metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60⋊9D4, C20⋊1D12, Dic5⋊1D12, (C2×D12)⋊4D5, C6.24(D4×D5), (C2×D60)⋊27C2, (C10×D12)⋊6C2, C12⋊7(C5⋊D4), C5⋊2(C4⋊D12), C4⋊1(C5⋊D12), C3⋊1(C20⋊D4), C15⋊1(C4⋊1D4), (C3×Dic5)⋊9D4, (C4×Dic5)⋊7S3, C30.61(C2×D4), C2.26(D5×D12), (C12×Dic5)⋊7C2, (C2×C20).131D6, C10.25(C2×D12), (C2×C12).307D10, (C2×C30).141C23, (C2×C60).151C22, (C2×Dic5).185D6, (C22×S3).17D10, (C6×Dic5).212C22, (C22×D15).48C22, (C2×C5⋊D12)⋊4C2, C6.16(C2×C5⋊D4), (C2×C4).164(S3×D5), C2.19(C2×C5⋊D12), C22.193(C2×S3×D5), (S3×C2×C10).32C22, (C2×C6).153(C22×D5), (C2×C10).153(C22×S3), SmallGroup(480,527)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C20⋊D12
G = < a,b,c | a20=b12=c2=1, bab-1=a9, cac=a-1, cbc=b-1 >
Subgroups: 1484 in 216 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C10, C10, C12, C12, D6, C2×C6, C15, C42, C2×D4, Dic5, C20, D10, C2×C10, C2×C10, D12, C2×C12, C2×C12, C22×S3, C22×S3, C5×S3, D15, C30, C30, C4⋊1D4, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×C10, C4×C12, C2×D12, C2×D12, C3×Dic5, C60, S3×C10, D30, C2×C30, C4×Dic5, C2×D20, C2×C5⋊D4, D4×C10, C4⋊D12, C5⋊D12, C6×Dic5, C5×D12, D60, C2×C60, S3×C2×C10, C22×D15, C20⋊D4, C12×Dic5, C2×C5⋊D12, C10×D12, C2×D60, C20⋊D12
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, D12, C22×S3, C4⋊1D4, C5⋊D4, C22×D5, C2×D12, S3×D5, D4×D5, C2×C5⋊D4, C4⋊D12, C5⋊D12, C2×S3×D5, C20⋊D4, D5×D12, C2×C5⋊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)
(1 182 114 78 174 134 229 210 156 23 93 60)(2 191 115 67 175 123 230 219 157 32 94 49)(3 200 116 76 176 132 231 208 158 21 95 58)(4 189 117 65 177 121 232 217 159 30 96 47)(5 198 118 74 178 130 233 206 160 39 97 56)(6 187 119 63 179 139 234 215 141 28 98 45)(7 196 120 72 180 128 235 204 142 37 99 54)(8 185 101 61 161 137 236 213 143 26 100 43)(9 194 102 70 162 126 237 202 144 35 81 52)(10 183 103 79 163 135 238 211 145 24 82 41)(11 192 104 68 164 124 239 220 146 33 83 50)(12 181 105 77 165 133 240 209 147 22 84 59)(13 190 106 66 166 122 221 218 148 31 85 48)(14 199 107 75 167 131 222 207 149 40 86 57)(15 188 108 64 168 140 223 216 150 29 87 46)(16 197 109 73 169 129 224 205 151 38 88 55)(17 186 110 62 170 138 225 214 152 27 89 44)(18 195 111 71 171 127 226 203 153 36 90 53)(19 184 112 80 172 136 227 212 154 25 91 42)(20 193 113 69 173 125 228 201 155 34 92 51)
(1 151)(2 150)(3 149)(4 148)(5 147)(6 146)(7 145)(8 144)(9 143)(10 142)(11 141)(12 160)(13 159)(14 158)(15 157)(16 156)(17 155)(18 154)(19 153)(20 152)(21 57)(22 56)(23 55)(24 54)(25 53)(26 52)(27 51)(28 50)(29 49)(30 48)(31 47)(32 46)(33 45)(34 44)(35 43)(36 42)(37 41)(38 60)(39 59)(40 58)(61 126)(62 125)(63 124)(64 123)(65 122)(66 121)(67 140)(68 139)(69 138)(70 137)(71 136)(72 135)(73 134)(74 133)(75 132)(76 131)(77 130)(78 129)(79 128)(80 127)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 237)(102 236)(103 235)(104 234)(105 233)(106 232)(107 231)(108 230)(109 229)(110 228)(111 227)(112 226)(113 225)(114 224)(115 223)(116 222)(117 221)(118 240)(119 239)(120 238)(161 162)(163 180)(164 179)(165 178)(166 177)(167 176)(168 175)(169 174)(170 173)(171 172)(181 206)(182 205)(183 204)(184 203)(185 202)(186 201)(187 220)(188 219)(189 218)(190 217)(191 216)(192 215)(193 214)(194 213)(195 212)(196 211)(197 210)(198 209)(199 208)(200 207)
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), (1,182,114,78,174,134,229,210,156,23,93,60)(2,191,115,67,175,123,230,219,157,32,94,49)(3,200,116,76,176,132,231,208,158,21,95,58)(4,189,117,65,177,121,232,217,159,30,96,47)(5,198,118,74,178,130,233,206,160,39,97,56)(6,187,119,63,179,139,234,215,141,28,98,45)(7,196,120,72,180,128,235,204,142,37,99,54)(8,185,101,61,161,137,236,213,143,26,100,43)(9,194,102,70,162,126,237,202,144,35,81,52)(10,183,103,79,163,135,238,211,145,24,82,41)(11,192,104,68,164,124,239,220,146,33,83,50)(12,181,105,77,165,133,240,209,147,22,84,59)(13,190,106,66,166,122,221,218,148,31,85,48)(14,199,107,75,167,131,222,207,149,40,86,57)(15,188,108,64,168,140,223,216,150,29,87,46)(16,197,109,73,169,129,224,205,151,38,88,55)(17,186,110,62,170,138,225,214,152,27,89,44)(18,195,111,71,171,127,226,203,153,36,90,53)(19,184,112,80,172,136,227,212,154,25,91,42)(20,193,113,69,173,125,228,201,155,34,92,51), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,60)(39,59)(40,58)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,237)(102,236)(103,235)(104,234)(105,233)(106,232)(107,231)(108,230)(109,229)(110,228)(111,227)(112,226)(113,225)(114,224)(115,223)(116,222)(117,221)(118,240)(119,239)(120,238)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,206)(182,205)(183,204)(184,203)(185,202)(186,201)(187,220)(188,219)(189,218)(190,217)(191,216)(192,215)(193,214)(194,213)(195,212)(196,211)(197,210)(198,209)(199,208)(200,207)>;
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), (1,182,114,78,174,134,229,210,156,23,93,60)(2,191,115,67,175,123,230,219,157,32,94,49)(3,200,116,76,176,132,231,208,158,21,95,58)(4,189,117,65,177,121,232,217,159,30,96,47)(5,198,118,74,178,130,233,206,160,39,97,56)(6,187,119,63,179,139,234,215,141,28,98,45)(7,196,120,72,180,128,235,204,142,37,99,54)(8,185,101,61,161,137,236,213,143,26,100,43)(9,194,102,70,162,126,237,202,144,35,81,52)(10,183,103,79,163,135,238,211,145,24,82,41)(11,192,104,68,164,124,239,220,146,33,83,50)(12,181,105,77,165,133,240,209,147,22,84,59)(13,190,106,66,166,122,221,218,148,31,85,48)(14,199,107,75,167,131,222,207,149,40,86,57)(15,188,108,64,168,140,223,216,150,29,87,46)(16,197,109,73,169,129,224,205,151,38,88,55)(17,186,110,62,170,138,225,214,152,27,89,44)(18,195,111,71,171,127,226,203,153,36,90,53)(19,184,112,80,172,136,227,212,154,25,91,42)(20,193,113,69,173,125,228,201,155,34,92,51), (1,151)(2,150)(3,149)(4,148)(5,147)(6,146)(7,145)(8,144)(9,143)(10,142)(11,141)(12,160)(13,159)(14,158)(15,157)(16,156)(17,155)(18,154)(19,153)(20,152)(21,57)(22,56)(23,55)(24,54)(25,53)(26,52)(27,51)(28,50)(29,49)(30,48)(31,47)(32,46)(33,45)(34,44)(35,43)(36,42)(37,41)(38,60)(39,59)(40,58)(61,126)(62,125)(63,124)(64,123)(65,122)(66,121)(67,140)(68,139)(69,138)(70,137)(71,136)(72,135)(73,134)(74,133)(75,132)(76,131)(77,130)(78,129)(79,128)(80,127)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,237)(102,236)(103,235)(104,234)(105,233)(106,232)(107,231)(108,230)(109,229)(110,228)(111,227)(112,226)(113,225)(114,224)(115,223)(116,222)(117,221)(118,240)(119,239)(120,238)(161,162)(163,180)(164,179)(165,178)(166,177)(167,176)(168,175)(169,174)(170,173)(171,172)(181,206)(182,205)(183,204)(184,203)(185,202)(186,201)(187,220)(188,219)(189,218)(190,217)(191,216)(192,215)(193,214)(194,213)(195,212)(196,211)(197,210)(198,209)(199,208)(200,207) );
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)], [(1,182,114,78,174,134,229,210,156,23,93,60),(2,191,115,67,175,123,230,219,157,32,94,49),(3,200,116,76,176,132,231,208,158,21,95,58),(4,189,117,65,177,121,232,217,159,30,96,47),(5,198,118,74,178,130,233,206,160,39,97,56),(6,187,119,63,179,139,234,215,141,28,98,45),(7,196,120,72,180,128,235,204,142,37,99,54),(8,185,101,61,161,137,236,213,143,26,100,43),(9,194,102,70,162,126,237,202,144,35,81,52),(10,183,103,79,163,135,238,211,145,24,82,41),(11,192,104,68,164,124,239,220,146,33,83,50),(12,181,105,77,165,133,240,209,147,22,84,59),(13,190,106,66,166,122,221,218,148,31,85,48),(14,199,107,75,167,131,222,207,149,40,86,57),(15,188,108,64,168,140,223,216,150,29,87,46),(16,197,109,73,169,129,224,205,151,38,88,55),(17,186,110,62,170,138,225,214,152,27,89,44),(18,195,111,71,171,127,226,203,153,36,90,53),(19,184,112,80,172,136,227,212,154,25,91,42),(20,193,113,69,173,125,228,201,155,34,92,51)], [(1,151),(2,150),(3,149),(4,148),(5,147),(6,146),(7,145),(8,144),(9,143),(10,142),(11,141),(12,160),(13,159),(14,158),(15,157),(16,156),(17,155),(18,154),(19,153),(20,152),(21,57),(22,56),(23,55),(24,54),(25,53),(26,52),(27,51),(28,50),(29,49),(30,48),(31,47),(32,46),(33,45),(34,44),(35,43),(36,42),(37,41),(38,60),(39,59),(40,58),(61,126),(62,125),(63,124),(64,123),(65,122),(66,121),(67,140),(68,139),(69,138),(70,137),(71,136),(72,135),(73,134),(74,133),(75,132),(76,131),(77,130),(78,129),(79,128),(80,127),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,237),(102,236),(103,235),(104,234),(105,233),(106,232),(107,231),(108,230),(109,229),(110,228),(111,227),(112,226),(113,225),(114,224),(115,223),(116,222),(117,221),(118,240),(119,239),(120,238),(161,162),(163,180),(164,179),(165,178),(166,177),(167,176),(168,175),(169,174),(170,173),(171,172),(181,206),(182,205),(183,204),(184,203),(185,202),(186,201),(187,220),(188,219),(189,218),(190,217),(191,216),(192,215),(193,214),(194,213),(195,212),(196,211),(197,210),(198,209),(199,208),(200,207)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 6A | 6B | 6C | 10A | ··· | 10F | 10G | ··· | 10N | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 15A | 15B | 20A | 20B | 20C | 20D | 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 | 10 | ··· | 10 | 10 | ··· | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 60 | 60 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
66 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 | D12 | D12 | C5⋊D4 | S3×D5 | D4×D5 | C5⋊D12 | C2×S3×D5 | D5×D12 |
| kernel | C20⋊D12 | C12×Dic5 | C2×C5⋊D12 | C10×D12 | C2×D60 | C4×Dic5 | C3×Dic5 | C60 | C2×D12 | C2×Dic5 | C2×C20 | C2×C12 | C22×S3 | Dic5 | C20 | C12 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 1 | 2 | 4 | 8 | 4 | 8 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of C20⋊D12 ►in GL4(𝔽61) generated by
| 38 | 15 | 0 | 0 |
| 46 | 23 | 0 | 0 |
| 0 | 0 | 17 | 1 |
| 0 | 0 | 16 | 1 |
| 15 | 38 | 0 | 0 |
| 23 | 38 | 0 | 0 |
| 0 | 0 | 39 | 30 |
| 0 | 0 | 55 | 22 |
| 46 | 23 | 0 | 0 |
| 38 | 15 | 0 | 0 |
| 0 | 0 | 44 | 60 |
| 0 | 0 | 44 | 17 |
G:=sub<GL(4,GF(61))| [38,46,0,0,15,23,0,0,0,0,17,16,0,0,1,1],[15,23,0,0,38,38,0,0,0,0,39,55,0,0,30,22],[46,38,0,0,23,15,0,0,0,0,44,44,0,0,60,17] >;
C20⋊D12 in GAP, Magma, Sage, TeX
C_{20}\rtimes D_{12} % in TeX
G:=Group("C20:D12"); // GroupNames label
G:=SmallGroup(480,527);
// by ID
G=gap.SmallGroup(480,527);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,56,141,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^20=b^12=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations