164,236)(135,205,165,235)(136,204,166,234)(137,203,167,233)(138,202,168,232)(139,201,169,231)(140,200,170,230)(141,199,171,229)(142,198,172,228)(143,197,173,227)(144,196,174,226)(145,195,175,225)(146,194,176,224)(147,193,177,223)(148,192,178,222)(149,191,179,221)(150,190,180,220) );
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)], [(1,210,160,81),(2,209,161,80),(3,208,162,79),(4,207,163,78),(5,206,164,77),(6,205,165,76),(7,204,166,75),(8,203,167,74),(9,202,168,73),(10,201,169,72),(11,200,170,71),(12,199,171,70),(13,198,172,69),(14,197,173,68),(15,196,174,67),(16,195,175,66),(17,194,176,65),(18,193,177,64),(19,192,178,63),(20,191,179,62),(21,190,180,61),(22,189,121,120),(23,188,122,119),(24,187,123,118),(25,186,124,117),(26,185,125,116),(27,184,126,115),(28,183,127,114),(29,182,128,113),(30,181,129,112),(31,240,130,111),(32,239,131,110),(33,238,132,109),(34,237,133,108),(35,236,134,107),(36,235,135,106),(37,234,136,105),(38,233,137,104),(39,232,138,103),(40,231,139,102),(41,230,140,101),(42,229,141,100),(43,228,142,99),(44,227,143,98),(45,226,144,97),(46,225,145,96),(47,224,146,95),(48,223,147,94),(49,222,148,93),(50,221,149,92),(51,220,150,91),(52,219,151,90),(53,218,152,89),(54,217,153,88),(55,216,154,87),(56,215,155,86),(57,214,156,85),(58,213,157,84),(59,212,158,83),(60,211,159,82)], [(1,111,31,81),(2,110,32,80),(3,109,33,79),(4,108,34,78),(5,107,35,77),(6,106,36,76),(7,105,37,75),(8,104,38,74),(9,103,39,73),(10,102,40,72),(11,101,41,71),(12,100,42,70),(13,99,43,69),(14,98,44,68),(15,97,45,67),(16,96,46,66),(17,95,47,65),(18,94,48,64),(19,93,49,63),(20,92,50,62),(21,91,51,61),(22,90,52,120),(23,89,53,119),(24,88,54,118),(25,87,55,117),(26,86,56,116),(27,85,57,115),(28,84,58,114),(29,83,59,113),(30,82,60,112),(121,219,151,189),(122,218,152,188),(123,217,153,187),(124,216,154,186),(125,215,155,185),(126,214,156,184),(127,213,157,183),(128,212,158,182),(129,211,159,181),(130,210,160,240),(131,209,161,239),(132,208,162,238),(133,207,163,237),(134,206,164,236),(135,205,165,235),(136,204,166,234),(137,203,167,233),(138,202,168,232),(139,201,169,231),(140,200,170,230),(141,199,171,229),(142,198,172,228),(143,197,173,227),(144,196,174,226),(145,195,175,225),(146,194,176,224),(147,193,177,223),(148,192,178,222),(149,191,179,221),(150,190,180,220)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 6A | ··· | 6G | 10A | ··· | 10N | 12A | ··· | 12H | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 30A | ··· | 30AB | 60A | ··· | 60AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 60 | 60 | 60 | 60 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | + | - | + | - | + | + | - | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | Dic6 | D15 | C5⋊D4 | Dic10 | C4○D12 | D30 | D30 | C4○D20 | C15⋊7D4 | Dic30 | D60⋊11C2 |
| kernel | C60.205D4 | C30.4Q8 | C60⋊5C4 | C30.38D4 | C2×Dic30 | C22×C60 | C22×C20 | C60 | C2×C30 | C22×C12 | C2×C20 | C22×C10 | C30 | C2×C12 | C22×C6 | C20 | C2×C10 | C22×C4 | C12 | C2×C6 | C10 | C2×C4 | C23 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 4 | 8 | 4 | 8 | 16 | 16 | 16 |
Matrix representation of C60.205D4 ►in GL4(𝔽61) generated by
| 50 | 0 | 0 | 0 |
| 0 | 11 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 49 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 0 |
| 0 | 1 | 0 | 0 |
| 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(61))| [50,0,0,0,0,11,0,0,0,0,5,0,0,0,0,49],[0,1,0,0,1,0,0,0,0,0,0,60,0,0,1,0],[0,60,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C60.205D4 in GAP, Magma, Sage, TeX
C_{60}._{205}D_4 % in TeX
G:=Group("C60.205D4"); // GroupNames label
G:=SmallGroup(480,889);
// by ID
G=gap.SmallGroup(480,889);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,253,120,254,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=1,c^2=a^30,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^30*b^-1>;
// generators/relations