metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊5D4, C12⋊1D20, Dic3⋊1D20, (C6×D20)⋊6C2, (C2×D20)⋊5S3, (C2×D60)⋊21C2, C5⋊1(C12⋊3D4), C20⋊7(C3⋊D4), C3⋊2(C20⋊4D4), C4⋊1(C3⋊D20), C15⋊2(C4⋊1D4), (C4×Dic3)⋊7D5, (C5×Dic3)⋊9D4, C2.24(S3×D20), C6.62(C2×D20), C10.22(S3×D4), (Dic3×C20)⋊7C2, C30.152(C2×D4), (C2×C20).302D6, (C2×C12).131D10, (C22×D5).19D6, (C2×C30).148C23, (C2×C60).121C22, (C2×Dic3).157D10, (C22×D15).51C22, (C10×Dic3).191C22, (C2×C3⋊D20)⋊5C2, (C2×C4).112(S3×D5), C10.17(C2×C3⋊D4), C2.20(C2×C3⋊D20), (D5×C2×C6).33C22, C22.200(C2×S3×D5), (C2×C6).160(C22×D5), (C2×C10).160(C22×S3), SmallGroup(480,534)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊D20
G = < a,b,c | a12=b20=c2=1, bab-1=a5, cac=a-1, cbc=b-1 >
Subgroups: 1564 in 216 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C42, C2×D4, C20, C20, D10, C2×C10, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C3×D5, D15, C30, C30, C4⋊1D4, D20, C2×C20, C2×C20, C22×D5, C22×D5, C4×Dic3, C2×D12, C2×C3⋊D4, C6×D4, C5×Dic3, C60, C6×D5, D30, C2×C30, C4×C20, C2×D20, C2×D20, C12⋊3D4, C3⋊D20, C3×D20, C10×Dic3, D60, C2×C60, D5×C2×C6, C22×D15, C20⋊4D4, Dic3×C20, C2×C3⋊D20, C6×D20, C2×D60, C12⋊D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C4⋊1D4, D20, C22×D5, S3×D4, C2×C3⋊D4, S3×D5, C2×D20, C12⋊3D4, C3⋊D20, C2×S3×D5, C20⋊4D4, S3×D20, C2×C3⋊D20, C12⋊D20
►On 240 points
Generators in S240
(1 101 142 203 23 89 67 45 132 196 237 164)(2 90 238 204 133 102 68 165 24 197 143 46)(3 103 144 205 25 91 69 47 134 198 239 166)(4 92 240 206 135 104 70 167 26 199 145 48)(5 105 146 207 27 93 71 49 136 200 221 168)(6 94 222 208 137 106 72 169 28 181 147 50)(7 107 148 209 29 95 73 51 138 182 223 170)(8 96 224 210 139 108 74 171 30 183 149 52)(9 109 150 211 31 97 75 53 140 184 225 172)(10 98 226 212 121 110 76 173 32 185 151 54)(11 111 152 213 33 99 77 55 122 186 227 174)(12 100 228 214 123 112 78 175 34 187 153 56)(13 113 154 215 35 81 79 57 124 188 229 176)(14 82 230 216 125 114 80 177 36 189 155 58)(15 115 156 217 37 83 61 59 126 190 231 178)(16 84 232 218 127 116 62 179 38 191 157 60)(17 117 158 219 39 85 63 41 128 192 233 180)(18 86 234 220 129 118 64 161 40 193 159 42)(19 119 160 201 21 87 65 43 130 194 235 162)(20 88 236 202 131 120 66 163 22 195 141 44)
(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)
(1 207)(2 206)(3 205)(4 204)(5 203)(6 202)(7 201)(8 220)(9 219)(10 218)(11 217)(12 216)(13 215)(14 214)(15 213)(16 212)(17 211)(18 210)(19 209)(20 208)(21 170)(22 169)(23 168)(24 167)(25 166)(26 165)(27 164)(28 163)(29 162)(30 161)(31 180)(32 179)(33 178)(34 177)(35 176)(36 175)(37 174)(38 173)(39 172)(40 171)(41 140)(42 139)(43 138)(44 137)(45 136)(46 135)(47 134)(48 133)(49 132)(50 131)(51 130)(52 129)(53 128)(54 127)(55 126)(56 125)(57 124)(58 123)(59 122)(60 121)(61 186)(62 185)(63 184)(64 183)(65 182)(66 181)(67 200)(68 199)(69 198)(70 197)(71 196)(72 195)(73 194)(74 193)(75 192)(76 191)(77 190)(78 189)(79 188)(80 187)(81 229)(82 228)(83 227)(84 226)(85 225)(86 224)(87 223)(88 222)(89 221)(90 240)(91 239)(92 238)(93 237)(94 236)(95 235)(96 234)(97 233)(98 232)(99 231)(100 230)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 160)(108 159)(109 158)(110 157)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)
G:=sub<Sym(240)| (1,101,142,203,23,89,67,45,132,196,237,164)(2,90,238,204,133,102,68,165,24,197,143,46)(3,103,144,205,25,91,69,47,134,198,239,166)(4,92,240,206,135,104,70,167,26,199,145,48)(5,105,146,207,27,93,71,49,136,200,221,168)(6,94,222,208,137,106,72,169,28,181,147,50)(7,107,148,209,29,95,73,51,138,182,223,170)(8,96,224,210,139,108,74,171,30,183,149,52)(9,109,150,211,31,97,75,53,140,184,225,172)(10,98,226,212,121,110,76,173,32,185,151,54)(11,111,152,213,33,99,77,55,122,186,227,174)(12,100,228,214,123,112,78,175,34,187,153,56)(13,113,154,215,35,81,79,57,124,188,229,176)(14,82,230,216,125,114,80,177,36,189,155,58)(15,115,156,217,37,83,61,59,126,190,231,178)(16,84,232,218,127,116,62,179,38,191,157,60)(17,117,158,219,39,85,63,41,128,192,233,180)(18,86,234,220,129,118,64,161,40,193,159,42)(19,119,160,201,21,87,65,43,130,194,235,162)(20,88,236,202,131,120,66,163,22,195,141,44), (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), (1,207)(2,206)(3,205)(4,204)(5,203)(6,202)(7,201)(8,220)(9,219)(10,218)(11,217)(12,216)(13,215)(14,214)(15,213)(16,212)(17,211)(18,210)(19,209)(20,208)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,180)(32,179)(33,178)(34,177)(35,176)(36,175)(37,174)(38,173)(39,172)(40,171)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,186)(62,185)(63,184)(64,183)(65,182)(66,181)(67,200)(68,199)(69,198)(70,197)(71,196)(72,195)(73,194)(74,193)(75,192)(76,191)(77,190)(78,189)(79,188)(80,187)(81,229)(82,228)(83,227)(84,226)(85,225)(86,224)(87,223)(88,222)(89,221)(90,240)(91,239)(92,238)(93,237)(94,236)(95,235)(96,234)(97,233)(98,232)(99,231)(100,230)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147)>;
G:=Group( (1,101,142,203,23,89,67,45,132,196,237,164)(2,90,238,204,133,102,68,165,24,197,143,46)(3,103,144,205,25,91,69,47,134,198,239,166)(4,92,240,206,135,104,70,167,26,199,145,48)(5,105,146,207,27,93,71,49,136,200,221,168)(6,94,222,208,137,106,72,169,28,181,147,50)(7,107,148,209,29,95,73,51,138,182,223,170)(8,96,224,210,139,108,74,171,30,183,149,52)(9,109,150,211,31,97,75,53,140,184,225,172)(10,98,226,212,121,110,76,173,32,185,151,54)(11,111,152,213,33,99,77,55,122,186,227,174)(12,100,228,214,123,112,78,175,34,187,153,56)(13,113,154,215,35,81,79,57,124,188,229,176)(14,82,230,216,125,114,80,177,36,189,155,58)(15,115,156,217,37,83,61,59,126,190,231,178)(16,84,232,218,127,116,62,179,38,191,157,60)(17,117,158,219,39,85,63,41,128,192,233,180)(18,86,234,220,129,118,64,161,40,193,159,42)(19,119,160,201,21,87,65,43,130,194,235,162)(20,88,236,202,131,120,66,163,22,195,141,44), (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), (1,207)(2,206)(3,205)(4,204)(5,203)(6,202)(7,201)(8,220)(9,219)(10,218)(11,217)(12,216)(13,215)(14,214)(15,213)(16,212)(17,211)(18,210)(19,209)(20,208)(21,170)(22,169)(23,168)(24,167)(25,166)(26,165)(27,164)(28,163)(29,162)(30,161)(31,180)(32,179)(33,178)(34,177)(35,176)(36,175)(37,174)(38,173)(39,172)(40,171)(41,140)(42,139)(43,138)(44,137)(45,136)(46,135)(47,134)(48,133)(49,132)(50,131)(51,130)(52,129)(53,128)(54,127)(55,126)(56,125)(57,124)(58,123)(59,122)(60,121)(61,186)(62,185)(63,184)(64,183)(65,182)(66,181)(67,200)(68,199)(69,198)(70,197)(71,196)(72,195)(73,194)(74,193)(75,192)(76,191)(77,190)(78,189)(79,188)(80,187)(81,229)(82,228)(83,227)(84,226)(85,225)(86,224)(87,223)(88,222)(89,221)(90,240)(91,239)(92,238)(93,237)(94,236)(95,235)(96,234)(97,233)(98,232)(99,231)(100,230)(101,146)(102,145)(103,144)(104,143)(105,142)(106,141)(107,160)(108,159)(109,158)(110,157)(111,156)(112,155)(113,154)(114,153)(115,152)(116,151)(117,150)(118,149)(119,148)(120,147) );
G=PermutationGroup([[(1,101,142,203,23,89,67,45,132,196,237,164),(2,90,238,204,133,102,68,165,24,197,143,46),(3,103,144,205,25,91,69,47,134,198,239,166),(4,92,240,206,135,104,70,167,26,199,145,48),(5,105,146,207,27,93,71,49,136,200,221,168),(6,94,222,208,137,106,72,169,28,181,147,50),(7,107,148,209,29,95,73,51,138,182,223,170),(8,96,224,210,139,108,74,171,30,183,149,52),(9,109,150,211,31,97,75,53,140,184,225,172),(10,98,226,212,121,110,76,173,32,185,151,54),(11,111,152,213,33,99,77,55,122,186,227,174),(12,100,228,214,123,112,78,175,34,187,153,56),(13,113,154,215,35,81,79,57,124,188,229,176),(14,82,230,216,125,114,80,177,36,189,155,58),(15,115,156,217,37,83,61,59,126,190,231,178),(16,84,232,218,127,116,62,179,38,191,157,60),(17,117,158,219,39,85,63,41,128,192,233,180),(18,86,234,220,129,118,64,161,40,193,159,42),(19,119,160,201,21,87,65,43,130,194,235,162),(20,88,236,202,131,120,66,163,22,195,141,44)], [(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)], [(1,207),(2,206),(3,205),(4,204),(5,203),(6,202),(7,201),(8,220),(9,219),(10,218),(11,217),(12,216),(13,215),(14,214),(15,213),(16,212),(17,211),(18,210),(19,209),(20,208),(21,170),(22,169),(23,168),(24,167),(25,166),(26,165),(27,164),(28,163),(29,162),(30,161),(31,180),(32,179),(33,178),(34,177),(35,176),(36,175),(37,174),(38,173),(39,172),(40,171),(41,140),(42,139),(43,138),(44,137),(45,136),(46,135),(47,134),(48,133),(49,132),(50,131),(51,130),(52,129),(53,128),(54,127),(55,126),(56,125),(57,124),(58,123),(59,122),(60,121),(61,186),(62,185),(63,184),(64,183),(65,182),(66,181),(67,200),(68,199),(69,198),(70,197),(71,196),(72,195),(73,194),(74,193),(75,192),(76,191),(77,190),(78,189),(79,188),(80,187),(81,229),(82,228),(83,227),(84,226),(85,225),(86,224),(87,223),(88,222),(89,221),(90,240),(91,239),(92,238),(93,237),(94,236),(95,235),(96,234),(97,233),(98,232),(99,231),(100,230),(101,146),(102,145),(103,144),(104,143),(105,142),(106,141),(107,160),(108,159),(109,158),(110,157),(111,156),(112,155),(113,154),(114,153),(115,152),(116,151),(117,150),(118,149),(119,148),(120,147)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 10A | ··· | 10F | 12A | 12B | 15A | 15B | 20A | ··· | 20H | 20I | ··· | 20X | 30A | ··· | 30F | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | ··· | 10 | 12 | 12 | 15 | 15 | 20 | ··· | 20 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 60 | 60 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 4 | ··· | 4 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | D20 | D20 | S3×D4 | S3×D5 | C3⋊D20 | C2×S3×D5 | S3×D20 |
| kernel | C12⋊D20 | Dic3×C20 | C2×C3⋊D20 | C6×D20 | C2×D60 | C2×D20 | C5×Dic3 | C60 | C4×Dic3 | C2×C20 | C22×D5 | C2×Dic3 | C2×C12 | C20 | Dic3 | C12 | C10 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 1 | 2 | 4 | 2 | 4 | 16 | 8 | 2 | 2 | 4 | 2 | 8 |
Matrix representation of C12⋊D20 ►in GL4(𝔽61) generated by
| 32 | 4 | 0 | 0 |
| 3 | 29 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 60 | 60 |
| 1 | 60 | 0 | 0 |
| 45 | 17 | 0 | 0 |
| 0 | 0 | 52 | 43 |
| 0 | 0 | 52 | 9 |
| 29 | 57 | 0 | 0 |
| 27 | 32 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 60 | 60 |
G:=sub<GL(4,GF(61))| [32,3,0,0,4,29,0,0,0,0,0,60,0,0,1,60],[1,45,0,0,60,17,0,0,0,0,52,52,0,0,43,9],[29,27,0,0,57,32,0,0,0,0,1,60,0,0,0,60] >;
C12⋊D20 in GAP, Magma, Sage, TeX
C_{12}\rtimes D_{20} % in TeX
G:=Group("C12:D20"); // GroupNames label
G:=SmallGroup(480,534);
// by ID
G=gap.SmallGroup(480,534);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=b^20=c^2=1,b*a*b^-1=a^5,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations