direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12×D20, C60⋊26D4, C5⋊2(D4×C12), C4⋊2(D5×C12), C20⋊5(C3×D4), C15⋊32(C4×D4), C20⋊7(C2×C12), (C4×C20)⋊13C6, (C4×C60)⋊15C2, C60⋊34(C2×C4), (C4×C12)⋊11D5, C12⋊12(C4×D5), C10.2(C6×D4), C2.1(C6×D20), C42⋊7(C3×D5), D10⋊2(C2×C12), C4⋊Dic5⋊16C6, C6.73(C2×D20), (C2×D20).11C6, (C6×D20).22C2, C30.275(C2×D4), D10⋊C4⋊17C6, (C2×C12).448D10, C30.183(C4○D4), C6.111(C4○D20), C10.17(C22×C12), (C2×C30).331C23, (C2×C60).548C22, C30.175(C22×C4), (C6×Dic5).233C22, (C2×C4×D5)⋊7C6, C2.6(D5×C2×C12), (D5×C2×C12)⋊23C2, C6.100(C2×C4×D5), (C6×D5)⋊22(C2×C4), C10.4(C3×C4○D4), C2.3(C3×C4○D20), (C2×C4).76(C6×D5), C22.11(D5×C2×C6), (C2×C20).75(C2×C6), (C3×C4⋊Dic5)⋊34C2, (C3×D10⋊C4)⋊39C2, (D5×C2×C6).123C22, (C2×C10).14(C22×C6), (C2×Dic5).25(C2×C6), (C22×D5).20(C2×C6), (C2×C6).327(C22×D5), SmallGroup(480,666)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×D20
G = < a,b,c | a12=b20=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 624 in 188 conjugacy classes, 90 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, D10, C2×C10, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, C4×D4, C4×D5, D20, C2×Dic5, C2×C20, C22×D5, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C3×Dic5, C60, C60, C6×D5, C6×D5, C2×C30, C4⋊Dic5, D10⋊C4, C4×C20, C2×C4×D5, C2×D20, D4×C12, D5×C12, C3×D20, C6×Dic5, C2×C60, D5×C2×C6, C4×D20, C3×C4⋊Dic5, C3×D10⋊C4, C4×C60, D5×C2×C12, C6×D20, C12×D20
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, D5, C12, C2×C6, C22×C4, C2×D4, C4○D4, D10, C2×C12, C3×D4, C22×C6, C3×D5, C4×D4, C4×D5, D20, C22×D5, C22×C12, C6×D4, C3×C4○D4, C6×D5, C2×C4×D5, C2×D20, C4○D20, D4×C12, D5×C12, C3×D20, D5×C2×C6, C4×D20, D5×C2×C12, C6×D20, C3×C4○D20, C12×D20
►On 240 points
Generators in S240
(1 38 134 168 203 151 106 54 97 235 193 70)(2 39 135 169 204 152 107 55 98 236 194 71)(3 40 136 170 205 153 108 56 99 237 195 72)(4 21 137 171 206 154 109 57 100 238 196 73)(5 22 138 172 207 155 110 58 81 239 197 74)(6 23 139 173 208 156 111 59 82 240 198 75)(7 24 140 174 209 157 112 60 83 221 199 76)(8 25 121 175 210 158 113 41 84 222 200 77)(9 26 122 176 211 159 114 42 85 223 181 78)(10 27 123 177 212 160 115 43 86 224 182 79)(11 28 124 178 213 141 116 44 87 225 183 80)(12 29 125 179 214 142 117 45 88 226 184 61)(13 30 126 180 215 143 118 46 89 227 185 62)(14 31 127 161 216 144 119 47 90 228 186 63)(15 32 128 162 217 145 120 48 91 229 187 64)(16 33 129 163 218 146 101 49 92 230 188 65)(17 34 130 164 219 147 102 50 93 231 189 66)(18 35 131 165 220 148 103 51 94 232 190 67)(19 36 132 166 201 149 104 52 95 233 191 68)(20 37 133 167 202 150 105 53 96 234 192 69)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 29)(22 28)(23 27)(24 26)(30 40)(31 39)(32 38)(33 37)(34 36)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 73)(62 72)(63 71)(64 70)(65 69)(66 68)(74 80)(75 79)(76 78)(81 87)(82 86)(83 85)(88 100)(89 99)(90 98)(91 97)(92 96)(93 95)(101 105)(102 104)(106 120)(107 119)(108 118)(109 117)(110 116)(111 115)(112 114)(122 140)(123 139)(124 138)(125 137)(126 136)(127 135)(128 134)(129 133)(130 132)(141 155)(142 154)(143 153)(144 152)(145 151)(146 150)(147 149)(156 160)(157 159)(161 169)(162 168)(163 167)(164 166)(170 180)(171 179)(172 178)(173 177)(174 176)(181 199)(182 198)(183 197)(184 196)(185 195)(186 194)(187 193)(188 192)(189 191)(201 219)(202 218)(203 217)(204 216)(205 215)(206 214)(207 213)(208 212)(209 211)(221 223)(224 240)(225 239)(226 238)(227 237)(228 236)(229 235)(230 234)(231 233)
