metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.31D4, C20.48D12, C12.48D20, (C2×C20).45D6, (C2×C12).46D10, C20.8(C3⋊D4), C4.Dic3.3D5, C4.Dic5.3S3, C10.18(D6⋊C4), C5⋊2(C12.47D4), C4.13(C5⋊D12), C3⋊1(C4.12D20), C4.13(C3⋊D20), C15⋊6(C4.10D4), C12.10(C5⋊D4), (C2×C60).92C22, (C2×Dic15).2C4, C6.3(D10⋊C4), C2.4(D30⋊4C4), C30.46(C22⋊C4), (C2×Dic30).11C2, C22.4(D30.C2), (C2×C4).6(S3×D5), (C2×C6).2(C4×D5), (C2×C10).25(C4×S3), (C2×C30).85(C2×C4), (C5×C4.Dic3).3C2, (C3×C4.Dic5).3C2, SmallGroup(480,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.31D4
G = < a,b,c | a60=1, b4=c2=a30, bab-1=a19, cac-1=a-1, cbc-1=a45b3 >
Subgroups: 380 in 76 conjugacy classes, 32 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, Dic3, C12, C2×C6, C15, M4(2), C2×Q8, Dic5, C20, C2×C10, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C30, C30, C4.10D4, C5⋊2C8, C40, Dic10, C2×Dic5, C2×C20, C4.Dic3, C3×M4(2), C2×Dic6, Dic15, C60, C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C12.47D4, C5×C3⋊C8, C3×C5⋊2C8, Dic30, C2×Dic15, C2×C60, C4.12D20, C3×C4.Dic5, C5×C4.Dic3, C2×Dic30, C60.31D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, D6, C22⋊C4, D10, C4×S3, D12, C3⋊D4, C4.10D4, C4×D5, D20, C5⋊D4, D6⋊C4, S3×D5, D10⋊C4, C12.47D4, D30.C2, C3⋊D20, C5⋊D12, C4.12D20, D30⋊4C4, C60.31D4
►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)
(1 144 46 159 31 174 16 129)(2 163 47 178 32 133 17 148)(3 122 48 137 33 152 18 167)(4 141 49 156 34 171 19 126)(5 160 50 175 35 130 20 145)(6 179 51 134 36 149 21 164)(7 138 52 153 37 168 22 123)(8 157 53 172 38 127 23 142)(9 176 54 131 39 146 24 161)(10 135 55 150 40 165 25 180)(11 154 56 169 41 124 26 139)(12 173 57 128 42 143 27 158)(13 132 58 147 43 162 28 177)(14 151 59 166 44 121 29 136)(15 170 60 125 45 140 30 155)(61 181 76 226 91 211 106 196)(62 200 77 185 92 230 107 215)(63 219 78 204 93 189 108 234)(64 238 79 223 94 208 109 193)(65 197 80 182 95 227 110 212)(66 216 81 201 96 186 111 231)(67 235 82 220 97 205 112 190)(68 194 83 239 98 224 113 209)(69 213 84 198 99 183 114 228)(70 232 85 217 100 202 115 187)(71 191 86 236 101 221 116 206)(72 210 87 195 102 240 117 225)(73 229 88 214 103 199 118 184)(74 188 89 233 104 218 119 203)(75 207 90 192 105 237 120 222)
(1 237 31 207)(2 236 32 206)(3 235 33 205)(4 234 34 204)(5 233 35 203)(6 232 36 202)(7 231 37 201)(8 230 38 200)(9 229 39 199)(10 228 40 198)(11 227 41 197)(12 226 42 196)(13 225 43 195)(14 224 44 194)(15 223 45 193)(16 222 46 192)(17 221 47 191)(18 220 48 190)(19 219 49 189)(20 218 50 188)(21 217 51 187)(22 216 52 186)(23 215 53 185)(24 214 54 184)(25 213 55 183)(26 212 56 182)(27 211 57 181)(28 210 58 240)(29 209 59 239)(30 208 60 238)(61 143 91 173)(62 142 92 172)(63 141 93 171)(64 140 94 170)(65 139 95 169)(66 138 96 168)(67 137 97 167)(68 136 98 166)(69 135 99 165)(70 134 100 164)(71 133 101 163)(72 132 102 162)(73 131 103 161)(74 130 104 160)(75 129 105 159)(76 128 106 158)(77 127 107 157)(78 126 108 156)(79 125 109 155)(80 124 110 154)(81 123 111 153)(82 122 112 152)(83 121 113 151)(84 180 114 150)(85 179 115 149)(86 178 116 148)(87 177 117 147)(88 176 118 146)(89 175 119 145)(90 174 120 144)
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), (1,144,46,159,31,174,16,129)(2,163,47,178,32,133,17,148)(3,122,48,137,33,152,18,167)(4,141,49,156,34,171,19,126)(5,160,50,175,35,130,20,145)(6,179,51,134,36,149,21,164)(7,138,52,153,37,168,22,123)(8,157,53,172,38,127,23,142)(9,176,54,131,39,146,24,161)(10,135,55,150,40,165,25,180)(11,154,56,169,41,124,26,139)(12,173,57,128,42,143,27,158)(13,132,58,147,43,162,28,177)(14,151,59,166,44,121,29,136)(15,170,60,125,45,140,30,155)(61,181,76,226,91,211,106,196)(62,200,77,185,92,230,107,215)(63,219,78,204,93,189,108,234)(64,238,79,223,94,208,109,193)(65,197,80,182,95,227,110,212)(66,216,81,201,96,186,111,231)(67,235,82,220,97,205,112,190)(68,194,83,239,98,224,113,209)(69,213,84,198,99,183,114,228)(70,232,85,217,100,202,115,187)(71,191,86,236,101,221,116,206)(72,210,87,195,102,240,117,225)(73,229,88,214,103,199,118,184)(74,188,89,233,104,218,119,203)(75,207,90,192,105,237,120,222), (1,237,31,207)(2,236,32,206)(3,235,33,205)(4,234,34,204)(5,233,35,203)(6,232,36,202)(7,231,37,201)(8,230,38,200)(9,229,39,199)(10,228,40,198)(11,227,41,197)(12,226,42,196)(13,225,43,195)(14,224,44,194)(15,223,45,193)(16,222,46,192)(17,221,47,191)(18,220,48,190)(19,219,49,189)(20,218,50,188)(21,217,51,187)(22,216,52,186)(23,215,53,185)(24,214,54,184)(25,213,55,183)(26,212,56,182)(27,211,57,181)(28,210,58,240)(29,209,59,239)(30,208,60,238)(61,143,91,173)(62,142,92,172)(63,141,93,171)(64,140,94,170)(65,139,95,169)(66,138,96,168)(67,137,97,167)(68,136,98,166)(69,135,99,165)(70,134,100,164)(71,133,101,163)(72,132,102,162)(73,131,103,161)(74,130,104,160)(75,129,105,159)(76,128,106,158)(77,127,107,157)(78,126,108,156)(79,125,109,155)(80,124,110,154)(81,123,111,153)(82,122,112,152)(83,121,113,151)(84,180,114,150)(85,179,115,149)(86,178,116,148)(87,177,117,147)(88,176,118,146)(89,175,119,145)(90,174,120,144)>;
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), (1,144,46,159,31,174,16,129)(2,163,47,178,32,133,17,148)(3,122,48,137,33,152,18,167)(4,141,49,156,34,171,19,126)(5,160,50,175,35,130,20,145)(6,179,51,134,36,149,21,164)(7,138,52,153,37,168,22,123)(8,157,53,172,38,127,23,142)(9,176,54,131,39,146,24,161)(10,135,55,150,40,165,25,180)(11,154,56,169,41,124,26,139)(12,173,57,128,42,143,27,158)(13,132,58,147,43,162,28,177)(14,151,59,166,44,121,29,136)(15,170,60,125,45,140,30,155)(61,181,76,226,91,211,106,196)(62,200,77,185,92,230,107,215)(63,219,78,204,93,189,108,234)(64,238,79,223,94,208,109,193)(65,197,80,182,95,227,110,212)(66,216,81,201,96,186,111,231)(67,235,82,220,97,205,112,190)(68,194,83,239,98,224,113,209)(69,213,84,198,99,183,114,228)(70,232,85,217,100,202,115,187)(71,191,86,236,101,221,116,206)(72,210,87,195,102,240,117,225)(73,229,88,214,103,199,118,184)(74,188,89,233,104,218,119,203)(75,207,90,192,105,237,120,222), (1,237,31,207)(2,236,32,206)(3,235,33,205)(4,234,34,204)(5,233,35,203)(6,232,36,202)(7,231,37,201)(8,230,38,200)(9,229,39,199)(10,228,40,198)(11,227,41,197)(12,226,42,196)(13,225,43,195)(14,224,44,194)(15,223,45,193)(16,222,46,192)(17,221,47,191)(18,220,48,190)(19,219,49,189)(20,218,50,188)(21,217,51,187)(22,216,52,186)(23,215,53,185)(24,214,54,184)(25,213,55,183)(26,212,56,182)(27,211,57,181)(28,210,58,240)(29,209,59,239)(30,208,60,238)(61,143,91,173)(62,142,92,172)(63,141,93,171)(64,140,94,170)(65,139,95,169)(66,138,96,168)(67,137,97,167)(68,136,98,166)(69,135,99,165)(70,134,100,164)(71,133,101,163)(72,132,102,162)(73,131,103,161)(74,130,104,160)(75,129,105,159)(76,128,106,158)(77,127,107,157)(78,126,108,156)(79,125,109,155)(80,124,110,154)(81,123,111,153)(82,122,112,152)(83,121,113,151)(84,180,114,150)(85,179,115,149)(86,178,116,148)(87,177,117,147)(88,176,118,146)(89,175,119,145)(90,174,120,144) );
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)], [(1,144,46,159,31,174,16,129),(2,163,47,178,32,133,17,148),(3,122,48,137,33,152,18,167),(4,141,49,156,34,171,19,126),(5,160,50,175,35,130,20,145),(6,179,51,134,36,149,21,164),(7,138,52,153,37,168,22,123),(8,157,53,172,38,127,23,142),(9,176,54,131,39,146,24,161),(10,135,55,150,40,165,25,180),(11,154,56,169,41,124,26,139),(12,173,57,128,42,143,27,158),(13,132,58,147,43,162,28,177),(14,151,59,166,44,121,29,136),(15,170,60,125,45,140,30,155),(61,181,76,226,91,211,106,196),(62,200,77,185,92,230,107,215),(63,219,78,204,93,189,108,234),(64,238,79,223,94,208,109,193),(65,197,80,182,95,227,110,212),(66,216,81,201,96,186,111,231),(67,235,82,220,97,205,112,190),(68,194,83,239,98,224,113,209),(69,213,84,198,99,183,114,228),(70,232,85,217,100,202,115,187),(71,191,86,236,101,221,116,206),(72,210,87,195,102,240,117,225),(73,229,88,214,103,199,118,184),(74,188,89,233,104,218,119,203),(75,207,90,192,105,237,120,222)], [(1,237,31,207),(2,236,32,206),(3,235,33,205),(4,234,34,204),(5,233,35,203),(6,232,36,202),(7,231,37,201),(8,230,38,200),(9,229,39,199),(10,228,40,198),(11,227,41,197),(12,226,42,196),(13,225,43,195),(14,224,44,194),(15,223,45,193),(16,222,46,192),(17,221,47,191),(18,220,48,190),(19,219,49,189),(20,218,50,188),(21,217,51,187),(22,216,52,186),(23,215,53,185),(24,214,54,184),(25,213,55,183),(26,212,56,182),(27,211,57,181),(28,210,58,240),(29,209,59,239),(30,208,60,238),(61,143,91,173),(62,142,92,172),(63,141,93,171),(64,140,94,170),(65,139,95,169),(66,138,96,168),(67,137,97,167),(68,136,98,166),(69,135,99,165),(70,134,100,164),(71,133,101,163),(72,132,102,162),(73,131,103,161),(74,130,104,160),(75,129,105,159),(76,128,106,158),(77,127,107,157),(78,126,108,156),(79,125,109,155),(80,124,110,154),(81,123,111,153),(82,122,112,152),(83,121,113,151),(84,180,114,150),(85,179,115,149),(86,178,116,148),(87,177,117,147),(88,176,118,146),(89,175,119,145),(90,174,120,144)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 60 | 60 | 2 | 2 | 2 | 4 | 12 | 12 | 20 | 20 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | + | - | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | D6 | D10 | D12 | C3⋊D4 | C4×S3 | D20 | C5⋊D4 | C4×D5 | C4.10D4 | S3×D5 | C12.47D4 | C3⋊D20 | C5⋊D12 | D30.C2 | C4.12D20 | C60.31D4 |
| kernel | C60.31D4 | C3×C4.Dic5 | C5×C4.Dic3 | C2×Dic30 | C2×Dic15 | C4.Dic5 | C60 | C4.Dic3 | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | C12 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.31D4 ►in GL8(𝔽241)
| 0 | 0 | 0 | 189 | 0 | 0 | 0 | 0 |
| 0 | 0 | 51 | 190 | 0 | 0 | 0 | 0 |
| 0 | 52 | 0 | 52 | 0 | 0 | 0 | 0 |
| 190 | 51 | 190 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 125 | 210 | 0 | 0 |
| 0 | 0 | 0 | 0 | 115 | 172 | 0 | 0 |
| 0 | 0 | 0 | 0 | 214 | 195 | 142 | 99 |
| 0 | 0 | 0 | 0 | 100 | 53 | 142 | 43 |
| 149 | 99 | 34 | 78 | 0 | 0 | 0 | 0 |
| 132 | 92 | 184 | 207 | 0 | 0 | 0 | 0 |
| 207 | 163 | 115 | 21 | 0 | 0 | 0 | 0 |
| 57 | 34 | 189 | 126 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 47 | 0 | 146 | 86 |
| 0 | 0 | 0 | 0 | 195 | 0 | 151 | 186 |
| 0 | 0 | 0 | 0 | 128 | 174 | 170 | 11 |
| 0 | 0 | 0 | 0 | 49 | 36 | 108 | 24 |
| 213 | 53 | 229 | 126 | 0 | 0 | 0 | 0 |
| 196 | 28 | 84 | 12 | 0 | 0 | 0 | 0 |
| 16 | 73 | 28 | 188 | 0 | 0 | 0 | 0 |
| 129 | 225 | 45 | 213 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 84 | 150 | 0 | 0 |
| 0 | 0 | 0 | 0 | 157 | 157 | 0 | 0 |
| 0 | 0 | 0 | 0 | 178 | 36 | 36 | 67 |
| 0 | 0 | 0 | 0 | 122 | 210 | 31 | 205 |
G:=sub<GL(8,GF(241))| [0,0,0,190,0,0,0,0,0,0,52,51,0,0,0,0,0,51,0,190,0,0,0,0,189,190,52,51,0,0,0,0,0,0,0,0,125,115,214,100,0,0,0,0,210,172,195,53,0,0,0,0,0,0,142,142,0,0,0,0,0,0,99,43],[149,132,207,57,0,0,0,0,99,92,163,34,0,0,0,0,34,184,115,189,0,0,0,0,78,207,21,126,0,0,0,0,0,0,0,0,47,195,128,49,0,0,0,0,0,0,174,36,0,0,0,0,146,151,170,108,0,0,0,0,86,186,11,24],[213,196,16,129,0,0,0,0,53,28,73,225,0,0,0,0,229,84,28,45,0,0,0,0,126,12,188,213,0,0,0,0,0,0,0,0,84,157,178,122,0,0,0,0,150,157,36,210,0,0,0,0,0,0,36,31,0,0,0,0,0,0,67,205] >;
C60.31D4 in GAP, Magma, Sage, TeX
C_{60}._{31}D_4 % in TeX
G:=Group("C60.31D4"); // GroupNames label
G:=SmallGroup(480,39);
// by ID
G=gap.SmallGroup(480,39);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,204,219,100,675,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=c^2=a^30,b*a*b^-1=a^19,c*a*c^-1=a^-1,c*b*c^-1=a^45*b^3>;
// generators/relations