metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.87D4, C20.18D12, C60.133C23, Dic6.34D10, D60.51C22, Dic30.53C22, C5⋊2C8.3D6, (C2×Dic6)⋊9D5, (C2×C30).64D4, C30.96(C2×D4), C5⋊4(C8.D6), C4.Dic5⋊9S3, (C10×Dic6)⋊2C2, C5⋊Dic12⋊13C2, C10.54(C2×D12), (C2×C20).103D6, (C2×C10).43D12, Dic6⋊D5⋊14C2, (C2×C12).104D10, C3⋊1(C20.C23), C4.24(C5⋊D12), C12.33(C5⋊D4), C15⋊11(C8.C22), (C2×C60).27C22, D60⋊11C2.2C2, C20.100(C22×S3), C12.156(C22×D5), C22.5(C5⋊D12), (C5×Dic6).39C22, C4.81(C2×S3×D5), C6.8(C2×C5⋊D4), (C2×C4).17(S3×D5), C2.12(C2×C5⋊D12), (C3×C4.Dic5)⋊3C2, (C2×C6).14(C5⋊D4), (C3×C5⋊2C8).21C22, SmallGroup(480,397)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20.D12
G = < a,b,c | a20=1, b12=c2=a10, bab-1=a-1, cac-1=a9, cbc-1=a10b11 >
Subgroups: 636 in 120 conjugacy classes, 44 normal (32 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, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, D15, C30, C30, C8.C22, C5⋊2C8, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×Q8, C24⋊C2, Dic12, C3×M4(2), C2×Dic6, C4○D12, C5×Dic3, Dic15, C60, D30, C2×C30, C4.Dic5, Q8⋊D5, C5⋊Q16, C4○D20, Q8×C10, C8.D6, C3×C5⋊2C8, C5×Dic6, C5×Dic6, C10×Dic3, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C20.C23, Dic6⋊D5, C5⋊Dic12, C3×C4.Dic5, C10×Dic6, D60⋊11C2, C20.D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C8.C22, C5⋊D4, C22×D5, C2×D12, S3×D5, C2×C5⋊D4, C8.D6, C5⋊D12, C2×S3×D5, C20.C23, C2×C5⋊D12, C20.D12
►On 240 points
Generators in S240
(1 73 105 129 229 19 91 99 123 223 13 85 117 141 217 7 79 111 135 235)(2 236 136 112 80 8 218 142 118 86 14 224 124 100 92 20 230 130 106 74)(3 75 107 131 231 21 93 101 125 225 15 87 119 143 219 9 81 113 137 237)(4 238 138 114 82 10 220 144 120 88 16 226 126 102 94 22 232 132 108 76)(5 77 109 133 233 23 95 103 127 227 17 89 97 121 221 11 83 115 139 239)(6 240 140 116 84 12 222 122 98 90 18 228 128 104 96 24 234 134 110 78)(25 56 185 198 146 31 62 191 204 152 37 68 173 210 158 43 50 179 216 164)(26 165 193 180 51 44 159 211 174 69 38 153 205 192 63 32 147 199 186 57)(27 58 187 200 148 33 64 169 206 154 39 70 175 212 160 45 52 181 194 166)(28 167 195 182 53 46 161 213 176 71 40 155 207 170 65 34 149 201 188 59)(29 60 189 202 150 35 66 171 208 156 41 72 177 214 162 47 54 183 196 168)(30 145 197 184 55 48 163 215 178 49 42 157 209 172 67 36 151 203 190 61)
(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 25 13 37)(2 48 14 36)(3 47 15 35)(4 46 16 34)(5 45 17 33)(6 44 18 32)(7 43 19 31)(8 42 20 30)(9 41 21 29)(10 40 22 28)(11 39 23 27)(12 38 24 26)(49 218 61 230)(50 217 62 229)(51 240 63 228)(52 239 64 227)(53 238 65 226)(54 237 66 225)(55 236 67 224)(56 235 68 223)(57 234 69 222)(58 233 70 221)(59 232 71 220)(60 231 72 219)(73 152 85 164)(74 151 86 163)(75 150 87 162)(76 149 88 161)(77 148 89 160)(78 147 90 159)(79 146 91 158)(80 145 92 157)(81 168 93 156)(82 167 94 155)(83 166 95 154)(84 165 96 153)(97 200 109 212)(98 199 110 211)(99 198 111 210)(100 197 112 209)(101 196 113 208)(102 195 114 207)(103 194 115 206)(104 193 116 205)(105 216 117 204)(106 215 118 203)(107 214 119 202)(108 213 120 201)(121 175 133 187)(122 174 134 186)(123 173 135 185)(124 172 136 184)(125 171 137 183)(126 170 138 182)(127 169 139 181)(128 192 140 180)(129 191 141 179)(130 190 142 178)(131 189 143 177)(132 188 144 176)
G:=sub<Sym(240)| (1,73,105,129,229,19,91,99,123,223,13,85,117,141,217,7,79,111,135,235)(2,236,136,112,80,8,218,142,118,86,14,224,124,100,92,20,230,130,106,74)(3,75,107,131,231,21,93,101,125,225,15,87,119,143,219,9,81,113,137,237)(4,238,138,114,82,10,220,144,120,88,16,226,126,102,94,22,232,132,108,76)(5,77,109,133,233,23,95,103,127,227,17,89,97,121,221,11,83,115,139,239)(6,240,140,116,84,12,222,122,98,90,18,228,128,104,96,24,234,134,110,78)(25,56,185,198,146,31,62,191,204,152,37,68,173,210,158,43,50,179,216,164)(26,165,193,180,51,44,159,211,174,69,38,153,205,192,63,32,147,199,186,57)(27,58,187,200,148,33,64,169,206,154,39,70,175,212,160,45,52,181,194,166)(28,167,195,182,53,46,161,213,176,71,40,155,207,170,65,34,149,201,188,59)(29,60,189,202,150,35,66,171,208,156,41,72,177,214,162,47,54,183,196,168)(30,145,197,184,55,48,163,215,178,49,42,157,209,172,67,36,151,203,190,61), (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), (1,25,13,37)(2,48,14,36)(3,47,15,35)(4,46,16,34)(5,45,17,33)(6,44,18,32)(7,43,19,31)(8,42,20,30)(9,41,21,29)(10,40,22,28)(11,39,23,27)(12,38,24,26)(49,218,61,230)(50,217,62,229)(51,240,63,228)(52,239,64,227)(53,238,65,226)(54,237,66,225)(55,236,67,224)(56,235,68,223)(57,234,69,222)(58,233,70,221)(59,232,71,220)(60,231,72,219)(73,152,85,164)(74,151,86,163)(75,150,87,162)(76,149,88,161)(77,148,89,160)(78,147,90,159)(79,146,91,158)(80,145,92,157)(81,168,93,156)(82,167,94,155)(83,166,95,154)(84,165,96,153)(97,200,109,212)(98,199,110,211)(99,198,111,210)(100,197,112,209)(101,196,113,208)(102,195,114,207)(103,194,115,206)(104,193,116,205)(105,216,117,204)(106,215,118,203)(107,214,119,202)(108,213,120,201)(121,175,133,187)(122,174,134,186)(123,173,135,185)(124,172,136,184)(125,171,137,183)(126,170,138,182)(127,169,139,181)(128,192,140,180)(129,191,141,179)(130,190,142,178)(131,189,143,177)(132,188,144,176)>;
G:=Group( (1,73,105,129,229,19,91,99,123,223,13,85,117,141,217,7,79,111,135,235)(2,236,136,112,80,8,218,142,118,86,14,224,124,100,92,20,230,130,106,74)(3,75,107,131,231,21,93,101,125,225,15,87,119,143,219,9,81,113,137,237)(4,238,138,114,82,10,220,144,120,88,16,226,126,102,94,22,232,132,108,76)(5,77,109,133,233,23,95,103,127,227,17,89,97,121,221,11,83,115,139,239)(6,240,140,116,84,12,222,122,98,90,18,228,128,104,96,24,234,134,110,78)(25,56,185,198,146,31,62,191,204,152,37,68,173,210,158,43,50,179,216,164)(26,165,193,180,51,44,159,211,174,69,38,153,205,192,63,32,147,199,186,57)(27,58,187,200,148,33,64,169,206,154,39,70,175,212,160,45,52,181,194,166)(28,167,195,182,53,46,161,213,176,71,40,155,207,170,65,34,149,201,188,59)(29,60,189,202,150,35,66,171,208,156,41,72,177,214,162,47,54,183,196,168)(30,145,197,184,55,48,163,215,178,49,42,157,209,172,67,36,151,203,190,61), (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), (1,25,13,37)(2,48,14,36)(3,47,15,35)(4,46,16,34)(5,45,17,33)(6,44,18,32)(7,43,19,31)(8,42,20,30)(9,41,21,29)(10,40,22,28)(11,39,23,27)(12,38,24,26)(49,218,61,230)(50,217,62,229)(51,240,63,228)(52,239,64,227)(53,238,65,226)(54,237,66,225)(55,236,67,224)(56,235,68,223)(57,234,69,222)(58,233,70,221)(59,232,71,220)(60,231,72,219)(73,152,85,164)(74,151,86,163)(75,150,87,162)(76,149,88,161)(77,148,89,160)(78,147,90,159)(79,146,91,158)(80,145,92,157)(81,168,93,156)(82,167,94,155)(83,166,95,154)(84,165,96,153)(97,200,109,212)(98,199,110,211)(99,198,111,210)(100,197,112,209)(101,196,113,208)(102,195,114,207)(103,194,115,206)(104,193,116,205)(105,216,117,204)(106,215,118,203)(107,214,119,202)(108,213,120,201)(121,175,133,187)(122,174,134,186)(123,173,135,185)(124,172,136,184)(125,171,137,183)(126,170,138,182)(127,169,139,181)(128,192,140,180)(129,191,141,179)(130,190,142,178)(131,189,143,177)(132,188,144,176) );
G=PermutationGroup([[(1,73,105,129,229,19,91,99,123,223,13,85,117,141,217,7,79,111,135,235),(2,236,136,112,80,8,218,142,118,86,14,224,124,100,92,20,230,130,106,74),(3,75,107,131,231,21,93,101,125,225,15,87,119,143,219,9,81,113,137,237),(4,238,138,114,82,10,220,144,120,88,16,226,126,102,94,22,232,132,108,76),(5,77,109,133,233,23,95,103,127,227,17,89,97,121,221,11,83,115,139,239),(6,240,140,116,84,12,222,122,98,90,18,228,128,104,96,24,234,134,110,78),(25,56,185,198,146,31,62,191,204,152,37,68,173,210,158,43,50,179,216,164),(26,165,193,180,51,44,159,211,174,69,38,153,205,192,63,32,147,199,186,57),(27,58,187,200,148,33,64,169,206,154,39,70,175,212,160,45,52,181,194,166),(28,167,195,182,53,46,161,213,176,71,40,155,207,170,65,34,149,201,188,59),(29,60,189,202,150,35,66,171,208,156,41,72,177,214,162,47,54,183,196,168),(30,145,197,184,55,48,163,215,178,49,42,157,209,172,67,36,151,203,190,61)], [(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)], [(1,25,13,37),(2,48,14,36),(3,47,15,35),(4,46,16,34),(5,45,17,33),(6,44,18,32),(7,43,19,31),(8,42,20,30),(9,41,21,29),(10,40,22,28),(11,39,23,27),(12,38,24,26),(49,218,61,230),(50,217,62,229),(51,240,63,228),(52,239,64,227),(53,238,65,226),(54,237,66,225),(55,236,67,224),(56,235,68,223),(57,234,69,222),(58,233,70,221),(59,232,71,220),(60,231,72,219),(73,152,85,164),(74,151,86,163),(75,150,87,162),(76,149,88,161),(77,148,89,160),(78,147,90,159),(79,146,91,158),(80,145,92,157),(81,168,93,156),(82,167,94,155),(83,166,95,154),(84,165,96,153),(97,200,109,212),(98,199,110,211),(99,198,111,210),(100,197,112,209),(101,196,113,208),(102,195,114,207),(103,194,115,206),(104,193,116,205),(105,216,117,204),(106,215,118,203),(107,214,119,202),(108,213,120,201),(121,175,133,187),(122,174,134,186),(123,173,135,185),(124,172,136,184),(125,171,137,183),(126,170,138,182),(127,169,139,181),(128,192,140,180),(129,191,141,179),(130,190,142,178),(131,189,143,177),(132,188,144,176)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 8A | 8B | 10A | ··· | 10F | 12A | 12B | 12C | 15A | 15B | 20A | 20B | 20C | 20D | 20E | ··· | 20L | 24A | 24B | 24C | 24D | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 8 | 8 | 10 | ··· | 10 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | ··· | 20 | 24 | 24 | 24 | 24 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 60 | 2 | 2 | 2 | 12 | 12 | 60 | 2 | 2 | 2 | 4 | 20 | 20 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 20 | 20 | 20 | 20 | 4 | ··· | 4 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 | D10 | D10 | D12 | D12 | C5⋊D4 | C5⋊D4 | C8.C22 | S3×D5 | C8.D6 | C5⋊D12 | C2×S3×D5 | C5⋊D12 | C20.C23 | C20.D12 |
| kernel | C20.D12 | Dic6⋊D5 | C5⋊Dic12 | C3×C4.Dic5 | C10×Dic6 | D60⋊11C2 | C4.Dic5 | C60 | C2×C30 | C2×Dic6 | C5⋊2C8 | C2×C20 | Dic6 | 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 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C20.D12 ►in GL8(𝔽241)
| 190 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 191 | 240 | 0 | 0 | 0 | 0 | 0 | 0 |
| 240 | 0 | 189 | 240 | 0 | 0 | 0 | 0 |
| 51 | 1 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 36 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 205 | 205 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 87 | 174 |
| 0 | 0 | 0 | 0 | 0 | 0 | 154 | 154 |
| 93 | 107 | 15 | 227 | 0 | 0 | 0 | 0 |
| 116 | 134 | 185 | 212 | 0 | 0 | 0 | 0 |
| 142 | 215 | 40 | 157 | 0 | 0 | 0 | 0 |
| 4 | 175 | 231 | 215 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 239 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 147 | 30 | 0 | 0 | 0 | 0 | 0 | 0 |
| 67 | 94 | 0 | 0 | 0 | 0 | 0 | 0 |
| 130 | 42 | 211 | 63 | 0 | 0 | 0 | 0 |
| 150 | 102 | 177 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 62 |
| 0 | 0 | 0 | 0 | 0 | 0 | 22 | 232 |
| 0 | 0 | 0 | 0 | 9 | 62 | 0 | 0 |
| 0 | 0 | 0 | 0 | 22 | 232 | 0 | 0 |
G:=sub<GL(8,GF(241))| [190,191,240,51,0,0,0,0,240,240,0,1,0,0,0,0,0,0,189,1,0,0,0,0,0,0,240,0,0,0,0,0,0,0,0,0,36,205,0,0,0,0,0,0,72,205,0,0,0,0,0,0,0,0,87,154,0,0,0,0,0,0,174,154],[93,116,142,4,0,0,0,0,107,134,215,175,0,0,0,0,15,185,40,231,0,0,0,0,227,212,157,215,0,0,0,0,0,0,0,0,0,0,240,1,0,0,0,0,0,0,239,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[147,67,130,150,0,0,0,0,30,94,42,102,0,0,0,0,0,0,211,177,0,0,0,0,0,0,63,30,0,0,0,0,0,0,0,0,0,0,9,22,0,0,0,0,0,0,62,232,0,0,0,0,9,22,0,0,0,0,0,0,62,232,0,0] >;
C20.D12 in GAP, Magma, Sage, TeX
C_{20}.D_{12} % in TeX
G:=Group("C20.D12"); // GroupNames label
G:=SmallGroup(480,397);
// by ID
G=gap.SmallGroup(480,397);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,120,422,346,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^20=1,b^12=c^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^9,c*b*c^-1=a^10*b^11>;
// generators/relations