metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C60.85D4, C12.17D20, C60.108C23, Dic10.34D6, D60.46C22, Dic30.48C22, C3⋊C8.4D10, (C2×C20).98D6, (C2×C6).42D20, C30.90(C2×D4), (C2×C30).58D4, C6.53(C2×D20), (C2×Dic10)⋊9S3, (C6×Dic10)⋊2C2, C3⋊Dic20⋊14C2, C3⋊4(C8.D10), C4.Dic3⋊10D5, (C2×C12).101D10, C15⋊SD16⋊14C2, C4.24(C3⋊D20), C20.31(C3⋊D4), C5⋊1(Q8.11D6), C15⋊10(C8.C22), (C2×C60).34C22, C12.99(C22×D5), D60⋊11C2.3C2, C20.158(C22×S3), C22.5(C3⋊D20), (C3×Dic10).39C22, C4.107(C2×S3×D5), (C2×C4).16(S3×D5), C10.8(C2×C3⋊D4), C2.12(C2×C3⋊D20), (C5×C3⋊C8).22C22, (C5×C4.Dic3)⋊3C2, (C2×C10).14(C3⋊D4), SmallGroup(480,391)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.D20
G = < a,b,c | a12=c2=1, b20=a6, bab-1=a-1, cac=a5, cbc=b19 >
Subgroups: 668 in 120 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, D5, C10, C10, Dic3, C12, C12, D6, C2×C6, C15, M4(2), SD16, Q16, C2×Q8, C4○D4, Dic5, C20, D10, C2×C10, C3⋊C8, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C2×C12, C3×Q8, D15, C30, C30, C8.C22, C40, Dic10, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C4.Dic3, Q8⋊2S3, C3⋊Q16, C4○D12, C6×Q8, C3×Dic5, Dic15, C60, D30, C2×C30, C40⋊C2, Dic20, C5×M4(2), C2×Dic10, C4○D20, Q8.11D6, C5×C3⋊C8, C3×Dic10, C3×Dic10, C6×Dic5, Dic30, C4×D15, D60, C15⋊7D4, C2×C60, C8.D10, C15⋊SD16, C3⋊Dic20, C5×C4.Dic3, C6×Dic10, D60⋊11C2, C12.D20
Quotients: C1, C2, C22, S3, D4, C23, D5, D6, C2×D4, D10, C3⋊D4, C22×S3, C8.C22, D20, C22×D5, C2×C3⋊D4, S3×D5, C2×D20, Q8.11D6, C3⋊D20, C2×S3×D5, C8.D10, C2×C3⋊D20, C12.D20
►On 240 points
Generators in S240
(1 163 131 31 193 121 21 183 151 11 173 141)(2 142 174 12 152 184 22 122 194 32 132 164)(3 165 133 33 195 123 23 185 153 13 175 143)(4 144 176 14 154 186 24 124 196 34 134 166)(5 167 135 35 197 125 25 187 155 15 177 145)(6 146 178 16 156 188 26 126 198 36 136 168)(7 169 137 37 199 127 27 189 157 17 179 147)(8 148 180 18 158 190 28 128 200 38 138 170)(9 171 139 39 161 129 29 191 159 19 181 149)(10 150 182 20 160 192 30 130 162 40 140 172)(41 93 211 51 103 221 61 113 231 71 83 201)(42 202 84 72 232 114 62 222 104 52 212 94)(43 95 213 53 105 223 63 115 233 73 85 203)(44 204 86 74 234 116 64 224 106 54 214 96)(45 97 215 55 107 225 65 117 235 75 87 205)(46 206 88 76 236 118 66 226 108 56 216 98)(47 99 217 57 109 227 67 119 237 77 89 207)(48 208 90 78 238 120 68 228 110 58 218 100)(49 101 219 59 111 229 69 81 239 79 91 209)(50 210 92 80 240 82 70 230 112 60 220 102)
(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 91)(2 110)(3 89)(4 108)(5 87)(6 106)(7 85)(8 104)(9 83)(10 102)(11 81)(12 100)(13 119)(14 98)(15 117)(16 96)(17 115)(18 94)(19 113)(20 92)(21 111)(22 90)(23 109)(24 88)(25 107)(26 86)(27 105)(28 84)(29 103)(30 82)(31 101)(32 120)(33 99)(34 118)(35 97)(36 116)(37 95)(38 114)(39 93)(40 112)(41 181)(42 200)(43 179)(44 198)(45 177)(46 196)(47 175)(48 194)(49 173)(50 192)(51 171)(52 190)(53 169)(54 188)(55 167)(56 186)(57 165)(58 184)(59 163)(60 182)(61 161)(62 180)(63 199)(64 178)(65 197)(66 176)(67 195)(68 174)(69 193)(70 172)(71 191)(72 170)(73 189)(74 168)(75 187)(76 166)(77 185)(78 164)(79 183)(80 162)(121 209)(122 228)(123 207)(124 226)(125 205)(126 224)(127 203)(128 222)(129 201)(130 220)(131 239)(132 218)(133 237)(134 216)(135 235)(136 214)(137 233)(138 212)(139 231)(140 210)(141 229)(142 208)(143 227)(144 206)(145 225)(146 204)(147 223)(148 202)(149 221)(150 240)(151 219)(152 238)(153 217)(154 236)(155 215)(156 234)(157 213)(158 232)(159 211)(160 230)
G:=sub<Sym(240)| (1,163,131,31,193,121,21,183,151,11,173,141)(2,142,174,12,152,184,22,122,194,32,132,164)(3,165,133,33,195,123,23,185,153,13,175,143)(4,144,176,14,154,186,24,124,196,34,134,166)(5,167,135,35,197,125,25,187,155,15,177,145)(6,146,178,16,156,188,26,126,198,36,136,168)(7,169,137,37,199,127,27,189,157,17,179,147)(8,148,180,18,158,190,28,128,200,38,138,170)(9,171,139,39,161,129,29,191,159,19,181,149)(10,150,182,20,160,192,30,130,162,40,140,172)(41,93,211,51,103,221,61,113,231,71,83,201)(42,202,84,72,232,114,62,222,104,52,212,94)(43,95,213,53,105,223,63,115,233,73,85,203)(44,204,86,74,234,116,64,224,106,54,214,96)(45,97,215,55,107,225,65,117,235,75,87,205)(46,206,88,76,236,118,66,226,108,56,216,98)(47,99,217,57,109,227,67,119,237,77,89,207)(48,208,90,78,238,120,68,228,110,58,218,100)(49,101,219,59,111,229,69,81,239,79,91,209)(50,210,92,80,240,82,70,230,112,60,220,102), (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), (1,91)(2,110)(3,89)(4,108)(5,87)(6,106)(7,85)(8,104)(9,83)(10,102)(11,81)(12,100)(13,119)(14,98)(15,117)(16,96)(17,115)(18,94)(19,113)(20,92)(21,111)(22,90)(23,109)(24,88)(25,107)(26,86)(27,105)(28,84)(29,103)(30,82)(31,101)(32,120)(33,99)(34,118)(35,97)(36,116)(37,95)(38,114)(39,93)(40,112)(41,181)(42,200)(43,179)(44,198)(45,177)(46,196)(47,175)(48,194)(49,173)(50,192)(51,171)(52,190)(53,169)(54,188)(55,167)(56,186)(57,165)(58,184)(59,163)(60,182)(61,161)(62,180)(63,199)(64,178)(65,197)(66,176)(67,195)(68,174)(69,193)(70,172)(71,191)(72,170)(73,189)(74,168)(75,187)(76,166)(77,185)(78,164)(79,183)(80,162)(121,209)(122,228)(123,207)(124,226)(125,205)(126,224)(127,203)(128,222)(129,201)(130,220)(131,239)(132,218)(133,237)(134,216)(135,235)(136,214)(137,233)(138,212)(139,231)(140,210)(141,229)(142,208)(143,227)(144,206)(145,225)(146,204)(147,223)(148,202)(149,221)(150,240)(151,219)(152,238)(153,217)(154,236)(155,215)(156,234)(157,213)(158,232)(159,211)(160,230)>;
