metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.63D4, D20.33D6, C12.52D20, C60.106C23, Dic10.33D6, Dic30.47C22, C3⋊C8.3D10, (C2×C6).7D20, C4○D20.5S3, (C2×C20).97D6, C6.51(C2×D20), (C2×C30).56D4, C30.88(C2×D4), C4.Dic3⋊9D5, C3⋊Dic20⋊13C2, (C2×C12).99D10, C5⋊1(Q8.14D6), C3⋊5(C8.D10), C15⋊9(C8.C22), (C2×Dic30)⋊18C2, C6.D20⋊14C2, C20.30(C3⋊D4), C4.17(C3⋊D20), (C2×C60).95C22, C12.97(C22×D5), C20.156(C22×S3), (C3×D20).38C22, C22.10(C3⋊D20), (C3×Dic10).38C22, C4.105(C2×S3×D5), (C2×C4).15(S3×D5), C10.6(C2×C3⋊D4), (C3×C4○D20).4C2, C2.10(C2×C3⋊D20), (C5×C3⋊C8).21C22, (C5×C4.Dic3)⋊9C2, (C2×C10).13(C3⋊D4), SmallGroup(480,389)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C60.63D4
G = < a,b,c | a60=1, b4=c2=a30, bab-1=a11, cac-1=a-1, cbc-1=b3 >
Subgroups: 620 in 120 conjugacy classes, 44 normal (34 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, C2×C6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×Q8, C3×D5, C30, C30, C8.C22, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, D4.S3, C3⋊Q16, C2×Dic6, C3×C4○D4, C3×Dic5, Dic15, C60, C6×D5, C2×C30, C40⋊C2, Dic20, C5×M4(2), C2×Dic10, C4○D20, Q8.14D6, C5×C3⋊C8, C3×Dic10, D5×C12, C3×D20, C3×C5⋊D4, Dic30, Dic30, C2×Dic15, C2×C60, C8.D10, C6.D20, C3⋊Dic20, C5×C4.Dic3, C3×C4○D20, C2×Dic30, C60.63D4
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, Q8.14D6, C3⋊D20, C2×S3×D5, C8.D10, C2×C3⋊D20, C60.63D4
►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)
(1 110 16 95 31 80 46 65)(2 61 17 106 32 91 47 76)(3 72 18 117 33 102 48 87)(4 83 19 68 34 113 49 98)(5 94 20 79 35 64 50 109)(6 105 21 90 36 75 51 120)(7 116 22 101 37 86 52 71)(8 67 23 112 38 97 53 82)(9 78 24 63 39 108 54 93)(10 89 25 74 40 119 55 104)(11 100 26 85 41 70 56 115)(12 111 27 96 42 81 57 66)(13 62 28 107 43 92 58 77)(14 73 29 118 44 103 59 88)(15 84 30 69 45 114 60 99)(121 221 166 236 151 191 136 206)(122 232 167 187 152 202 137 217)(123 183 168 198 153 213 138 228)(124 194 169 209 154 224 139 239)(125 205 170 220 155 235 140 190)(126 216 171 231 156 186 141 201)(127 227 172 182 157 197 142 212)(128 238 173 193 158 208 143 223)(129 189 174 204 159 219 144 234)(130 200 175 215 160 230 145 185)(131 211 176 226 161 181 146 196)(132 222 177 237 162 192 147 207)(133 233 178 188 163 203 148 218)(134 184 179 199 164 214 149 229)(135 195 180 210 165 225 150 240)
(1 190 31 220)(2 189 32 219)(3 188 33 218)(4 187 34 217)(5 186 35 216)(6 185 36 215)(7 184 37 214)(8 183 38 213)(9 182 39 212)(10 181 40 211)(11 240 41 210)(12 239 42 209)(13 238 43 208)(14 237 44 207)(15 236 45 206)(16 235 46 205)(17 234 47 204)(18 233 48 203)(19 232 49 202)(20 231 50 201)(21 230 51 200)(22 229 52 199)(23 228 53 198)(24 227 54 197)(25 226 55 196)(26 225 56 195)(27 224 57 194)(28 223 58 193)(29 222 59 192)(30 221 60 191)(61 159 91 129)(62 158 92 128)(63 157 93 127)(64 156 94 126)(65 155 95 125)(66 154 96 124)(67 153 97 123)(68 152 98 122)(69 151 99 121)(70 150 100 180)(71 149 101 179)(72 148 102 178)(73 147 103 177)(74 146 104 176)(75 145 105 175)(76 144 106 174)(77 143 107 173)(78 142 108 172)(79 141 109 171)(80 140 110 170)(81 139 111 169)(82 138 112 168)(83 137 113 167)(84 136 114 166)(85 135 115 165)(86 134 116 164)(87 133 117 163)(88 132 118 162)(89 131 119 161)(90 130 120 160)
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), (1,110,16,95,31,80,46,65)(2,61,17,106,32,91,47,76)(3,72,18,117,33,102,48,87)(4,83,19,68,34,113,49,98)(5,94,20,79,35,64,50,109)(6,105,21,90,36,75,51,120)(7,116,22,101,37,86,52,71)(8,67,23,112,38,97,53,82)(9,78,24,63,39,108,54,93)(10,89,25,74,40,119,55,104)(11,100,26,85,41,70,56,115)(12,111,27,96,42,81,57,66)(13,62,28,107,43,92,58,77)(14,73,29,118,44,103,59,88)(15,84,30,69,45,114,60,99)(121,221,166,236,151,191,136,206)(122,232,167,187,152,202,137,217)(123,183,168,198,153,213,138,228)(124,194,169,209,154,224,139,239)(125,205,170,220,155,235,140,190)(126,216,171,231,156,186,141,201)(127,227,172,182,157,197,142,212)(128,238,173,193,158,208,143,223)(129,189,174,204,159,219,144,234)(130,200,175,215,160,230,145,185)(131,211,176,226,161,181,146,196)(132,222,177,237,162,192,147,207)(133,233,178,188,163,203,148,218)(134,184,179,199,164,214,149,229)(135,195,180,210,165,225,150,240), (1,190,31,220)(2,189,32,219)(3,188,33,218)(4,187,34,217)(5,186,35,216)(6,185,36,215)(7,184,37,214)(8,183,38,213)(9,182,39,212)(10,181,40,211)(11,240,41,210)(12,239,42,209)(13,238,43,208)(14,237,44,207)(15,236,45,206)(16,235,46,205)(17,234,47,204)(18,233,48,203)(19,232,49,202)(20,231,50,201)(21,230,51,200)(22,229,52,199)(23,228,53,198)(24,227,54,197)(25,226,55,196)(26,225,56,195)(27,224,57,194)(28,223,58,193)(29,222,59,192)(30,221,60,191)(61,159,91,129)(62,158,92,128)(63,157,93,127)(64,156,94,126)(65,155,95,125)(66,154,96,124)(67,153,97,123)(68,152,98,122)(69,151,99,121)(70,150,100,180)(71,149,101,179)(72,148,102,178)(73,147,103,177)(74,146,104,176)(75,145,105,175)(76,144,106,174)(77,143,107,173)(78,142,108,172)(79,141,109,171)(80,140,110,170)(81,139,111,169)(82,138,112,168)(83,137,113,167)(84,136,114,166)(85,135,115,165)(86,134,116,164)(87,133,117,163)(88,132,118,162)(89,131,119,161)(90,130,120,160)>;
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), (1,110,16,95,31,80,46,65)(2,61,17,106,32,91,47,76)(3,72,18,117,33,102,48,87)(4,83,19,68,34,113,49,98)(5,94,20,79,35,64,50,109)(6,105,21,90,36,75,51,120)(7,116,22,101,37,86,52,71)(8,67,23,112,38,97,53,82)(9,78,24,63,39,108,54,93)(10,89,25,74,40,119,55,104)(11,100,26,85,41,70,56,115)(12,111,27,96,42,81,57,66)(13,62,28,107,43,92,58,77)(14,73,29,118,44,103,59,88)(15,84,30,69,45,114,60,99)(121,221,166,236,151,191,136,206)(122,232,167,187,152,202,137,217)(123,183,168,198,153,213,138,228)(124,194,169,209,154,224,139,239)(125,205,170,220,155,235,140,190)(126,216,171,231,156,186,141,201)(127,227,172,182,157,197,142,212)(128,238,173,193,158,208,143,223)(129,189,174,204,159,219,144,234)(130,200,175,215,160,230,145,185)(131,211,176,226,161,181,146,196)(132,222,177,237,162,192,147,207)(133,233,178,188,163,203,148,218)(134,184,179,199,164,214,149,229)(135,195,180,210,165,225,150,240), (1,190,31,220)(2,189,32,219)(3,188,33,218)(4,187,34,217)(5,186,35,216)(6,185,36,215)(7,184,37,214)(8,183,38,213)(9,182,39,212)(10,181,40,211)(11,240,41,210)(12,239,42,209)(13,238,43,208)(14,237,44,207)(15,236,45,206)(16,235,46,205)(17,234,47,204)(18,233,48,203)(19,232,49,202)(20,231,50,201)(21,230,51,200)(22,229,52,199)(23,228,53,198)(24,227,54,197)(25,226,55,196)(26,225,56,195)(27,224,57,194)(28,223,58,193)(29,222,59,192)(30,221,60,191)(61,159,91,129)(62,158,92,128)(63,157,93,127)(64,156,94,126)(65,155,95,125)(66,154,96,124)(67,153,97,123)(68,152,98,122)(69,151,99,121)(70,150,100,180)(71,149,101,179)(72,148,102,178)(73,147,103,177)(74,146,104,176)(75,145,105,175)(76,144,106,174)(77,143,107,173)(78,142,108,172)(79,141,109,171)(80,140,110,170)(81,139,111,169)(82,138,112,168)(83,137,113,167)(84,136,114,166)(85,135,115,165)(86,134,116,164)(87,133,117,163)(88,132,118,162)(89,131,119,161)(90,130,120,160) );
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)], [(1,110,16,95,31,80,46,65),(2,61,17,106,32,91,47,76),(3,72,18,117,33,102,48,87),(4,83,19,68,34,113,49,98),(5,94,20,79,35,64,50,109),(6,105,21,90,36,75,51,120),(7,116,22,101,37,86,52,71),(8,67,23,112,38,97,53,82),(9,78,24,63,39,108,54,93),(10,89,25,74,40,119,55,104),(11,100,26,85,41,70,56,115),(12,111,27,96,42,81,57,66),(13,62,28,107,43,92,58,77),(14,73,29,118,44,103,59,88),(15,84,30,69,45,114,60,99),(121,221,166,236,151,191,136,206),(122,232,167,187,152,202,137,217),(123,183,168,198,153,213,138,228),(124,194,169,209,154,224,139,239),(125,205,170,220,155,235,140,190),(126,216,171,231,156,186,141,201),(127,227,172,182,157,197,142,212),(128,238,173,193,158,208,143,223),(129,189,174,204,159,219,144,234),(130,200,175,215,160,230,145,185),(131,211,176,226,161,181,146,196),(132,222,177,237,162,192,147,207),(133,233,178,188,163,203,148,218),(134,184,179,199,164,214,149,229),(135,195,180,210,165,225,150,240)], [(1,190,31,220),(2,189,32,219),(3,188,33,218),(4,187,34,217),(5,186,35,216),(6,185,36,215),(7,184,37,214),(8,183,38,213),(9,182,39,212),(10,181,40,211),(11,240,41,210),(12,239,42,209),(13,238,43,208),(14,237,44,207),(15,236,45,206),(16,235,46,205),(17,234,47,204),(18,233,48,203),(19,232,49,202),(20,231,50,201),(21,230,51,200),(22,229,52,199),(23,228,53,198),(24,227,54,197),(25,226,55,196),(26,225,56,195),(27,224,57,194),(28,223,58,193),(29,222,59,192),(30,221,60,191),(61,159,91,129),(62,158,92,128),(63,157,93,127),(64,156,94,126),(65,155,95,125),(66,154,96,124),(67,153,97,123),(68,152,98,122),(69,151,99,121),(70,150,100,180),(71,149,101,179),(72,148,102,178),(73,147,103,177),(74,146,104,176),(75,145,105,175),(76,144,106,174),(77,143,107,173),(78,142,108,172),(79,141,109,171),(80,140,110,170),(81,139,111,169),(82,138,112,168),(83,137,113,167),(84,136,114,166),(85,135,115,165),(86,134,116,164),(87,133,117,163),(88,132,118,162),(89,131,119,161),(90,130,120,160)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 6D | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 20 | 2 | 2 | 2 | 20 | 60 | 60 | 2 | 2 | 2 | 4 | 20 | 20 | 12 | 12 | 2 | 2 | 4 | 4 | 2 | 2 | 4 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 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 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | + | + | + | - | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | D20 | D20 | C8.C22 | S3×D5 | Q8.14D6 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | C8.D10 | C60.63D4 |
| kernel | C60.63D4 | C6.D20 | C3⋊Dic20 | C5×C4.Dic3 | C3×C4○D20 | C2×Dic30 | C4○D20 | C60 | C2×C30 | C4.Dic3 | Dic10 | D20 | C2×C20 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C60.63D4 ►in GL4(𝔽241) generated by
| 78 | 220 | 0 | 0 |
| 149 | 170 | 0 | 0 |
| 21 | 0 | 220 | 57 |
| 6 | 57 | 184 | 235 |
| 0 | 0 | 240 | 1 |
| 190 | 1 | 239 | 51 |
| 83 | 178 | 240 | 0 |
| 124 | 181 | 240 | 0 |
| 14 | 59 | 0 | 0 |
| 13 | 227 | 0 | 0 |
| 223 | 212 | 33 | 102 |
| 227 | 4 | 98 | 208 |
G:=sub<GL(4,GF(241))| [78,149,21,6,220,170,0,57,0,0,220,184,0,0,57,235],[0,190,83,124,0,1,178,181,240,239,240,240,1,51,0,0],[14,13,223,227,59,227,212,4,0,0,33,98,0,0,102,208] >;
C60.63D4 in GAP, Magma, Sage, TeX
C_{60}._{63}D_4 % in TeX
G:=Group("C60.63D4"); // GroupNames label
G:=SmallGroup(480,389);
// by ID
G=gap.SmallGroup(480,389);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,219,100,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=1,b^4=c^2=a^30,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=b^3>;
// generators/relations