metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20.60D12, C60.109D4, D12.30D10, C60.127C23, Dic6.32D10, D60.48C22, Dic30.51C22, C4○D12⋊1D5, C5⋊5(C4○D24), C15⋊11(C4○D8), C5⋊D24⋊15C2, (C2×C10).5D12, (C2×C30).46D4, C30.78(C2×D4), (C2×C20).89D6, C5⋊2C8.35D6, C5⋊Dic12⋊15C2, C10.48(C2×D12), Dic6⋊D5⋊15C2, D12.D5⋊15C2, (C2×C12).318D10, C3⋊1(D4.8D10), C4.32(C5⋊D12), C12.68(C5⋊D4), (C2×C60).49C22, C20.89(C22×S3), D60⋊11C2⋊10C2, (C5×D12).35C22, C12.150(C22×D5), C22.1(C5⋊D12), (C5×Dic6).37C22, (C6×C5⋊2C8)⋊3C2, C4.75(C2×S3×D5), (C2×C5⋊2C8)⋊6S3, C6.2(C2×C5⋊D4), (C5×C4○D12)⋊4C2, C2.6(C2×C5⋊D12), (C2×C4).146(S3×D5), (C2×C6).31(C5⋊D4), (C3×C5⋊2C8).39C22, SmallGroup(480,379)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.60D12
G = < a,b,c | a20=c2=1, b12=a10, bab-1=cac=a9, cbc=a10b11 >
Subgroups: 668 in 124 conjugacy classes, 44 normal (all characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, D6, C2×C6, C15, C2×C8, D8, SD16, Q16, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C24, Dic6, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C5×S3, D15, C30, C30, C4○D8, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C24⋊C2, D24, Dic12, C2×C24, C4○D12, C4○D12, C5×Dic3, Dic15, C60, S3×C10, D30, C2×C30, C2×C5⋊2C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C4○D20, C5×C4○D4, C4○D24, C3×C5⋊2C8, C5×Dic6, S3×C20, C5×D12, C5×C3⋊D4, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, D4.8D10, C5⋊D24, D12.D5, Dic6⋊D5, C5⋊Dic12, C6×C5⋊2C8, C5×C4○D12, D60⋊11C2, C20.60D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C4○D8, C5⋊D4, C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C4○D24, C5⋊D12, C2×S3×D5, D4.8D10, C2×C5⋊D12, C20.60D12
►On 240 points
Generators in S240
(1 223 142 34 195 7 229 124 40 201 13 235 130 46 207 19 217 136 28 213)(2 202 29 125 218 8 208 35 131 224 14 214 41 137 230 20 196 47 143 236)(3 225 144 36 197 9 231 126 42 203 15 237 132 48 209 21 219 138 30 215)(4 204 31 127 220 10 210 37 133 226 16 216 43 139 232 22 198 25 121 238)(5 227 122 38 199 11 233 128 44 205 17 239 134 26 211 23 221 140 32 193)(6 206 33 129 222 12 212 39 135 228 18 194 45 141 234 24 200 27 123 240)(49 109 170 146 87 67 103 188 164 81 61 97 182 158 75 55 115 176 152 93)(50 82 153 189 116 68 76 147 183 110 62 94 165 177 104 56 88 159 171 98)(51 111 172 148 89 69 105 190 166 83 63 99 184 160 77 57 117 178 154 95)(52 84 155 191 118 70 78 149 185 112 64 96 167 179 106 58 90 161 173 100)(53 113 174 150 91 71 107 192 168 85 65 101 186 162 79 59 119 180 156 73)(54 86 157 169 120 72 80 151 187 114 66 74 145 181 108 60 92 163 175 102)
(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 55)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 72)(9 71)(10 70)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 60)(21 59)(22 58)(23 57)(24 56)(25 179)(26 178)(27 177)(28 176)(29 175)(30 174)(31 173)(32 172)(33 171)(34 170)(35 169)(36 192)(37 191)(38 190)(39 189)(40 188)(41 187)(42 186)(43 185)(44 184)(45 183)(46 182)(47 181)(48 180)(73 237)(74 236)(75 235)(76 234)(77 233)(78 232)(79 231)(80 230)(81 229)(82 228)(83 227)(84 226)(85 225)(86 224)(87 223)(88 222)(89 221)(90 220)(91 219)(92 218)(93 217)(94 240)(95 239)(96 238)(97 195)(98 194)(99 193)(100 216)(101 215)(102 214)(103 213)(104 212)(105 211)(106 210)(107 209)(108 208)(109 207)(110 206)(111 205)(112 204)(113 203)(114 202)(115 201)(116 200)(117 199)(118 198)(119 197)(120 196)(121 155)(122 154)(123 153)(124 152)(125 151)(126 150)(127 149)(128 148)(129 147)(130 146)(131 145)(132 168)(133 167)(134 166)(135 165)(136 164)(137 163)(138 162)(139 161)(140 160)(141 159)(142 158)(143 157)(144 156)
G:=sub<Sym(240)| (1,223,142,34,195,7,229,124,40,201,13,235,130,46,207,19,217,136,28,213)(2,202,29,125,218,8,208,35,131,224,14,214,41,137,230,20,196,47,143,236)(3,225,144,36,197,9,231,126,42,203,15,237,132,48,209,21,219,138,30,215)(4,204,31,127,220,10,210,37,133,226,16,216,43,139,232,22,198,25,121,238)(5,227,122,38,199,11,233,128,44,205,17,239,134,26,211,23,221,140,32,193)(6,206,33,129,222,12,212,39,135,228,18,194,45,141,234,24,200,27,123,240)(49,109,170,146,87,67,103,188,164,81,61,97,182,158,75,55,115,176,152,93)(50,82,153,189,116,68,76,147,183,110,62,94,165,177,104,56,88,159,171,98)(51,111,172,148,89,69,105,190,166,83,63,99,184,160,77,57,117,178,154,95)(52,84,155,191,118,70,78,149,185,112,64,96,167,179,106,58,90,161,173,100)(53,113,174,150,91,71,107,192,168,85,65,101,186,162,79,59,119,180,156,73)(54,86,157,169,120,72,80,151,187,114,66,74,145,181,108,60,92,163,175,102), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,179)(26,178)(27,177)(28,176)(29,175)(30,174)(31,173)(32,172)(33,171)(34,170)(35,169)(36,192)(37,191)(38,190)(39,189)(40,188)(41,187)(42,186)(43,185)(44,184)(45,183)(46,182)(47,181)(48,180)(73,237)(74,236)(75,235)(76,234)(77,233)(78,232)(79,231)(80,230)(81,229)(82,228)(83,227)(84,226)(85,225)(86,224)(87,223)(88,222)(89,221)(90,220)(91,219)(92,218)(93,217)(94,240)(95,239)(96,238)(97,195)(98,194)(99,193)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)(107,209)(108,208)(109,207)(110,206)(111,205)(112,204)(113,203)(114,202)(115,201)(116,200)(117,199)(118,198)(119,197)(120,196)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,168)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156)>;
G:=Group( (1,223,142,34,195,7,229,124,40,201,13,235,130,46,207,19,217,136,28,213)(2,202,29,125,218,8,208,35,131,224,14,214,41,137,230,20,196,47,143,236)(3,225,144,36,197,9,231,126,42,203,15,237,132,48,209,21,219,138,30,215)(4,204,31,127,220,10,210,37,133,226,16,216,43,139,232,22,198,25,121,238)(5,227,122,38,199,11,233,128,44,205,17,239,134,26,211,23,221,140,32,193)(6,206,33,129,222,12,212,39,135,228,18,194,45,141,234,24,200,27,123,240)(49,109,170,146,87,67,103,188,164,81,61,97,182,158,75,55,115,176,152,93)(50,82,153,189,116,68,76,147,183,110,62,94,165,177,104,56,88,159,171,98)(51,111,172,148,89,69,105,190,166,83,63,99,184,160,77,57,117,178,154,95)(52,84,155,191,118,70,78,149,185,112,64,96,167,179,106,58,90,161,173,100)(53,113,174,150,91,71,107,192,168,85,65,101,186,162,79,59,119,180,156,73)(54,86,157,169,120,72,80,151,187,114,66,74,145,181,108,60,92,163,175,102), (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), (1,55)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,72)(9,71)(10,70)(11,69)(12,68)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,60)(21,59)(22,58)(23,57)(24,56)(25,179)(26,178)(27,177)(28,176)(29,175)(30,174)(31,173)(32,172)(33,171)(34,170)(35,169)(36,192)(37,191)(38,190)(39,189)(40,188)(41,187)(42,186)(43,185)(44,184)(45,183)(46,182)(47,181)(48,180)(73,237)(74,236)(75,235)(76,234)(77,233)(78,232)(79,231)(80,230)(81,229)(82,228)(83,227)(84,226)(85,225)(86,224)(87,223)(88,222)(89,221)(90,220)(91,219)(92,218)(93,217)(94,240)(95,239)(96,238)(97,195)(98,194)(99,193)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)(106,210)(107,209)(108,208)(109,207)(110,206)(111,205)(112,204)(113,203)(114,202)(115,201)(116,200)(117,199)(118,198)(119,197)(120,196)(121,155)(122,154)(123,153)(124,152)(125,151)(126,150)(127,149)(128,148)(129,147)(130,146)(131,145)(132,168)(133,167)(134,166)(135,165)(136,164)(137,163)(138,162)(139,161)(140,160)(141,159)(142,158)(143,157)(144,156) );
G=PermutationGroup([[(1,223,142,34,195,7,229,124,40,201,13,235,130,46,207,19,217,136,28,213),(2,202,29,125,218,8,208,35,131,224,14,214,41,137,230,20,196,47,143,236),(3,225,144,36,197,9,231,126,42,203,15,237,132,48,209,21,219,138,30,215),(4,204,31,127,220,10,210,37,133,226,16,216,43,139,232,22,198,25,121,238),(5,227,122,38,199,11,233,128,44,205,17,239,134,26,211,23,221,140,32,193),(6,206,33,129,222,12,212,39,135,228,18,194,45,141,234,24,200,27,123,240),(49,109,170,146,87,67,103,188,164,81,61,97,182,158,75,55,115,176,152,93),(50,82,153,189,116,68,76,147,183,110,62,94,165,177,104,56,88,159,171,98),(51,111,172,148,89,69,105,190,166,83,63,99,184,160,77,57,117,178,154,95),(52,84,155,191,118,70,78,149,185,112,64,96,167,179,106,58,90,161,173,100),(53,113,174,150,91,71,107,192,168,85,65,101,186,162,79,59,119,180,156,73),(54,86,157,169,120,72,80,151,187,114,66,74,145,181,108,60,92,163,175,102)], [(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)], [(1,55),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,72),(9,71),(10,70),(11,69),(12,68),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,60),(21,59),(22,58),(23,57),(24,56),(25,179),(26,178),(27,177),(28,176),(29,175),(30,174),(31,173),(32,172),(33,171),(34,170),(35,169),(36,192),(37,191),(38,190),(39,189),(40,188),(41,187),(42,186),(43,185),(44,184),(45,183),(46,182),(47,181),(48,180),(73,237),(74,236),(75,235),(76,234),(77,233),(78,232),(79,231),(80,230),(81,229),(82,228),(83,227),(84,226),(85,225),(86,224),(87,223),(88,222),(89,221),(90,220),(91,219),(92,218),(93,217),(94,240),(95,239),(96,238),(97,195),(98,194),(99,193),(100,216),(101,215),(102,214),(103,213),(104,212),(105,211),(106,210),(107,209),(108,208),(109,207),(110,206),(111,205),(112,204),(113,203),(114,202),(115,201),(116,200),(117,199),(118,198),(119,197),(120,196),(121,155),(122,154),(123,153),(124,152),(125,151),(126,150),(127,149),(128,148),(129,147),(130,146),(131,145),(132,168),(133,167),(134,166),(135,165),(136,164),(137,163),(138,162),(139,161),(140,160),(141,159),(142,158),(143,157),(144,156)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 10A | 10B | 10C | 10D | 10E | 10F | 10G | 10H | 12A | 12B | 12C | 12D | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 20G | 20H | 20I | 20J | 24A | ··· | 24H | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 20 | 24 | ··· | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 12 | 60 | 2 | 1 | 1 | 2 | 12 | 60 | 2 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 12 | 10 | ··· | 10 | 4 | ··· | 4 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | D10 | D12 | D12 | C4○D8 | C5⋊D4 | C5⋊D4 | C4○D24 | S3×D5 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | D4.8D10 | C20.60D12 |
| kernel | C20.60D12 | C5⋊D24 | D12.D5 | Dic6⋊D5 | C5⋊Dic12 | C6×C5⋊2C8 | C5×C4○D12 | D60⋊11C2 | C2×C5⋊2C8 | C60 | C2×C30 | C4○D12 | C5⋊2C8 | C2×C20 | Dic6 | D12 | C2×C12 | C20 | C2×C10 | C15 | C12 | C2×C6 | C5 | C2×C4 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C20.60D12 ►in GL4(𝔽241) generated by
| 177 | 0 | 0 | 0 |
| 0 | 177 | 0 | 0 |
| 0 | 0 | 1 | 1 |
| 0 | 0 | 50 | 51 |
| 113 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 |
| 0 | 0 | 141 | 63 |
| 0 | 0 | 44 | 100 |
| 0 | 120 | 0 | 0 |
| 239 | 0 | 0 | 0 |
| 0 | 0 | 176 | 95 |
| 0 | 0 | 194 | 65 |
G:=sub<GL(4,GF(241))| [177,0,0,0,0,177,0,0,0,0,1,50,0,0,1,51],[113,0,0,0,0,32,0,0,0,0,141,44,0,0,63,100],[0,239,0,0,120,0,0,0,0,0,176,194,0,0,95,65] >;
C20.60D12 in GAP, Magma, Sage, TeX
C_{20}._{60}D_{12} % in TeX
G:=Group("C20.60D12"); // GroupNames label
G:=SmallGroup(480,379);
// by ID
G=gap.SmallGroup(480,379);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,100,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^20=c^2=1,b^12=a^10,b*a*b^-1=c*a*c=a^9,c*b*c=a^10*b^11>;
// generators/relations