metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.106D4, (C2×C30).3Q8, C15⋊3C8.7C4, C20.44(C4×S3), C12.12(C4×D5), (C2×C20).63D6, C30.32(C4⋊C4), C15⋊8(C8.C4), C60.128(C2×C4), (C2×C12).63D10, (C2×C6).1Dic10, (C2×C10).2Dic6, C4.Dic3.4D5, C4.Dic5.4S3, C22.3(C15⋊Q8), C20.89(C3⋊D4), C12.89(C5⋊D4), C3⋊2(C20.53D4), C5⋊4(C12.53D4), C4.31(C15⋊D4), (C2×C60).212C22, C4.13(D30.C2), C6.8(C10.D4), C2.5(Dic15⋊5C4), C10.14(Dic3⋊C4), (C2×C4).192(S3×D5), (C2×C15⋊3C8).16C2, (C5×C4.Dic3).4C2, (C3×C4.Dic5).4C2, SmallGroup(480,68)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.D4
G = < a,b,c | a60=1, b4=a30, c2=a45, bab-1=a29, cac-1=a49, cbc-1=b3 >
Subgroups: 188 in 60 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C5, C6, C6, C8, C2×C4, C10, C10, C12, C2×C6, C15, C2×C8, M4(2), C20, C2×C10, C3⋊C8, C24, C2×C12, C30, C30, C8.C4, C5⋊2C8, C40, C2×C20, C2×C3⋊C8, C4.Dic3, C3×M4(2), C60, C2×C30, C2×C5⋊2C8, C4.Dic5, C5×M4(2), C12.53D4, C5×C3⋊C8, C3×C5⋊2C8, C15⋊3C8, C2×C60, C20.53D4, C3×C4.Dic5, C5×C4.Dic3, C2×C15⋊3C8, C60.D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, D5, D6, C4⋊C4, D10, Dic6, C4×S3, C3⋊D4, C8.C4, Dic10, C4×D5, C5⋊D4, Dic3⋊C4, S3×D5, C10.D4, C12.53D4, D30.C2, C15⋊D4, C15⋊Q8, C20.53D4, Dic15⋊5C4, C60.D4
►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)
(1 224 16 239 31 194 46 209)(2 193 17 208 32 223 47 238)(3 222 18 237 33 192 48 207)(4 191 19 206 34 221 49 236)(5 220 20 235 35 190 50 205)(6 189 21 204 36 219 51 234)(7 218 22 233 37 188 52 203)(8 187 23 202 38 217 53 232)(9 216 24 231 39 186 54 201)(10 185 25 200 40 215 55 230)(11 214 26 229 41 184 56 199)(12 183 27 198 42 213 57 228)(13 212 28 227 43 182 58 197)(14 181 29 196 44 211 59 226)(15 210 30 225 45 240 60 195)(61 148 106 133 91 178 76 163)(62 177 107 162 92 147 77 132)(63 146 108 131 93 176 78 161)(64 175 109 160 94 145 79 130)(65 144 110 129 95 174 80 159)(66 173 111 158 96 143 81 128)(67 142 112 127 97 172 82 157)(68 171 113 156 98 141 83 126)(69 140 114 125 99 170 84 155)(70 169 115 154 100 139 85 124)(71 138 116 123 101 168 86 153)(72 167 117 152 102 137 87 122)(73 136 118 121 103 166 88 151)(74 165 119 150 104 135 89 180)(75 134 120 179 105 164 90 149)
(1 121 46 166 31 151 16 136)(2 170 47 155 32 140 17 125)(3 159 48 144 33 129 18 174)(4 148 49 133 34 178 19 163)(5 137 50 122 35 167 20 152)(6 126 51 171 36 156 21 141)(7 175 52 160 37 145 22 130)(8 164 53 149 38 134 23 179)(9 153 54 138 39 123 24 168)(10 142 55 127 40 172 25 157)(11 131 56 176 41 161 26 146)(12 180 57 165 42 150 27 135)(13 169 58 154 43 139 28 124)(14 158 59 143 44 128 29 173)(15 147 60 132 45 177 30 162)(61 206 106 191 91 236 76 221)(62 195 107 240 92 225 77 210)(63 184 108 229 93 214 78 199)(64 233 109 218 94 203 79 188)(65 222 110 207 95 192 80 237)(66 211 111 196 96 181 81 226)(67 200 112 185 97 230 82 215)(68 189 113 234 98 219 83 204)(69 238 114 223 99 208 84 193)(70 227 115 212 100 197 85 182)(71 216 116 201 101 186 86 231)(72 205 117 190 102 235 87 220)(73 194 118 239 103 224 88 209)(74 183 119 228 104 213 89 198)(75 232 120 217 105 202 90 187)
