metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.10D4, (C2×C4).4D30, (C2×C4).Dic15, (C6×Q8).2D5, (C2×C60).25C4, (C2×C20).75D6, (Q8×C30).2C2, (Q8×C10).6S3, (C2×Q8).4D15, (C2×C12).76D10, (C2×C12).2Dic5, C60.7C4.8C2, C5⋊3(C12.10D4), C20.22(C3⋊D4), C4.15(C15⋊7D4), C3⋊2(C20.10D4), C12.24(C5⋊D4), (C2×C60).61C22, (C2×C20).13Dic3, C15⋊10(C4.10D4), C6.18(C23.D5), C22.4(C2×Dic15), C30.106(C22⋊C4), C2.7(C30.38D4), C10.29(C6.D4), (C2×C30).175(C2×C4), (C2×C6).32(C2×Dic5), (C2×C10).52(C2×Dic3), SmallGroup(480,196)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.10D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a-1, cac-1=a29, cbc-1=a45b3 >
Subgroups: 244 in 76 conjugacy classes, 39 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, M4(2), C2×Q8, C20, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C5⋊2C8, C2×C20, C2×C20, C5×Q8, C4.Dic3, C6×Q8, C60, C60, C2×C30, C4.Dic5, Q8×C10, C12.10D4, C15⋊3C8, C2×C60, C2×C60, Q8×C15, C20.10D4, C60.7C4, Q8×C30, C60.10D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, Dic5, D10, C2×Dic3, C3⋊D4, D15, C4.10D4, C2×Dic5, C5⋊D4, C6.D4, Dic15, D30, C23.D5, C12.10D4, C2×Dic15, C15⋊7D4, C20.10D4, C30.38D4, 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)
(1 215 16 200 31 185 46 230)(2 214 17 199 32 184 47 229)(3 213 18 198 33 183 48 228)(4 212 19 197 34 182 49 227)(5 211 20 196 35 181 50 226)(6 210 21 195 36 240 51 225)(7 209 22 194 37 239 52 224)(8 208 23 193 38 238 53 223)(9 207 24 192 39 237 54 222)(10 206 25 191 40 236 55 221)(11 205 26 190 41 235 56 220)(12 204 27 189 42 234 57 219)(13 203 28 188 43 233 58 218)(14 202 29 187 44 232 59 217)(15 201 30 186 45 231 60 216)(61 180 76 165 91 150 106 135)(62 179 77 164 92 149 107 134)(63 178 78 163 93 148 108 133)(64 177 79 162 94 147 109 132)(65 176 80 161 95 146 110 131)(66 175 81 160 96 145 111 130)(67 174 82 159 97 144 112 129)(68 173 83 158 98 143 113 128)(69 172 84 157 99 142 114 127)(70 171 85 156 100 141 115 126)(71 170 86 155 101 140 116 125)(72 169 87 154 102 139 117 124)(73 168 88 153 103 138 118 123)(74 167 89 152 104 137 119 122)(75 166 90 151 105 136 120 121)
(1 165 46 150 31 135 16 180)(2 134 47 179 32 164 17 149)(3 163 48 148 33 133 18 178)(4 132 49 177 34 162 19 147)(5 161 50 146 35 131 20 176)(6 130 51 175 36 160 21 145)(7 159 52 144 37 129 22 174)(8 128 53 173 38 158 23 143)(9 157 54 142 39 127 24 172)(10 126 55 171 40 156 25 141)(11 155 56 140 41 125 26 170)(12 124 57 169 42 154 27 139)(13 153 58 138 43 123 28 168)(14 122 59 167 44 152 29 137)(15 151 60 136 45 121 30 166)(61 230 106 215 91 200 76 185)(62 199 107 184 92 229 77 214)(63 228 108 213 93 198 78 183)(64 197 109 182 94 227 79 212)(65 226 110 211 95 196 80 181)(66 195 111 240 96 225 81 210)(67 224 112 209 97 194 82 239)(68 193 113 238 98 223 83 208)(69 222 114 207 99 192 84 237)(70 191 115 236 100 221 85 206)(71 220 116 205 101 190 86 235)(72 189 117 234 102 219 87 204)(73 218 118 203 103 188 88 233)(74 187 119 232 104 217 89 202)(75 216 120 201 105 186 90 231)
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), (1,215,16,200,31,185,46,230)(2,214,17,199,32,184,47,229)(3,213,18,198,33,183,48,228)(4,212,19,197,34,182,49,227)(5,211,20,196,35,181,50,226)(6,210,21,195,36,240,51,225)(7,209,22,194,37,239,52,224)(8,208,23,193,38,238,53,223)(9,207,24,192,39,237,54,222)(10,206,25,191,40,236,55,221)(11,205,26,190,41,235,56,220)(12,204,27,189,42,234,57,219)(13,203,28,188,43,233,58,218)(14,202,29,187,44,232,59,217)(15,201,30,186,45,231,60,216)(61,180,76,165,91,150,106,135)(62,179,77,164,92,149,107,134)(63,178,78,163,93,148,108,133)(64,177,79,162,94,147,109,132)(65,176,80,161,95,146,110,131)(66,175,81,160,96,145,111,130)(67,174,82,159,97,144,112,129)(68,173,83,158,98,143,113,128)(69,172,84,157,99,142,114,127)(70,171,85,156,100,141,115,126)(71,170,86,155,101,140,116,125)(72,169,87,154,102,139,117,124)(73,168,88,153,103,138,118,123)(74,167,89,152,104,137,119,122)(75,166,90,151,105,136,120,121), (1,165,46,150,31,135,16,180)(2,134,47,179,32,164,17,149)(3,163,48,148,33,133,18,178)(4,132,49,177,34,162,19,147)(5,161,50,146,35,131,20,176)(6,130,51,175,36,160,21,145)(7,159,52,144,37,129,22,174)(8,128,53,173,38,158,23,143)(9,157,54,142,39,127,24,172)(10,126,55,171,40,156,25,141)(11,155,56,140,41,125,26,170)(12,124,57,169,42,154,27,139)(13,153,58,138,43,123,28,168)(14,122,59,167,44,152,29,137)(15,151,60,136,45,121,30,166)(61,230,106,215,91,200,76,185)(62,199,107,184,92,229,77,214)(63,228,108,213,93,198,78,183)(64,197,109,182,94,227,79,212)(65,226,110,211,95,196,80,181)(66,195,111,240,96,225,81,210)(67,224,112,209,97,194,82,239)(68,193,113,238,98,223,83,208)(69,222,114,207,99,192,84,237)(70,191,115,236,100,221,85,206)(71,220,116,205,101,190,86,235)(72,189,117,234,102,219,87,204)(73,218,118,203,103,188,88,233)(74,187,119,232,104,217,89,202)(75,216,120,201,105,186,90,231)>;
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), (1,215,16,200,31,185,46,230)(2,214,17,199,32,184,47,229)(3,213,18,198,33,183,48,228)(4,212,19,197,34,182,49,227)(5,211,20,196,35,181,50,226)(6,210,21,195,36,240,51,225)(7,209,22,194,37,239,52,224)(8,208,23,193,38,238,53,223)(9,207,24,192,39,237,54,222)(10,206,25,191,40,236,55,221)(11,205,26,190,41,235,56,220)(12,204,27,189,42,234,57,219)(13,203,28,188,43,233,58,218)(14,202,29,187,44,232,59,217)(15,201,30,186,45,231,60,216)(61,180,76,165,91,150,106,135)(62,179,77,164,92,149,107,134)(63,178,78,163,93,148,108,133)(64,177,79,162,94,147,109,132)(65,176,80,161,95,146,110,131)(66,175,81,160,96,145,111,130)(67,174,82,159,97,144,112,129)(68,173,83,158,98,143,113,128)(69,172,84,157,99,142,114,127)(70,171,85,156,100,141,115,126)(71,170,86,155,101,140,116,125)(72,169,87,154,102,139,117,124)(73,168,88,153,103,138,118,123)(74,167,89,152,104,137,119,122)(75,166,90,151,105,136,120,121), (1,165,46,150,31,135,16,180)(2,134,47,179,32,164,17,149)(3,163,48,148,33,133,18,178)(4,132,49,177,34,162,19,147)(5,161,50,146,35,131,20,176)(6,130,51,175,36,160,21,145)(7,159,52,144,37,129,22,174)(8,128,53,173,38,158,23,143)(9,157,54,142,39,127,24,172)(10,126,55,171,40,156,25,141)(11,155,56,140,41,125,26,170)(12,124,57,169,42,154,27,139)(13,153,58,138,43,123,28,168)(14,122,59,167,44,152,29,137)(15,151,60,136,45,121,30,166)(61,230,106,215,91,200,76,185)(62,199,107,184,92,229,77,214)(63,228,108,213,93,198,78,183)(64,197,109,182,94,227,79,212)(65,226,110,211,95,196,80,181)(66,195,111,240,96,225,81,210)(67,224,112,209,97,194,82,239)(68,193,113,238,98,223,83,208)(69,222,114,207,99,192,84,237)(70,191,115,236,100,221,85,206)(71,220,116,205,101,190,86,235)(72,189,117,234,102,219,87,204)(73,218,118,203,103,188,88,233)(74,187,119,232,104,217,89,202)(75,216,120,201,105,186,90,231) );