G:=sub<Sym(240)| (1,38,134,168,203,151,106,54,97,235,193,70)(2,39,135,169,204,152,107,55,98,236,194,71)(3,40,136,170,205,153,108,56,99,237,195,72)(4,21,137,171,206,154,109,57,100,238,196,73)(5,22,138,172,207,155,110,58,81,239,197,74)(6,23,139,173,208,156,111,59,82,240,198,75)(7,24,140,174,209,157,112,60,83,221,199,76)(8,25,121,175,210,158,113,41,84,222,200,77)(9,26,122,176,211,159,114,42,85,223,181,78)(10,27,123,177,212,160,115,43,86,224,182,79)(11,28,124,178,213,141,116,44,87,225,183,80)(12,29,125,179,214,142,117,45,88,226,184,61)(13,30,126,180,215,143,118,46,89,227,185,62)(14,31,127,161,216,144,119,47,90,228,186,63)(15,32,128,162,217,145,120,48,91,229,187,64)(16,33,129,163,218,146,101,49,92,230,188,65)(17,34,130,164,219,147,102,50,93,231,189,66)(18,35,131,165,220,148,103,51,94,232,190,67)(19,36,132,166,201,149,104,52,95,233,191,68)(20,37,133,167,202,150,105,53,96,234,192,69), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159)(161,169)(162,168)(163,167)(164,166)(170,180)(171,179)(172,178)(173,177)(174,176)(181,199)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191)(201,219)(202,218)(203,217)(204,216)(205,215)(206,214)(207,213)(208,212)(209,211)(221,223)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233)>;
G:=Group( (1,38,134,168,203,151,106,54,97,235,193,70)(2,39,135,169,204,152,107,55,98,236,194,71)(3,40,136,170,205,153,108,56,99,237,195,72)(4,21,137,171,206,154,109,57,100,238,196,73)(5,22,138,172,207,155,110,58,81,239,197,74)(6,23,139,173,208,156,111,59,82,240,198,75)(7,24,140,174,209,157,112,60,83,221,199,76)(8,25,121,175,210,158,113,41,84,222,200,77)(9,26,122,176,211,159,114,42,85,223,181,78)(10,27,123,177,212,160,115,43,86,224,182,79)(11,28,124,178,213,141,116,44,87,225,183,80)(12,29,125,179,214,142,117,45,88,226,184,61)(13,30,126,180,215,143,118,46,89,227,185,62)(14,31,127,161,216,144,119,47,90,228,186,63)(15,32,128,162,217,145,120,48,91,229,187,64)(16,33,129,163,218,146,101,49,92,230,188,65)(17,34,130,164,219,147,102,50,93,231,189,66)(18,35,131,165,220,148,103,51,94,232,190,67)(19,36,132,166,201,149,104,52,95,233,191,68)(20,37,133,167,202,150,105,53,96,234,192,69), (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), (1,15)(2,14)(3,13)(4,12)(5,11)(6,10)(7,9)(16,20)(17,19)(21,29)(22,28)(23,27)(24,26)(30,40)(31,39)(32,38)(33,37)(34,36)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(61,73)(62,72)(63,71)(64,70)(65,69)(66,68)(74,80)(75,79)(76,78)(81,87)(82,86)(83,85)(88,100)(89,99)(90,98)(91,97)(92,96)(93,95)(101,105)(102,104)(106,120)(107,119)(108,118)(109,117)(110,116)(111,115)(112,114)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)(130,132)(141,155)(142,154)(143,153)(144,152)(145,151)(146,150)(147,149)(156,160)(157,159)(161,169)(162,168)(163,167)(164,166)(170,180)(171,179)(172,178)(173,177)(174,176)(181,199)(182,198)(183,197)(184,196)(185,195)(186,194)(187,193)(188,192)(189,191)(201,219)(202,218)(203,217)(204,216)(205,215)(206,214)(207,213)(208,212)(209,211)(221,223)(224,240)(225,239)(226,238)(227,237)(228,236)(229,235)(230,234)(231,233) );
G=PermutationGroup([[(1,38,134,168,203,151,106,54,97,235,193,70),(2,39,135,169,204,152,107,55,98,236,194,71),(3,40,136,170,205,153,108,56,99,237,195,72),(4,21,137,171,206,154,109,57,100,238,196,73),(5,22,138,172,207,155,110,58,81,239,197,74),(6,23,139,173,208,156,111,59,82,240,198,75),(7,24,140,174,209,157,112,60,83,221,199,76),(8,25,121,175,210,158,113,41,84,222,200,77),(9,26,122,176,211,159,114,42,85,223,181,78),(10,27,123,177,212,160,115,43,86,224,182,79),(11,28,124,178,213,141,116,44,87,225,183,80),(12,29,125,179,214,142,117,45,88,226,184,61),(13,30,126,180,215,143,118,46,89,227,185,62),(14,31,127,161,216,144,119,47,90,228,186,63),(15,32,128,162,217,145,120,48,91,229,187,64),(16,33,129,163,218,146,101,49,92,230,188,65),(17,34,130,164,219,147,102,50,93,231,189,66),(18,35,131,165,220,148,103,51,94,232,190,67),(19,36,132,166,201,149,104,52,95,233,191,68),(20,37,133,167,202,150,105,53,96,234,192,69)], [(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)], [(1,15),(2,14),(3,13),(4,12),(5,11),(6,10),(7,9),(16,20),(17,19),(21,29),(22,28),(23,27),(24,26),(30,40),(31,39),(32,38),(33,37),(34,36),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52),(61,73),(62,72),(63,71),(64,70),(65,69),(66,68),(74,80),(75,79),(76,78),(81,87),(82,86),(83,85),(88,100),(89,99),(90,98),(91,97),(92,96),(93,95),(101,105),(102,104),(106,120),(107,119),(108,118),(109,117),(110,116),(111,115),(112,114),(122,140),(123,139),(124,138),(125,137),(126,136),(127,135),(128,134),(129,133),(130,132),(141,155),(142,154),(143,153),(144,152),(145,151),(146,150),(147,149),(156,160),(157,159),(161,169),(162,168),(163,167),(164,166),(170,180),(171,179),(172,178),(173,177),(174,176),(181,199),(182,198),(183,197),(184,196),(185,195),(186,194),(187,193),(188,192),(189,191),(201,219),(202,218),(203,217),(204,216),(205,215),(206,214),(207,213),(208,212),(209,211),(221,223),(224,240),(225,239),(226,238),(227,237),(228,236),(229,235),(230,234),(231,233)]])
156 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 5A | 5B | 6A | ··· | 6F | 6G | ··· | 6N | 10A | ··· | 10F | 12A | ··· | 12H | 12I | ··· | 12P | 12Q | ··· | 12X | 15A | 15B | 15C | 15D | 20A | ··· | 20X | 30A | ··· | 30L | 60A | ··· | 60AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | ··· | 6 | 6 | ··· | 6 | 10 | ··· | 10 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 15 | 15 | 15 | 15 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 10 | 10 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 10 | 10 | 10 | 10 | 2 | 2 | 1 | ··· | 1 | 10 | ··· | 10 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 10 | ··· | 10 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
156 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 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | D5 | C4○D4 | D10 | C3×D4 | C3×D5 | C4×D5 | D20 | C3×C4○D4 | C6×D5 | C4○D20 | D5×C12 | C3×D20 | C3×C4○D20 |
| kernel | C12×D20 | C3×C4⋊Dic5 | C3×D10⋊C4 | C4×C60 | D5×C2×C12 | C6×D20 | C4×D20 | C3×D20 | C4⋊Dic5 | D10⋊C4 | C4×C20 | C2×C4×D5 | C2×D20 | D20 | C60 | C4×C12 | C30 | C2×C12 | C20 | C42 | C12 | C12 | C10 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 8 | 2 | 4 | 2 | 4 | 2 | 16 | 2 | 2 | 2 | 6 | 4 | 4 | 8 | 8 | 4 | 12 | 8 | 16 | 16 | 16 |
Matrix representation of C12×D20 ►in GL4(𝔽61) generated by
| 21 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 18 | 60 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 29 | 36 |
| 0 | 0 | 27 | 2 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 45 | 60 |
G:=sub<GL(4,GF(61))| [21,0,0,0,0,21,0,0,0,0,1,0,0,0,0,1],[18,1,0,0,60,0,0,0,0,0,29,27,0,0,36,2],[0,1,0,0,1,0,0,0,0,0,1,45,0,0,0,60] >;
C12×D20 in GAP, Magma, Sage, TeX
C_{12}\times D_{20} % in TeX
G:=Group("C12xD20"); // GroupNames label
G:=SmallGroup(480,666);
// by ID
G=gap.SmallGroup(480,666);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-5,701,344,142,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=b^20=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations