direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C20×D12, C60⋊27D4, C3⋊1(D4×C20), C4⋊2(S3×C20), C12⋊5(C5×D4), C15⋊33(C4×D4), C20⋊16(C4×S3), (C4×C12)⋊7C10, (C4×C20)⋊13S3, C12⋊4(C2×C20), (C4×C60)⋊19C2, C60⋊35(C2×C4), D6⋊1(C2×C20), C42⋊5(C5×S3), C6.2(D4×C10), D6⋊C4⋊17C10, C2.1(C10×D12), C4⋊Dic3⋊16C10, (C2×C20).428D6, C30.289(C2×D4), C10.73(C2×D12), C6.4(C22×C20), (C2×D12).10C10, (C10×D12).20C2, C30.200(C4○D4), (C2×C30).393C23, C30.195(C22×C4), (C2×C60).452C22, C10.111(C4○D12), (C10×Dic3).213C22, (S3×C2×C4)⋊7C10, C2.6(S3×C2×C20), (S3×C2×C20)⋊23C2, C6.4(C5×C4○D4), (C5×D6⋊C4)⋊39C2, C10.131(S3×C2×C4), C2.3(C5×C4○D12), (S3×C10)⋊23(C2×C4), (C2×C4).77(S3×C10), (C5×C4⋊Dic3)⋊34C2, C22.11(S3×C2×C10), (C2×C12).72(C2×C10), (S3×C2×C10).106C22, (C2×C6).14(C22×C10), (C22×S3).15(C2×C10), (C2×C10).327(C22×S3), (C2×Dic3).19(C2×C10), SmallGroup(480,752)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20×D12
G = < a,b,c | a20=b12=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 436 in 188 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C2×C4, C2×C4, D4, C23, C10, C10, Dic3, C12, C12, D6, D6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C20, C20, C2×C10, C2×C10, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C5×S3, C30, C4×D4, C2×C20, C2×C20, C5×D4, C22×C10, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C5×Dic3, C60, C60, S3×C10, S3×C10, C2×C30, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C22×C20, D4×C10, C4×D12, S3×C20, C5×D12, C10×Dic3, C2×C60, S3×C2×C10, D4×C20, C5×C4⋊Dic3, C5×D6⋊C4, C4×C60, S3×C2×C20, C10×D12, C20×D12
Quotients: C1, C2, C4, C22, C5, S3, C2×C4, D4, C23, C10, D6, C22×C4, C2×D4, C4○D4, C20, C2×C10, C4×S3, D12, C22×S3, C5×S3, C4×D4, C2×C20, C5×D4, C22×C10, S3×C2×C4, C2×D12, C4○D12, S3×C10, C22×C20, D4×C10, C5×C4○D4, C4×D12, S3×C20, C5×D12, S3×C2×C10, D4×C20, S3×C2×C20, C10×D12, C5×C4○D12, C20×D12
►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 53 215 115 95 235 32 141 197 129 62 163)(2 54 216 116 96 236 33 142 198 130 63 164)(3 55 217 117 97 237 34 143 199 131 64 165)(4 56 218 118 98 238 35 144 200 132 65 166)(5 57 219 119 99 239 36 145 181 133 66 167)(6 58 220 120 100 240 37 146 182 134 67 168)(7 59 201 101 81 221 38 147 183 135 68 169)(8 60 202 102 82 222 39 148 184 136 69 170)(9 41 203 103 83 223 40 149 185 137 70 171)(10 42 204 104 84 224 21 150 186 138 71 172)(11 43 205 105 85 225 22 151 187 139 72 173)(12 44 206 106 86 226 23 152 188 140 73 174)(13 45 207 107 87 227 24 153 189 121 74 175)(14 46 208 108 88 228 25 154 190 122 75 176)(15 47 209 109 89 229 26 155 191 123 76 177)(16 48 210 110 90 230 27 156 192 124 77 178)(17 49 211 111 91 231 28 157 193 125 78 179)(18 50 212 112 92 232 29 158 194 126 79 180)(19 51 213 113 93 233 30 159 195 127 80 161)(20 52 214 114 94 234 31 160 196 128 61 162)
(1 215)(2 216)(3 217)(4 218)(5 219)(6 220)(7 201)(8 202)(9 203)(10 204)(11 205)(12 206)(13 207)(14 208)(15 209)(16 210)(17 211)(18 212)(19 213)(20 214)(21 186)(22 187)(23 188)(24 189)(25 190)(26 191)(27 192)(28 193)(29 194)(30 195)(31 196)(32 197)(33 198)(34 199)(35 200)(36 181)(37 182)(38 183)(39 184)(40 185)(61 94)(62 95)(63 96)(64 97)(65 98)(66 99)(67 100)(68 81)(69 82)(70 83)(71 84)(72 85)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(101 169)(102 170)(103 171)(104 172)(105 173)(106 174)(107 175)(108 176)(109 177)(110 178)(111 179)(112 180)(113 161)(114 162)(115 163)(116 164)(117 165)(118 166)(119 167)(120 168)(121 227)(122 228)(123 229)(124 230)(125 231)(126 232)(127 233)(128 234)(129 235)(130 236)(131 237)(132 238)(133 239)(134 240)(135 221)(136 222)(137 223)(138 224)(139 225)(140 226)
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), (1,53,215,115,95,235,32,141,197,129,62,163)(2,54,216,116,96,236,33,142,198,130,63,164)(3,55,217,117,97,237,34,143,199,131,64,165)(4,56,218,118,98,238,35,144,200,132,65,166)(5,57,219,119,99,239,36,145,181,133,66,167)(6,58,220,120,100,240,37,146,182,134,67,168)(7,59,201,101,81,221,38,147,183,135,68,169)(8,60,202,102,82,222,39,148,184,136,69,170)(9,41,203,103,83,223,40,149,185,137,70,171)(10,42,204,104,84,224,21,150,186,138,71,172)(11,43,205,105,85,225,22,151,187,139,72,173)(12,44,206,106,86,226,23,152,188,140,73,174)(13,45,207,107,87,227,24,153,189,121,74,175)(14,46,208,108,88,228,25,154,190,122,75,176)(15,47,209,109,89,229,26,155,191,123,76,177)(16,48,210,110,90,230,27,156,192,124,77,178)(17,49,211,111,91,231,28,157,193,125,78,179)(18,50,212,112,92,232,29,158,194,126,79,180)(19,51,213,113,93,233,30,159,195,127,80,161)(20,52,214,114,94,234,31,160,196,128,61,162), (1,215)(2,216)(3,217)(4,218)(5,219)(6,220)(7,201)(8,202)(9,203)(10,204)(11,205)(12,206)(13,207)(14,208)(15,209)(16,210)(17,211)(18,212)(19,213)(20,214)(21,186)(22,187)(23,188)(24,189)(25,190)(26,191)(27,192)(28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,181)(37,182)(38,183)(39,184)(40,185)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,227)(122,228)(123,229)(124,230)(125,231)(126,232)(127,233)(128,234)(129,235)(130,236)(131,237)(132,238)(133,239)(134,240)(135,221)(136,222)(137,223)(138,224)(139,225)(140,226)>;
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), (1,53,215,115,95,235,32,141,197,129,62,163)(2,54,216,116,96,236,33,142,198,130,63,164)(3,55,217,117,97,237,34,143,199,131,64,165)(4,56,218,118,98,238,35,144,200,132,65,166)(5,57,219,119,99,239,36,145,181,133,66,167)(6,58,220,120,100,240,37,146,182,134,67,168)(7,59,201,101,81,221,38,147,183,135,68,169)(8,60,202,102,82,222,39,148,184,136,69,170)(9,41,203,103,83,223,40,149,185,137,70,171)(10,42,204,104,84,224,21,150,186,138,71,172)(11,43,205,105,85,225,22,151,187,139,72,173)(12,44,206,106,86,226,23,152,188,140,73,174)(13,45,207,107,87,227,24,153,189,121,74,175)(14,46,208,108,88,228,25,154,190,122,75,176)(15,47,209,109,89,229,26,155,191,123,76,177)(16,48,210,110,90,230,27,156,192,124,77,178)(17,49,211,111,91,231,28,157,193,125,78,179)(18,50,212,112,92,232,29,158,194,126,79,180)(19,51,213,113,93,233,30,159,195,127,80,161)(20,52,214,114,94,234,31,160,196,128,61,162), (1,215)(2,216)(3,217)(4,218)(5,219)(6,220)(7,201)(8,202)(9,203)(10,204)(11,205)(12,206)(13,207)(14,208)(15,209)(16,210)(17,211)(18,212)(19,213)(20,214)(21,186)(22,187)(23,188)(24,189)(25,190)(26,191)(27,192)(28,193)(29,194)(30,195)(31,196)(32,197)(33,198)(34,199)(35,200)(36,181)(37,182)(38,183)(39,184)(40,185)(61,94)(62,95)(63,96)(64,97)(65,98)(66,99)(67,100)(68,81)(69,82)(70,83)(71,84)(72,85)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(101,169)(102,170)(103,171)(104,172)(105,173)(106,174)(107,175)(108,176)(109,177)(110,178)(111,179)(112,180)(113,161)(114,162)(115,163)(116,164)(117,165)(118,166)(119,167)(120,168)(121,227)(122,228)(123,229)(124,230)(125,231)(126,232)(127,233)(128,234)(129,235)(130,236)(131,237)(132,238)(133,239)(134,240)(135,221)(136,222)(137,223)(138,224)(139,225)(140,226) );
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)], [(1,53,215,115,95,235,32,141,197,129,62,163),(2,54,216,116,96,236,33,142,198,130,63,164),(3,55,217,117,97,237,34,143,199,131,64,165),(4,56,218,118,98,238,35,144,200,132,65,166),(5,57,219,119,99,239,36,145,181,133,66,167),(6,58,220,120,100,240,37,146,182,134,67,168),(7,59,201,101,81,221,38,147,183,135,68,169),(8,60,202,102,82,222,39,148,184,136,69,170),(9,41,203,103,83,223,40,149,185,137,70,171),(10,42,204,104,84,224,21,150,186,138,71,172),(11,43,205,105,85,225,22,151,187,139,72,173),(12,44,206,106,86,226,23,152,188,140,73,174),(13,45,207,107,87,227,24,153,189,121,74,175),(14,46,208,108,88,228,25,154,190,122,75,176),(15,47,209,109,89,229,26,155,191,123,76,177),(16,48,210,110,90,230,27,156,192,124,77,178),(17,49,211,111,91,231,28,157,193,125,78,179),(18,50,212,112,92,232,29,158,194,126,79,180),(19,51,213,113,93,233,30,159,195,127,80,161),(20,52,214,114,94,234,31,160,196,128,61,162)], [(1,215),(2,216),(3,217),(4,218),(5,219),(6,220),(7,201),(8,202),(9,203),(10,204),(11,205),(12,206),(13,207),(14,208),(15,209),(16,210),(17,211),(18,212),(19,213),(20,214),(21,186),(22,187),(23,188),(24,189),(25,190),(26,191),(27,192),(28,193),(29,194),(30,195),(31,196),(32,197),(33,198),(34,199),(35,200),(36,181),(37,182),(38,183),(39,184),(40,185),(61,94),(62,95),(63,96),(64,97),(65,98),(66,99),(67,100),(68,81),(69,82),(70,83),(71,84),(72,85),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(101,169),(102,170),(103,171),(104,172),(105,173),(106,174),(107,175),(108,176),(109,177),(110,178),(111,179),(112,180),(113,161),(114,162),(115,163),(116,164),(117,165),(118,166),(119,167),(120,168),(121,227),(122,228),(123,229),(124,230),(125,231),(126,232),(127,233),(128,234),(129,235),(130,236),(131,237),(132,238),(133,239),(134,240),(135,221),(136,222),(137,223),(138,224),(139,225),(140,226)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 5C | 5D | 6A | 6B | 6C | 10A | ··· | 10L | 10M | ··· | 10AB | 12A | ··· | 12L | 15A | 15B | 15C | 15D | 20A | ··· | 20P | 20Q | ··· | 20AF | 20AG | ··· | 20AV | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | 5 | 6 | 6 | 6 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C5 | C10 | C10 | C10 | C10 | C10 | C20 | S3 | D4 | D6 | C4○D4 | C4×S3 | D12 | C5×S3 | C5×D4 | C4○D12 | S3×C10 | C5×C4○D4 | S3×C20 | C5×D12 | C5×C4○D12 |
| kernel | C20×D12 | C5×C4⋊Dic3 | C5×D6⋊C4 | C4×C60 | S3×C2×C20 | C10×D12 | C5×D12 | C4×D12 | C4⋊Dic3 | D6⋊C4 | C4×C12 | S3×C2×C4 | C2×D12 | D12 | C4×C20 | C60 | C2×C20 | C30 | C20 | C20 | C42 | C12 | C10 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 8 | 4 | 32 | 1 | 2 | 3 | 2 | 4 | 4 | 4 | 8 | 4 | 12 | 8 | 16 | 16 | 16 |
Matrix representation of C20×D12 ►in GL5(𝔽61)
| 50 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 0 | 52 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 59 | 0 | 0 |
| 0 | 1 | 60 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 1 | 60 |
| 60 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 |
| 0 | 60 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 60 |
| 0 | 0 | 0 | 0 | 60 |
G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,3,0,0,0,0,0,3,0,0,0,0,0,52,0,0,0,0,0,52],[1,0,0,0,0,0,1,1,0,0,0,59,60,0,0,0,0,0,0,1,0,0,0,60,60],[60,0,0,0,0,0,60,60,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,60,60] >;
C20×D12 in GAP, Magma, Sage, TeX
C_{20}\times D_{12} % in TeX
G:=Group("C20xD12"); // GroupNames label
G:=SmallGroup(480,752);
// by ID
G=gap.SmallGroup(480,752);
# by ID
G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-3,1149,568,226,15686]);
// Polycyclic
G:=Group<a,b,c|a^20=b^12=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations