metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊2D20, C60⋊11D4, D30⋊13D4, C6.49(D4×D5), (C2×D20)⋊11S3, (C6×D20)⋊11C2, C5⋊1(D6⋊3D4), C4⋊3(C3⋊D20), C3⋊5(C4⋊D20), C20⋊4(C3⋊D4), C4⋊Dic3⋊15D5, C6.63(C2×D20), C10.50(S3×D4), C15⋊14(C4⋊D4), C30.158(C2×D4), (C2×C20).133D6, C30.95(C4○D4), (C2×C12).134D10, (C22×D5).25D6, D10⋊Dic3⋊21C2, C2.25(C20⋊D6), (C2×C60).204C22, (C2×C30).155C23, C6.37(Q8⋊2D5), (C2×Dic3).49D10, C10.16(D4⋊2S3), C2.19(D20⋊S3), (C10×Dic3).94C22, (C2×Dic15).216C22, (C22×D15).109C22, (C2×C4×D15)⋊22C2, (C2×C3⋊D20)⋊8C2, (C5×C4⋊Dic3)⋊12C2, (C2×C4).214(S3×D5), C2.21(C2×C3⋊D20), C10.18(C2×C3⋊D4), (D5×C2×C6).39C22, C22.207(C2×S3×D5), (C2×C6).167(C22×D5), (C2×C10).167(C22×S3), SmallGroup(480,541)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊2D20
G = < a,b,c | a60=b4=c2=1, bab-1=a11, cac=a29, cbc=b-1 >
Subgroups: 1180 in 188 conjugacy classes, 54 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C2×C4, C2×C4, D4, C23, D5, C10, Dic3, C12, D6, C2×C6, C2×C6, C15, C22⋊C4, C4⋊C4, C22×C4, C2×D4, Dic5, C20, C20, D10, C2×C10, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C3×D5, D15, C30, C4⋊D4, C4×D5, D20, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C4⋊Dic3, C6.D4, S3×C2×C4, C2×C3⋊D4, C6×D4, C5×Dic3, Dic15, C60, C6×D5, D30, D30, C2×C30, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, C2×D20, D6⋊3D4, C3⋊D20, C3×D20, C10×Dic3, C4×D15, C2×Dic15, C2×C60, D5×C2×C6, C22×D15, C4⋊D20, D10⋊Dic3, C5×C4⋊Dic3, C2×C3⋊D20, C6×D20, C2×C4×D15, C12⋊2D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, C4○D4, D10, C3⋊D4, C22×S3, C4⋊D4, D20, C22×D5, S3×D4, D4⋊2S3, C2×C3⋊D4, S3×D5, C2×D20, D4×D5, Q8⋊2D5, D6⋊3D4, C3⋊D20, C2×S3×D5, C4⋊D20, D20⋊S3, C20⋊D6, C2×C3⋊D20, C12⋊2D20
►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)
(1 166 198 107)(2 177 199 118)(3 128 200 69)(4 139 201 80)(5 150 202 91)(6 161 203 102)(7 172 204 113)(8 123 205 64)(9 134 206 75)(10 145 207 86)(11 156 208 97)(12 167 209 108)(13 178 210 119)(14 129 211 70)(15 140 212 81)(16 151 213 92)(17 162 214 103)(18 173 215 114)(19 124 216 65)(20 135 217 76)(21 146 218 87)(22 157 219 98)(23 168 220 109)(24 179 221 120)(25 130 222 71)(26 141 223 82)(27 152 224 93)(28 163 225 104)(29 174 226 115)(30 125 227 66)(31 136 228 77)(32 147 229 88)(33 158 230 99)(34 169 231 110)(35 180 232 61)(36 131 233 72)(37 142 234 83)(38 153 235 94)(39 164 236 105)(40 175 237 116)(41 126 238 67)(42 137 239 78)(43 148 240 89)(44 159 181 100)(45 170 182 111)(46 121 183 62)(47 132 184 73)(48 143 185 84)(49 154 186 95)(50 165 187 106)(51 176 188 117)(52 127 189 68)(53 138 190 79)(54 149 191 90)(55 160 192 101)(56 171 193 112)(57 122 194 63)(58 133 195 74)(59 144 196 85)(60 155 197 96)
