metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.6D20, C60.30D4, (C2×C20).43D6, (C2×C12).44D10, C12.8(C5⋊D4), (C6×Dic5).1C4, C4.Dic3.2D5, C60.7C4.3C2, C20.78(C3⋊D4), C15⋊4(C4.10D4), C4.20(C3⋊D20), C3⋊3(C4.12D20), C4.20(C15⋊D4), C5⋊2(C12.10D4), (C2×C60).30C22, (C6×Dic10).1C2, (C2×Dic10).6S3, C22.4(D5×Dic3), C30.44(C22⋊C4), (C2×Dic5).1Dic3, C6.25(D10⋊C4), C2.4(D10⋊Dic3), C10.14(C6.D4), (C2×C4).4(S3×D5), (C2×C6).46(C4×D5), (C2×C30).83(C2×C4), (C5×C4.Dic3).1C2, (C2×C10).22(C2×Dic3), SmallGroup(480,37)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.6D20
G = < a,b,c | a12=1, b20=a6, c2=a3, bab-1=a-1, cac-1=a5, cbc-1=a3b19 >
Subgroups: 284 in 76 conjugacy classes, 34 normal (30 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C2×C4, C2×C4, Q8, C10, C10, C12, C12, C2×C6, C15, M4(2), C2×Q8, Dic5, C20, C2×C10, C3⋊C8, C2×C12, C2×C12, C3×Q8, C30, C30, C4.10D4, C5⋊2C8, C40, Dic10, C2×Dic5, C2×C20, C4.Dic3, C4.Dic3, C6×Q8, C3×Dic5, C60, C2×C30, C4.Dic5, C5×M4(2), C2×Dic10, C12.10D4, C5×C3⋊C8, C15⋊3C8, C3×Dic10, C6×Dic5, C2×C60, C4.12D20, C5×C4.Dic3, C60.7C4, C6×Dic10, C12.6D20
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D5, Dic3, D6, C22⋊C4, D10, C2×Dic3, C3⋊D4, C4.10D4, C4×D5, D20, C5⋊D4, C6.D4, S3×D5, D10⋊C4, C12.10D4, D5×Dic3, C15⋊D4, C3⋊D20, C4.12D20, D10⋊Dic3, C12.6D20
►On 240 points
Generators in S240
(1 228 138 31 218 128 21 208 158 11 238 148)(2 149 239 12 159 209 22 129 219 32 139 229)(3 230 140 33 220 130 23 210 160 13 240 150)(4 151 201 14 121 211 24 131 221 34 141 231)(5 232 142 35 222 132 25 212 122 15 202 152)(6 153 203 16 123 213 26 133 223 36 143 233)(7 234 144 37 224 134 27 214 124 17 204 154)(8 155 205 18 125 215 28 135 225 38 145 235)(9 236 146 39 226 136 29 216 126 19 206 156)(10 157 207 20 127 217 30 137 227 40 147 237)(41 187 103 71 177 93 61 167 83 51 197 113)(42 114 198 52 84 168 62 94 178 72 104 188)(43 189 105 73 179 95 63 169 85 53 199 115)(44 116 200 54 86 170 64 96 180 74 106 190)(45 191 107 75 181 97 65 171 87 55 161 117)(46 118 162 56 88 172 66 98 182 76 108 192)(47 193 109 77 183 99 67 173 89 57 163 119)(48 120 164 58 90 174 68 100 184 78 110 194)(49 195 111 79 185 101 69 175 91 59 165 81)(50 82 166 60 92 176 70 102 186 80 112 196)
(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)
(1 162 31 172 21 182 11 192)(2 171 12 161 22 191 32 181)(3 200 33 170 23 180 13 190)(4 169 14 199 24 189 34 179)(5 198 35 168 25 178 15 188)(6 167 16 197 26 187 36 177)(7 196 37 166 27 176 17 186)(8 165 18 195 28 185 38 175)(9 194 39 164 29 174 19 184)(10 163 20 193 30 183 40 173)(41 123 71 133 61 143 51 153)(42 132 52 122 62 152 72 142)(43 121 73 131 63 141 53 151)(44 130 54 160 64 150 74 140)(45 159 75 129 65 139 55 149)(46 128 56 158 66 148 76 138)(47 157 77 127 67 137 57 147)(48 126 58 156 68 146 78 136)(49 155 79 125 69 135 59 145)(50 124 60 154 70 144 80 134)(81 225 111 235 101 205 91 215)(82 234 92 224 102 214 112 204)(83 223 113 233 103 203 93 213)(84 232 94 222 104 212 114 202)(85 221 115 231 105 201 95 211)(86 230 96 220 106 210 116 240)(87 219 117 229 107 239 97 209)(88 228 98 218 108 208 118 238)(89 217 119 227 109 237 99 207)(90 226 100 216 110 206 120 236)
G:=sub<Sym(240)| (1,228,138,31,218,128,21,208,158,11,238,148)(2,149,239,12,159,209,22,129,219,32,139,229)(3,230,140,33,220,130,23,210,160,13,240,150)(4,151,201,14,121,211,24,131,221,34,141,231)(5,232,142,35,222,132,25,212,122,15,202,152)(6,153,203,16,123,213,26,133,223,36,143,233)(7,234,144,37,224,134,27,214,124,17,204,154)(8,155,205,18,125,215,28,135,225,38,145,235)(9,236,146,39,226,136,29,216,126,19,206,156)(10,157,207,20,127,217,30,137,227,40,147,237)(41,187,103,71,177,93,61,167,83,51,197,113)(42,114,198,52,84,168,62,94,178,72,104,188)(43,189,105,73,179,95,63,169,85,53,199,115)(44,116,200,54,86,170,64,96,180,74,106,190)(45,191,107,75,181,97,65,171,87,55,161,117)(46,118,162,56,88,172,66,98,182,76,108,192)(47,193,109,77,183,99,67,173,89,57,163,119)(48,120,164,58,90,174,68,100,184,78,110,194)(49,195,111,79,185,101,69,175,91,59,165,81)(50,82,166,60,92,176,70,102,186,80,112,196), (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), (1,162,31,172,21,182,11,192)(2,171,12,161,22,191,32,181)(3,200,33,170,23,180,13,190)(4,169,14,199,24,189,34,179)(5,198,35,168,25,178,15,188)(6,167,16,197,26,187,36,177)(7,196,37,166,27,176,17,186)(8,165,18,195,28,185,38,175)(9,194,39,164,29,174,19,184)(10,163,20,193,30,183,40,173)(41,123,71,133,61,143,51,153)(42,132,52,122,62,152,72,142)(43,121,73,131,63,141,53,151)(44,130,54,160,64,150,74,140)(45,159,75,129,65,139,55,149)(46,128,56,158,66,148,76,138)(47,157,77,127,67,137,57,147)(48,126,58,156,68,146,78,136)(49,155,79,125,69,135,59,145)(50,124,60,154,70,144,80,134)(81,225,111,235,101,205,91,215)(82,234,92,224,102,214,112,204)(83,223,113,233,103,203,93,213)(84,232,94,222,104,212,114,202)(85,221,115,231,105,201,95,211)(86,230,96,220,106,210,116,240)(87,219,117,229,107,239,97,209)(88,228,98,218,108,208,118,238)(89,217,119,227,109,237,99,207)(90,226,100,216,110,206,120,236)>;
G:=Group( (1,228,138,31,218,128,21,208,158,11,238,148)(2,149,239,12,159,209,22,129,219,32,139,229)(3,230,140,33,220,130,23,210,160,13,240,150)(4,151,201,14,121,211,24,131,221,34,141,231)(5,232,142,35,222,132,25,212,122,15,202,152)(6,153,203,16,123,213,26,133,223,36,143,233)(7,234,144,37,224,134,27,214,124,17,204,154)(8,155,205,18,125,215,28,135,225,38,145,235)(9,236,146,39,226,136,29,216,126,19,206,156)(10,157,207,20,127,217,30,137,227,40,147,237)(41,187,103,71,177,93,61,167,83,51,197,113)(42,114,198,52,84,168,62,94,178,72,104,188)(43,189,105,73,179,95,63,169,85,53,199,115)(44,116,200,54,86,170,64,96,180,74,106,190)(45,191,107,75,181,97,65,171,87,55,161,117)(46,118,162,56,88,172,66,98,182,76,108,192)(47,193,109,77,183,99,67,173,89,57,163,119)(48,120,164,58,90,174,68,100,184,78,110,194)(49,195,111,79,185,101,69,175,91,59,165,81)(50,82,166,60,92,176,70,102,186,80,112,196), (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), (1,162,31,172,21,182,11,192)(2,171,12,161,22,191,32,181)(3,200,33,170,23,180,13,190)(4,169,14,199,24,189,34,179)(5,198,35,168,25,178,15,188)(6,167,16,197,26,187,36,177)(7,196,37,166,27,176,17,186)(8,165,18,195,28,185,38,175)(9,194,39,164,29,174,19,184)(10,163,20,193,30,183,40,173)(41,123,71,133,61,143,51,153)(42,132,52,122,62,152,72,142)(43,121,73,131,63,141,53,151)(44,130,54,160,64,150,74,140)(45,159,75,129,65,139,55,149)(46,128,56,158,66,148,76,138)(47,157,77,127,67,137,57,147)(48,126,58,156,68,146,78,136)(49,155,79,125,69,135,59,145)(50,124,60,154,70,144,80,134)(81,225,111,235,101,205,91,215)(82,234,92,224,102,214,112,204)(83,223,113,233,103,203,93,213)(84,232,94,222,104,212,114,202)(85,221,115,231,105,201,95,211)(86,230,96,220,106,210,116,240)(87,219,117,229,107,239,97,209)(88,228,98,218,108,208,118,238)(89,217,119,227,109,237,99,207)(90,226,100,216,110,206,120,236) );
G=PermutationGroup([[(1,228,138,31,218,128,21,208,158,11,238,148),(2,149,239,12,159,209,22,129,219,32,139,229),(3,230,140,33,220,130,23,210,160,13,240,150),(4,151,201,14,121,211,24,131,221,34,141,231),(5,232,142,35,222,132,25,212,122,15,202,152),(6,153,203,16,123,213,26,133,223,36,143,233),(7,234,144,37,224,134,27,214,124,17,204,154),(8,155,205,18,125,215,28,135,225,38,145,235),(9,236,146,39,226,136,29,216,126,19,206,156),(10,157,207,20,127,217,30,137,227,40,147,237),(41,187,103,71,177,93,61,167,83,51,197,113),(42,114,198,52,84,168,62,94,178,72,104,188),(43,189,105,73,179,95,63,169,85,53,199,115),(44,116,200,54,86,170,64,96,180,74,106,190),(45,191,107,75,181,97,65,171,87,55,161,117),(46,118,162,56,88,172,66,98,182,76,108,192),(47,193,109,77,183,99,67,173,89,57,163,119),(48,120,164,58,90,174,68,100,184,78,110,194),(49,195,111,79,185,101,69,175,91,59,165,81),(50,82,166,60,92,176,70,102,186,80,112,196)], [(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)], [(1,162,31,172,21,182,11,192),(2,171,12,161,22,191,32,181),(3,200,33,170,23,180,13,190),(4,169,14,199,24,189,34,179),(5,198,35,168,25,178,15,188),(6,167,16,197,26,187,36,177),(7,196,37,166,27,176,17,186),(8,165,18,195,28,185,38,175),(9,194,39,164,29,174,19,184),(10,163,20,193,30,183,40,173),(41,123,71,133,61,143,51,153),(42,132,52,122,62,152,72,142),(43,121,73,131,63,141,53,151),(44,130,54,160,64,150,74,140),(45,159,75,129,65,139,55,149),(46,128,56,158,66,148,76,138),(47,157,77,127,67,137,57,147),(48,126,58,156,68,146,78,136),(49,155,79,125,69,135,59,145),(50,124,60,154,70,144,80,134),(81,225,111,235,101,205,91,215),(82,234,92,224,102,214,112,204),(83,223,113,233,103,203,93,213),(84,232,94,222,104,212,114,202),(85,221,115,231,105,201,95,211),(86,230,96,220,106,210,116,240),(87,219,117,229,107,239,97,209),(88,228,98,218,108,208,118,238),(89,217,119,227,109,237,99,207),(90,226,100,216,110,206,120,236)]])
57 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 5A | 5B | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 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 | 3 | 4 | 4 | 4 | 4 | 5 | 5 | 6 | 6 | 6 | 8 | 8 | 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 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 60 | 60 | 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 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | + | + | - | + | - | + | - | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D5 | Dic3 | D6 | D10 | C3⋊D4 | D20 | C5⋊D4 | C4×D5 | C4.10D4 | S3×D5 | C12.10D4 | C15⋊D4 | C3⋊D20 | D5×Dic3 | C4.12D20 | C12.6D20 |
| kernel | C12.6D20 | C5×C4.Dic3 | C60.7C4 | C6×Dic10 | C6×Dic5 | C2×Dic10 | C60 | C4.Dic3 | C2×Dic5 | C2×C20 | C2×C12 | C20 | C12 | C12 | C2×C6 | C15 | C2×C4 | C5 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 8 |
Matrix representation of C12.6D20 ►in GL6(𝔽241)
| 226 | 0 | 0 | 0 | 0 | 0 |
| 0 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 197 | 238 | 0 | 0 |
| 0 | 0 | 3 | 44 | 0 | 0 |
| 0 | 0 | 0 | 0 | 44 | 3 |
| 0 | 0 | 0 | 0 | 238 | 197 |
| 0 | 108 | 0 | 0 | 0 | 0 |
| 212 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 240 | 51 |
| 0 | 0 | 3 | 44 | 0 | 0 |
| 0 | 0 | 197 | 78 | 0 | 0 |
| 0 | 108 | 0 | 0 | 0 | 0 |
| 29 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 35 | 66 |
| 0 | 0 | 0 | 0 | 164 | 206 |
| 0 | 0 | 104 | 148 | 0 | 0 |
| 0 | 0 | 150 | 137 | 0 | 0 |
G:=sub<GL(6,GF(241))| [226,0,0,0,0,0,0,16,0,0,0,0,0,0,197,3,0,0,0,0,238,44,0,0,0,0,0,0,44,238,0,0,0,0,3,197],[0,212,0,0,0,0,108,0,0,0,0,0,0,0,0,0,3,197,0,0,0,0,44,78,0,0,0,240,0,0,0,0,1,51,0,0],[0,29,0,0,0,0,108,0,0,0,0,0,0,0,0,0,104,150,0,0,0,0,148,137,0,0,35,164,0,0,0,0,66,206,0,0] >;
C12.6D20 in GAP, Magma, Sage, TeX
C_{12}._6D_{20} % in TeX
G:=Group("C12.6D20"); // GroupNames label
G:=SmallGroup(480,37);
// by ID
G=gap.SmallGroup(480,37);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,112,141,36,422,100,346,1356,18822]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^20=a^6,c^2=a^3,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^3*b^19>;
// generators/relations