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), (1,224,16,239,31,194,46,209)(2,193,17,208,32,223,47,238)(3,222,18,237,33,192,48,207)(4,191,19,206,34,221,49,236)(5,220,20,235,35,190,50,205)(6,189,21,204,36,219,51,234)(7,218,22,233,37,188,52,203)(8,187,23,202,38,217,53,232)(9,216,24,231,39,186,54,201)(10,185,25,200,40,215,55,230)(11,214,26,229,41,184,56,199)(12,183,27,198,42,213,57,228)(13,212,28,227,43,182,58,197)(14,181,29,196,44,211,59,226)(15,210,30,225,45,240,60,195)(61,148,106,133,91,178,76,163)(62,177,107,162,92,147,77,132)(63,146,108,131,93,176,78,161)(64,175,109,160,94,145,79,130)(65,144,110,129,95,174,80,159)(66,173,111,158,96,143,81,128)(67,142,112,127,97,172,82,157)(68,171,113,156,98,141,83,126)(69,140,114,125,99,170,84,155)(70,169,115,154,100,139,85,124)(71,138,116,123,101,168,86,153)(72,167,117,152,102,137,87,122)(73,136,118,121,103,166,88,151)(74,165,119,150,104,135,89,180)(75,134,120,179,105,164,90,149), (1,121,46,166,31,151,16,136)(2,170,47,155,32,140,17,125)(3,159,48,144,33,129,18,174)(4,148,49,133,34,178,19,163)(5,137,50,122,35,167,20,152)(6,126,51,171,36,156,21,141)(7,175,52,160,37,145,22,130)(8,164,53,149,38,134,23,179)(9,153,54,138,39,123,24,168)(10,142,55,127,40,172,25,157)(11,131,56,176,41,161,26,146)(12,180,57,165,42,150,27,135)(13,169,58,154,43,139,28,124)(14,158,59,143,44,128,29,173)(15,147,60,132,45,177,30,162)(61,206,106,191,91,236,76,221)(62,195,107,240,92,225,77,210)(63,184,108,229,93,214,78,199)(64,233,109,218,94,203,79,188)(65,222,110,207,95,192,80,237)(66,211,111,196,96,181,81,226)(67,200,112,185,97,230,82,215)(68,189,113,234,98,219,83,204)(69,238,114,223,99,208,84,193)(70,227,115,212,100,197,85,182)(71,216,116,201,101,186,86,231)(72,205,117,190,102,235,87,220)(73,194,118,239,103,224,88,209)(74,183,119,228,104,213,89,198)(75,232,120,217,105,202,90,187)>;
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), (1,224,16,239,31,194,46,209)(2,193,17,208,32,223,47,238)(3,222,18,237,33,192,48,207)(4,191,19,206,34,221,49,236)(5,220,20,235,35,190,50,205)(6,189,21,204,36,219,51,234)(7,218,22,233,37,188,52,203)(8,187,23,202,38,217,53,232)(9,216,24,231,39,186,54,201)(10,185,25,200,40,215,55,230)(11,214,26,229,41,184,56,199)(12,183,27,198,42,213,57,228)(13,212,28,227,43,182,58,197)(14,181,29,196,44,211,59,226)(15,210,30,225,45,240,60,195)(61,148,106,133,91,178,76,163)(62,177,107,162,92,147,77,132)(63,146,108,131,93,176,78,161)(64,175,109,160,94,145,79,130)(65,144,110,129,95,174,80,159)(66,173,111,158,96,143,81,128)(67,142,112,127,97,172,82,157)(68,171,113,156,98,141,83,126)(69,140,114,125,99,170,84,155)(70,169,115,154,100,139,85,124)(71,138,116,123,101,168,86,153)(72,167,117,152,102,137,87,122)(73,136,118,121,103,166,88,151)(74,165,119,150,104,135,89,180)(75,134,120,179,105,164,90,149), (1,121,46,166,31,151,16,136)(2,170,47,155,32,140,17,125)(3,159,48,144,33,129,18,174)(4,148,49,133,34,178,19,163)(5,137,50,122,35,167,20,152)(6,126,51,171,36,156,21,141)(7,175,52,160,37,145,22,130)(8,164,53,149,38,134,23,179)(9,153,54,138,39,123,24,168)(10,142,55,127,40,172,25,157)(11,131,56,176,41,161,26,146)(12,180,57,165,42,150,27,135)(13,169,58,154,43,139,28,124)(14,158,59,143,44,128,29,173)(15,147,60,132,45,177,30,162)(61,206,106,191,91,236,76,221)(62,195,107,240,92,225,77,210)(63,184,108,229,93,214,78,199)(64,233,109,218,94,203,79,188)(65,222,110,207,95,192,80,237)(66,211,111,196,96,181,81,226)(67,200,112,185,97,230,82,215)(68,189,113,234,98,219,83,204)(69,238,114,223,99,208,84,193)(70,227,115,212,100,197,85,182)(71,216,116,201,101,186,86,231)(72,205,117,190,102,235,87,220)(73,194,118,239,103,224,88,209)(74,183,119,228,104,213,89,198)(75,232,120,217,105,202,90,187) );
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)], [(1,224,16,239,31,194,46,209),(2,193,17,208,32,223,47,238),(3,222,18,237,33,192,48,207),(4,191,19,206,34,221,49,236),(5,220,20,235,35,190,50,205),(6,189,21,204,36,219,51,234),(7,218,22,233,37,188,52,203),(8,187,23,202,38,217,53,232),(9,216,24,231,39,186,54,201),(10,185,25,200,40,215,55,230),(11,214,26,229,41,184,56,199),(12,183,27,198,42,213,57,228),(13,212,28,227,43,182,58,197),(14,181,29,196,44,211,59,226),(15,210,30,225,45,240,60,195),(61,148,106,133,91,178,76,163),(62,177,107,162,92,147,77,132),(63,146,108,131,93,176,78,161),(64,175,109,160,94,145,79,130),(65,144,110,129,95,174,80,159),(66,173,111,158,96,143,81,128),(67,142,112,127,97,172,82,157),(68,171,113,156,98,141,83,126),(69,140,114,125,99,170,84,155),(70,169,115,154,100,139,85,124),(71,138,116,123,101,168,86,153),(72,167,117,152,102,137,87,122),(73,136,118,121,103,166,88,151),(74,165,119,150,104,135,89,180),(75,134,120,179,105,164,90,149)], [(1,121,46,166,31,151,16,136),(2,170,47,155,32,140,17,125),(3,159,48,144,33,129,18,174),(4,148,49,133,34,178,19,163),(5,137,50,122,35,167,20,152),(6,126,51,171,36,156,21,141),(7,175,52,160,37,145,22,130),(8,164,53,149,38,134,23,179),(9,153,54,138,39,123,24,168),(10,142,55,127,40,172,25,157),(11,131,56,176,41,161,26,146),(12,180,57,165,42,150,27,135),(13,169,58,154,43,139,28,124),(14,158,59,143,44,128,29,173),(15,147,60,132,45,177,30,162),(61,206,106,191,91,236,76,221),(62,195,107,240,92,225,77,210),(63,184,108,229,93,214,78,199),(64,233,109,218,94,203,79,188),(65,222,110,207,95,192,80,237),(66,211,111,196,96,181,81,226),(67,200,112,185,97,230,82,215),(68,189,113,234,98,219,83,204),(69,238,114,223,99,208,84,193),(70,227,115,212,100,197,85,182),(71,216,116,201,101,186,86,231),(72,205,117,190,102,235,87,220),(73,194,118,239,103,224,88,209),(74,183,119,228,104,213,89,198),(75,232,120,217,105,202,90,187)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 5A | 5B | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 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 | 5 | 5 | 6 | 6 | 8 | 8 | 8 | 8 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 12 | 12 | 20 | 20 | 30 | 30 | 30 | 30 | 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 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | - | + | + | - | - | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | Q8 | D5 | D6 | D10 | C4×S3 | C3⋊D4 | Dic6 | C8.C4 | C4×D5 | C5⋊D4 | Dic10 | S3×D5 | C12.53D4 | D30.C2 | C15⋊D4 | C15⋊Q8 | C20.53D4 | C60.D4 |
| kernel | C60.D4 | C3×C4.Dic5 | C5×C4.Dic3 | C2×C15⋊3C8 | C15⋊3C8 | C4.Dic5 | C60 | C2×C30 | C4.Dic3 | C2×C20 | C2×C12 | C20 | C20 | C2×C10 | C15 | C12 | C12 | C2×C6 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.D4 ►in GL6(𝔽241)
| 64 | 0 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 51 | 0 | 0 |
| 0 | 0 | 189 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 240 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 102 | 56 | 0 | 0 |
| 0 | 0 | 180 | 139 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 216 |
| 0 | 0 | 0 | 0 | 225 | 232 |
| 0 | 211 | 0 | 0 | 0 | 0 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 143 | 62 | 0 | 0 |
| 0 | 0 | 20 | 98 | 0 | 0 |
| 0 | 0 | 0 | 0 | 99 | 43 |
| 0 | 0 | 0 | 0 | 198 | 142 |
G:=sub<GL(6,GF(241))| [64,0,0,0,0,0,0,64,0,0,0,0,0,0,0,189,0,0,0,0,51,52,0,0,0,0,0,0,1,1,0,0,0,0,240,0],[30,0,0,0,0,0,0,8,0,0,0,0,0,0,102,180,0,0,0,0,56,139,0,0,0,0,0,0,9,225,0,0,0,0,216,232],[0,30,0,0,0,0,211,0,0,0,0,0,0,0,143,20,0,0,0,0,62,98,0,0,0,0,0,0,99,198,0,0,0,0,43,142] >;
C60.D4 in GAP, Magma, Sage, TeX
C_{60}.D_4 % in TeX
G:=Group("C60.D4"); // GroupNames label
G:=SmallGroup(480,68);
// by ID
G=gap.SmallGroup(480,68);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,36,100,675,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=a^30,c^2=a^45,b*a*b^-1=a^29,c*a*c^-1=a^49,c*b*c^-1=b^3>;
// generators/relations