G:=Group( (1,163,131,31,193,121,21,183,151,11,173,141)(2,142,174,12,152,184,22,122,194,32,132,164)(3,165,133,33,195,123,23,185,153,13,175,143)(4,144,176,14,154,186,24,124,196,34,134,166)(5,167,135,35,197,125,25,187,155,15,177,145)(6,146,178,16,156,188,26,126,198,36,136,168)(7,169,137,37,199,127,27,189,157,17,179,147)(8,148,180,18,158,190,28,128,200,38,138,170)(9,171,139,39,161,129,29,191,159,19,181,149)(10,150,182,20,160,192,30,130,162,40,140,172)(41,93,211,51,103,221,61,113,231,71,83,201)(42,202,84,72,232,114,62,222,104,52,212,94)(43,95,213,53,105,223,63,115,233,73,85,203)(44,204,86,74,234,116,64,224,106,54,214,96)(45,97,215,55,107,225,65,117,235,75,87,205)(46,206,88,76,236,118,66,226,108,56,216,98)(47,99,217,57,109,227,67,119,237,77,89,207)(48,208,90,78,238,120,68,228,110,58,218,100)(49,101,219,59,111,229,69,81,239,79,91,209)(50,210,92,80,240,82,70,230,112,60,220,102), (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), (1,91)(2,110)(3,89)(4,108)(5,87)(6,106)(7,85)(8,104)(9,83)(10,102)(11,81)(12,100)(13,119)(14,98)(15,117)(16,96)(17,115)(18,94)(19,113)(20,92)(21,111)(22,90)(23,109)(24,88)(25,107)(26,86)(27,105)(28,84)(29,103)(30,82)(31,101)(32,120)(33,99)(34,118)(35,97)(36,116)(37,95)(38,114)(39,93)(40,112)(41,181)(42,200)(43,179)(44,198)(45,177)(46,196)(47,175)(48,194)(49,173)(50,192)(51,171)(52,190)(53,169)(54,188)(55,167)(56,186)(57,165)(58,184)(59,163)(60,182)(61,161)(62,180)(63,199)(64,178)(65,197)(66,176)(67,195)(68,174)(69,193)(70,172)(71,191)(72,170)(73,189)(74,168)(75,187)(76,166)(77,185)(78,164)(79,183)(80,162)(121,209)(122,228)(123,207)(124,226)(125,205)(126,224)(127,203)(128,222)(129,201)(130,220)(131,239)(132,218)(133,237)(134,216)(135,235)(136,214)(137,233)(138,212)(139,231)(140,210)(141,229)(142,208)(143,227)(144,206)(145,225)(146,204)(147,223)(148,202)(149,221)(150,240)(151,219)(152,238)(153,217)(154,236)(155,215)(156,234)(157,213)(158,232)(159,211)(160,230) );
G=PermutationGroup([[(1,163,131,31,193,121,21,183,151,11,173,141),(2,142,174,12,152,184,22,122,194,32,132,164),(3,165,133,33,195,123,23,185,153,13,175,143),(4,144,176,14,154,186,24,124,196,34,134,166),(5,167,135,35,197,125,25,187,155,15,177,145),(6,146,178,16,156,188,26,126,198,36,136,168),(7,169,137,37,199,127,27,189,157,17,179,147),(8,148,180,18,158,190,28,128,200,38,138,170),(9,171,139,39,161,129,29,191,159,19,181,149),(10,150,182,20,160,192,30,130,162,40,140,172),(41,93,211,51,103,221,61,113,231,71,83,201),(42,202,84,72,232,114,62,222,104,52,212,94),(43,95,213,53,105,223,63,115,233,73,85,203),(44,204,86,74,234,116,64,224,106,54,214,96),(45,97,215,55,107,225,65,117,235,75,87,205),(46,206,88,76,236,118,66,226,108,56,216,98),(47,99,217,57,109,227,67,119,237,77,89,207),(48,208,90,78,238,120,68,228,110,58,218,100),(49,101,219,59,111,229,69,81,239,79,91,209),(50,210,92,80,240,82,70,230,112,60,220,102)], [(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)], [(1,91),(2,110),(3,89),(4,108),(5,87),(6,106),(7,85),(8,104),(9,83),(10,102),(11,81),(12,100),(13,119),(14,98),(15,117),(16,96),(17,115),(18,94),(19,113),(20,92),(21,111),(22,90),(23,109),(24,88),(25,107),(26,86),(27,105),(28,84),(29,103),(30,82),(31,101),(32,120),(33,99),(34,118),(35,97),(36,116),(37,95),(38,114),(39,93),(40,112),(41,181),(42,200),(43,179),(44,198),(45,177),(46,196),(47,175),(48,194),(49,173),(50,192),(51,171),(52,190),(53,169),(54,188),(55,167),(56,186),(57,165),(58,184),(59,163),(60,182),(61,161),(62,180),(63,199),(64,178),(65,197),(66,176),(67,195),(68,174),(69,193),(70,172),(71,191),(72,170),(73,189),(74,168),(75,187),(76,166),(77,185),(78,164),(79,183),(80,162),(121,209),(122,228),(123,207),(124,226),(125,205),(126,224),(127,203),(128,222),(129,201),(130,220),(131,239),(132,218),(133,237),(134,216),(135,235),(136,214),(137,233),(138,212),(139,231),(140,210),(141,229),(142,208),(143,227),(144,206),(145,225),(146,204),(147,223),(148,202),(149,221),(150,240),(151,219),(152,238),(153,217),(154,236),(155,215),(156,234),(157,213),(158,232),(159,211),(160,230)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 10A | 10B | 10C | 10D | 12A | 12B | 12C | 12D | 12E | 12F | 15A | 15B | 20A | 20B | 20C | 20D | 20E | 20F | 30A | ··· | 30F | 40A | ··· | 40H | 60A | ··· | 60H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 10 | 10 | 10 | 10 | 12 | 12 | 12 | 12 | 12 | 12 | 15 | 15 | 20 | 20 | 20 | 20 | 20 | 20 | 30 | ··· | 30 | 40 | ··· | 40 | 60 | ··· | 60 |
| size | 1 | 1 | 2 | 60 | 2 | 2 | 2 | 20 | 20 | 60 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 | 12 | ··· | 12 | 4 | ··· | 4 |
57 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | + | - | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D5 | D6 | D6 | D10 | D10 | C3⋊D4 | C3⋊D4 | D20 | D20 | C8.C22 | S3×D5 | Q8.11D6 | C3⋊D20 | C2×S3×D5 | C3⋊D20 | C8.D10 | C12.D20 |
| kernel | C12.D20 | C15⋊SD16 | C3⋊Dic20 | C5×C4.Dic3 | C6×Dic10 | D60⋊11C2 | C2×Dic10 | C60 | C2×C30 | C4.Dic3 | Dic10 | C2×C20 | C3⋊C8 | C2×C12 | C20 | C2×C10 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C12.D20 ►in GL6(𝔽241)
| 240 | 240 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 44 | 3 | 0 | 0 |
| 0 | 0 | 238 | 197 | 0 | 0 |
| 0 | 0 | 0 | 0 | 197 | 238 |
| 0 | 0 | 0 | 0 | 3 | 44 |
| 204 | 2 | 0 | 0 | 0 | 0 |
| 39 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 239 |
| 0 | 0 | 0 | 0 | 2 | 139 |
| 0 | 0 | 122 | 22 | 0 | 0 |
| 0 | 0 | 219 | 39 | 0 | 0 |
| 204 | 2 | 0 | 0 | 0 | 0 |
| 39 | 37 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 235 | 77 |
| 0 | 0 | 0 | 0 | 12 | 6 |
| 0 | 0 | 122 | 41 | 0 | 0 |
| 0 | 0 | 238 | 119 | 0 | 0 |
G:=sub<GL(6,GF(241))| [240,1,0,0,0,0,240,0,0,0,0,0,0,0,44,238,0,0,0,0,3,197,0,0,0,0,0,0,197,3,0,0,0,0,238,44],[204,39,0,0,0,0,2,37,0,0,0,0,0,0,0,0,122,219,0,0,0,0,22,39,0,0,0,2,0,0,0,0,239,139,0,0],[204,39,0,0,0,0,2,37,0,0,0,0,0,0,0,0,122,238,0,0,0,0,41,119,0,0,235,12,0,0,0,0,77,6,0,0] >;
C12.D20 in GAP, Magma, Sage, TeX
C_{12}.D_{20} % in TeX
G:=Group("C12.D20"); // GroupNames label
G:=SmallGroup(480,391);
// by ID
G=gap.SmallGroup(480,391);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,64,219,675,80,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=c^2=1,b^20=a^6,b*a*b^-1=a^-1,c*a*c=a^5,c*b*c=b^19>;
// generators/relations