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)], [(1,215,16,200,31,185,46,230),(2,214,17,199,32,184,47,229),(3,213,18,198,33,183,48,228),(4,212,19,197,34,182,49,227),(5,211,20,196,35,181,50,226),(6,210,21,195,36,240,51,225),(7,209,22,194,37,239,52,224),(8,208,23,193,38,238,53,223),(9,207,24,192,39,237,54,222),(10,206,25,191,40,236,55,221),(11,205,26,190,41,235,56,220),(12,204,27,189,42,234,57,219),(13,203,28,188,43,233,58,218),(14,202,29,187,44,232,59,217),(15,201,30,186,45,231,60,216),(61,180,76,165,91,150,106,135),(62,179,77,164,92,149,107,134),(63,178,78,163,93,148,108,133),(64,177,79,162,94,147,109,132),(65,176,80,161,95,146,110,131),(66,175,81,160,96,145,111,130),(67,174,82,159,97,144,112,129),(68,173,83,158,98,143,113,128),(69,172,84,157,99,142,114,127),(70,171,85,156,100,141,115,126),(71,170,86,155,101,140,116,125),(72,169,87,154,102,139,117,124),(73,168,88,153,103,138,118,123),(74,167,89,152,104,137,119,122),(75,166,90,151,105,136,120,121)], [(1,165,46,150,31,135,16,180),(2,134,47,179,32,164,17,149),(3,163,48,148,33,133,18,178),(4,132,49,177,34,162,19,147),(5,161,50,146,35,131,20,176),(6,130,51,175,36,160,21,145),(7,159,52,144,37,129,22,174),(8,128,53,173,38,158,23,143),(9,157,54,142,39,127,24,172),(10,126,55,171,40,156,25,141),(11,155,56,140,41,125,26,170),(12,124,57,169,42,154,27,139),(13,153,58,138,43,123,28,168),(14,122,59,167,44,152,29,137),(15,151,60,136,45,121,30,166),(61,230,106,215,91,200,76,185),(62,199,107,184,92,229,77,214),(63,228,108,213,93,198,78,183),(64,197,109,182,94,227,79,212),(65,226,110,211,95,196,80,181),(66,195,111,240,96,225,81,210),(67,224,112,209,97,194,82,239),(68,193,113,238,98,223,83,208),(69,222,114,207,99,192,84,237),(70,191,115,236,100,221,85,206),(71,220,116,205,101,190,86,235),(72,189,117,234,102,219,87,204),(73,218,118,203,103,188,88,233),(74,187,119,232,104,217,89,202),(75,216,120,201,105,186,90,231)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 12A | ··· | 12F | 15A | 15B | 15C | 15D | 20A | ··· | 20L | 30A | ··· | 30L | 60A | ··· | 60X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 60 | 60 | 60 | 60 | 2 | ··· | 2 | 4 | ··· | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
81 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | + | - | + | - | |||||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D5 | Dic3 | D6 | Dic5 | D10 | C3⋊D4 | D15 | C5⋊D4 | Dic15 | D30 | C15⋊7D4 | C4.10D4 | C12.10D4 | C20.10D4 | C60.10D4 |
| kernel | C60.10D4 | C60.7C4 | Q8×C30 | C2×C60 | Q8×C10 | C60 | C6×Q8 | C2×C20 | C2×C20 | C2×C12 | C2×C12 | C20 | C2×Q8 | C12 | C2×C4 | C2×C4 | C4 | C15 | C5 | C3 | C1 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 2 | 2 | 1 | 4 | 2 | 4 | 4 | 8 | 8 | 4 | 16 | 1 | 2 | 4 | 8 |
Matrix representation of C60.10D4 ►in GL6(𝔽241)
| 15 | 0 | 0 | 0 | 0 | 0 |
| 0 | 225 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 91 | 0 | 0 |
| 0 | 0 | 150 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 98 |
| 0 | 0 | 0 | 0 | 143 | 0 |
| 0 | 197 | 0 | 0 | 0 | 0 |
| 126 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 50 | 106 |
| 0 | 0 | 0 | 0 | 106 | 191 |
| 0 | 0 | 135 | 50 | 0 | 0 |
| 0 | 0 | 50 | 106 | 0 | 0 |
| 0 | 197 | 0 | 0 | 0 | 0 |
| 115 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 240 | 0 | 0 | 0 |
G:=sub<GL(6,GF(241))| [15,0,0,0,0,0,0,225,0,0,0,0,0,0,0,150,0,0,0,0,91,0,0,0,0,0,0,0,0,143,0,0,0,0,98,0],[0,126,0,0,0,0,197,0,0,0,0,0,0,0,0,0,135,50,0,0,0,0,50,106,0,0,50,106,0,0,0,0,106,191,0,0],[0,115,0,0,0,0,197,0,0,0,0,0,0,0,0,0,0,240,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C60.10D4 in GAP, Magma, Sage, TeX
C_{60}._{10}D_4 % in TeX
G:=Group("C60.10D4"); // GroupNames label
G:=SmallGroup(480,196);
// by ID
G=gap.SmallGroup(480,196);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,120,219,100,675,2693,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^45*b^3>;
// generators/relations