(1 198)(2 227)(3 196)(4 225)(5 194)(6 223)(7 192)(8 221)(9 190)(10 219)(11 188)(12 217)(13 186)(14 215)(15 184)(16 213)(17 182)(18 211)(19 240)(20 209)(21 238)(22 207)(23 236)(24 205)(25 234)(26 203)(27 232)(28 201)(29 230)(30 199)(31 228)(32 197)(33 226)(34 195)(35 224)(36 193)(37 222)(38 191)(39 220)(40 189)(41 218)(42 187)(43 216)(44 185)(45 214)(46 183)(47 212)(48 181)(49 210)(50 239)(51 208)(52 237)(53 206)(54 235)(55 204)(56 233)(57 202)(58 231)(59 200)(60 229)(61 93)(63 91)(64 120)(65 89)(66 118)(67 87)(68 116)(69 85)(70 114)(71 83)(72 112)(73 81)(74 110)(75 79)(76 108)(78 106)(80 104)(82 102)(84 100)(86 98)(88 96)(90 94)(95 119)(97 117)(99 115)(101 113)(103 111)(105 109)(122 150)(123 179)(124 148)(125 177)(126 146)(127 175)(128 144)(129 173)(130 142)(131 171)(132 140)(133 169)(134 138)(135 167)(137 165)(139 163)(141 161)(143 159)(145 157)(147 155)(149 153)(152 180)(154 178)(156 176)(158 174)(160 172)(162 170)(164 168)
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), (1,166,198,107)(2,177,199,118)(3,128,200,69)(4,139,201,80)(5,150,202,91)(6,161,203,102)(7,172,204,113)(8,123,205,64)(9,134,206,75)(10,145,207,86)(11,156,208,97)(12,167,209,108)(13,178,210,119)(14,129,211,70)(15,140,212,81)(16,151,213,92)(17,162,214,103)(18,173,215,114)(19,124,216,65)(20,135,217,76)(21,146,218,87)(22,157,219,98)(23,168,220,109)(24,179,221,120)(25,130,222,71)(26,141,223,82)(27,152,224,93)(28,163,225,104)(29,174,226,115)(30,125,227,66)(31,136,228,77)(32,147,229,88)(33,158,230,99)(34,169,231,110)(35,180,232,61)(36,131,233,72)(37,142,234,83)(38,153,235,94)(39,164,236,105)(40,175,237,116)(41,126,238,67)(42,137,239,78)(43,148,240,89)(44,159,181,100)(45,170,182,111)(46,121,183,62)(47,132,184,73)(48,143,185,84)(49,154,186,95)(50,165,187,106)(51,176,188,117)(52,127,189,68)(53,138,190,79)(54,149,191,90)(55,160,192,101)(56,171,193,112)(57,122,194,63)(58,133,195,74)(59,144,196,85)(60,155,197,96), (1,198)(2,227)(3,196)(4,225)(5,194)(6,223)(7,192)(8,221)(9,190)(10,219)(11,188)(12,217)(13,186)(14,215)(15,184)(16,213)(17,182)(18,211)(19,240)(20,209)(21,238)(22,207)(23,236)(24,205)(25,234)(26,203)(27,232)(28,201)(29,230)(30,199)(31,228)(32,197)(33,226)(34,195)(35,224)(36,193)(37,222)(38,191)(39,220)(40,189)(41,218)(42,187)(43,216)(44,185)(45,214)(46,183)(47,212)(48,181)(49,210)(50,239)(51,208)(52,237)(53,206)(54,235)(55,204)(56,233)(57,202)(58,231)(59,200)(60,229)(61,93)(63,91)(64,120)(65,89)(66,118)(67,87)(68,116)(69,85)(70,114)(71,83)(72,112)(73,81)(74,110)(75,79)(76,108)(78,106)(80,104)(82,102)(84,100)(86,98)(88,96)(90,94)(95,119)(97,117)(99,115)(101,113)(103,111)(105,109)(122,150)(123,179)(124,148)(125,177)(126,146)(127,175)(128,144)(129,173)(130,142)(131,171)(132,140)(133,169)(134,138)(135,167)(137,165)(139,163)(141,161)(143,159)(145,157)(147,155)(149,153)(152,180)(154,178)(156,176)(158,174)(160,172)(162,170)(164,168)>;
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), (1,166,198,107)(2,177,199,118)(3,128,200,69)(4,139,201,80)(5,150,202,91)(6,161,203,102)(7,172,204,113)(8,123,205,64)(9,134,206,75)(10,145,207,86)(11,156,208,97)(12,167,209,108)(13,178,210,119)(14,129,211,70)(15,140,212,81)(16,151,213,92)(17,162,214,103)(18,173,215,114)(19,124,216,65)(20,135,217,76)(21,146,218,87)(22,157,219,98)(23,168,220,109)(24,179,221,120)(25,130,222,71)(26,141,223,82)(27,152,224,93)(28,163,225,104)(29,174,226,115)(30,125,227,66)(31,136,228,77)(32,147,229,88)(33,158,230,99)(34,169,231,110)(35,180,232,61)(36,131,233,72)(37,142,234,83)(38,153,235,94)(39,164,236,105)(40,175,237,116)(41,126,238,67)(42,137,239,78)(43,148,240,89)(44,159,181,100)(45,170,182,111)(46,121,183,62)(47,132,184,73)(48,143,185,84)(49,154,186,95)(50,165,187,106)(51,176,188,117)(52,127,189,68)(53,138,190,79)(54,149,191,90)(55,160,192,101)(56,171,193,112)(57,122,194,63)(58,133,195,74)(59,144,196,85)(60,155,197,96), (1,198)(2,227)(3,196)(4,225)(5,194)(6,223)(7,192)(8,221)(9,190)(10,219)(11,188)(12,217)(13,186)(14,215)(15,184)(16,213)(17,182)(18,211)(19,240)(20,209)(21,238)(22,207)(23,236)(24,205)(25,234)(26,203)(27,232)(28,201)(29,230)(30,199)(31,228)(32,197)(33,226)(34,195)(35,224)(36,193)(37,222)(38,191)(39,220)(40,189)(41,218)(42,187)(43,216)(44,185)(45,214)(46,183)(47,212)(48,181)(49,210)(50,239)(51,208)(52,237)(53,206)(54,235)(55,204)(56,233)(57,202)(58,231)(59,200)(60,229)(61,93)(63,91)(64,120)(65,89)(66,118)(67,87)(68,116)(69,85)(70,114)(71,83)(72,112)(73,81)(74,110)(75,79)(76,108)(78,106)(80,104)(82,102)(84,100)(86,98)(88,96)(90,94)(95,119)(97,117)(99,115)(101,113)(103,111)(105,109)(122,150)(123,179)(124,148)(125,177)(126,146)(127,175)(128,144)(129,173)(130,142)(131,171)(132,140)(133,169)(134,138)(135,167)(137,165)(139,163)(141,161)(143,159)(145,157)(147,155)(149,153)(152,180)(154,178)(156,176)(158,174)(160,172)(162,170)(164,168) );
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)], [(1,166,198,107),(2,177,199,118),(3,128,200,69),(4,139,201,80),(5,150,202,91),(6,161,203,102),(7,172,204,113),(8,123,205,64),(9,134,206,75),(10,145,207,86),(11,156,208,97),(12,167,209,108),(13,178,210,119),(14,129,211,70),(15,140,212,81),(16,151,213,92),(17,162,214,103),(18,173,215,114),(19,124,216,65),(20,135,217,76),(21,146,218,87),(22,157,219,98),(23,168,220,109),(24,179,221,120),(25,130,222,71),(26,141,223,82),(27,152,224,93),(28,163,225,104),(29,174,226,115),(30,125,227,66),(31,136,228,77),(32,147,229,88),(33,158,230,99),(34,169,231,110),(35,180,232,61),(36,131,233,72),(37,142,234,83),(38,153,235,94),(39,164,236,105),(40,175,237,116),(41,126,238,67),(42,137,239,78),(43,148,240,89),(44,159,181,100),(45,170,182,111),(46,121,183,62),(47,132,184,73),(48,143,185,84),(49,154,186,95),(50,165,187,106),(51,176,188,117),(52,127,189,68),(53,138,190,79),(54,149,191,90),(55,160,192,101),(56,171,193,112),(57,122,194,63),(58,133,195,74),(59,144,196,85),(60,155,197,96)], [(1,198),(2,227),(3,196),(4,225),(5,194),(6,223),(7,192),(8,221),(9,190),(10,219),(11,188),(12,217),(13,186),(14,215),(15,184),(16,213),(17,182),(18,211),(19,240),(20,209),(21,238),(22,207),(23,236),(24,205),(25,234),(26,203),(27,232),(28,201),(29,230),(30,199),(31,228),(32,197),(33,226),(34,195),(35,224),(36,193),(37,222),(38,191),(39,220),(40,189),(41,218),(42,187),(43,216),(44,185),(45,214),(46,183),(47,212),(48,181),(49,210),(50,239),(51,208),(52,237),(53,206),(54,235),(55,204),(56,233),(57,202),(58,231),(59,200),(60,229),(61,93),(63,91),(64,120),(65,89),(66,118),(67,87),(68,116),(69,85),(70,114),(71,83),(72,112),(73,81),(74,110),(75,79),(76,108),(78,106),(80,104),(82,102),(84,100),(86,98),(88,96),(90,94),(95,119),(97,117),(99,115),(101,113),(103,111),(105,109),(122,150),(123,179),(124,148),(125,177),(126,146),(127,175),(128,144),(129,173),(130,142),(131,171),(132,140),(133,169),(134,138),(135,167),(137,165),(139,163),(141,161),(143,159),(145,157),(147,155),(149,153),(152,180),(154,178),(156,176),(158,174),(160,172),(162,170),(164,168)]])
60 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 | 20B | 20C | 20D | 20E | ··· | 20L | 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 | 20 | ··· | 20 | 30 | ··· | 30 | 60 | ··· | 60 |
| size | 1 | 1 | 1 | 1 | 20 | 20 | 30 | 30 | 2 | 2 | 2 | 12 | 12 | 30 | 30 | 2 | 2 | 2 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | C4○D4 | D10 | D10 | C3⋊D4 | D20 | S3×D4 | D4⋊2S3 | S3×D5 | D4×D5 | Q8⋊2D5 | C3⋊D20 | C2×S3×D5 | D20⋊S3 | C20⋊D6 |
| kernel | C12⋊2D20 | D10⋊Dic3 | C5×C4⋊Dic3 | C2×C3⋊D20 | C6×D20 | C2×C4×D15 | C2×D20 | C60 | D30 | C4⋊Dic3 | C2×C20 | C22×D5 | C30 | C2×Dic3 | C2×C12 | C20 | C12 | C10 | C10 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 2 | 4 | 4 |
Matrix representation of C12⋊2D20 ►in GL6(𝔽61)
| 1 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 30 | 0 | 0 |
| 0 | 0 | 0 | 11 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 18 |
| 0 | 0 | 0 | 0 | 43 | 60 |
| 9 | 9 | 0 | 0 | 0 | 0 |
| 18 | 52 | 0 | 0 | 0 | 0 |
| 0 | 0 | 35 | 37 | 0 | 0 |
| 0 | 0 | 46 | 26 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 36 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 43 | 60 |
G:=sub<GL(6,GF(61))| [1,1,0,0,0,0,60,0,0,0,0,0,0,0,50,0,0,0,0,0,30,11,0,0,0,0,0,0,18,43,0,0,0,0,18,60],[9,18,0,0,0,0,9,52,0,0,0,0,0,0,35,46,0,0,0,0,37,26,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[60,0,0,0,0,0,1,1,0,0,0,0,0,0,60,0,0,0,0,0,36,1,0,0,0,0,0,0,1,43,0,0,0,0,0,60] >;
C12⋊2D20 in GAP, Magma, Sage, TeX
C_{12}\rtimes_2D_{20} % in TeX
G:=Group("C12:2D20"); // GroupNames label
G:=SmallGroup(480,541);
// by ID
G=gap.SmallGroup(480,541);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,141,64,422,219,100,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^60=b^4=c^2=1,b*a*b^-1=a^11,c*a*c=a^29,c*b*c=b^-1>;
// generators/relations