metacyclic, supersoluble, monomial, Z-group
Aliases: C57.C6, C19⋊3C18, D19⋊2C9, C19⋊2C9⋊C2, C3.(C19⋊C6), (C3×D19).C3, SmallGroup(342,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C19 — C57.C6 |
Generators and relations for C57.C6
G = < a,b | a57=1, b6=a19, bab-1=a31 >
Character table of C57.C6
| class | 1 | 2 | 3A | 3B | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | 19A | 19B | 19C | 57A | 57B | 57C | 57D | 57E | 57F | |
| size | 1 | 19 | 1 | 1 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 19 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ97 | ζ92 | ζ95 | ζ94 | ζ9 | ζ98 | ζ95 | ζ94 | ζ97 | ζ9 | ζ98 | ζ92 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 9 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ94 | ζ95 | ζ98 | ζ9 | ζ97 | ζ92 | ζ98 | ζ9 | ζ94 | ζ97 | ζ92 | ζ95 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 9 |
| ρ9 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ98 | ζ9 | ζ97 | ζ92 | ζ95 | ζ94 | -ζ97 | -ζ92 | -ζ98 | -ζ95 | -ζ94 | -ζ9 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 18 |
| ρ10 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ94 | ζ95 | ζ98 | ζ9 | ζ97 | ζ92 | -ζ98 | -ζ9 | -ζ94 | -ζ97 | -ζ92 | -ζ95 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 18 |
| ρ11 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ98 | ζ9 | ζ97 | ζ92 | ζ95 | ζ94 | ζ97 | ζ92 | ζ98 | ζ95 | ζ94 | ζ9 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 9 |
| ρ12 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ95 | ζ94 | ζ9 | ζ98 | ζ92 | ζ97 | ζ9 | ζ98 | ζ95 | ζ92 | ζ97 | ζ94 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 9 |
| ρ13 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ92 | ζ97 | ζ94 | ζ95 | ζ98 | ζ9 | -ζ94 | -ζ95 | -ζ92 | -ζ98 | -ζ9 | -ζ97 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 18 |
| ρ14 | 1 | -1 | ζ3 | ζ32 | ζ65 | ζ6 | ζ95 | ζ94 | ζ9 | ζ98 | ζ92 | ζ97 | -ζ9 | -ζ98 | -ζ95 | -ζ92 | -ζ97 | -ζ94 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 18 |
| ρ15 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ9 | ζ98 | ζ92 | ζ97 | ζ94 | ζ95 | ζ92 | ζ97 | ζ9 | ζ94 | ζ95 | ζ98 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 9 |
| ρ16 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ97 | ζ92 | ζ95 | ζ94 | ζ9 | ζ98 | -ζ95 | -ζ94 | -ζ97 | -ζ9 | -ζ98 | -ζ92 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 18 |
| ρ17 | 1 | -1 | ζ32 | ζ3 | ζ6 | ζ65 | ζ9 | ζ98 | ζ92 | ζ97 | ζ94 | ζ95 | -ζ92 | -ζ97 | -ζ9 | -ζ94 | -ζ95 | -ζ98 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 18 |
| ρ18 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ92 | ζ97 | ζ94 | ζ95 | ζ98 | ζ9 | ζ94 | ζ95 | ζ92 | ζ98 | ζ9 | ζ97 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 9 |
| ρ19 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | orthogonal lifted from C19⋊C6 |
| ρ20 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | orthogonal lifted from C19⋊C6 |
| ρ21 | 6 | 0 | 6 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | orthogonal lifted from C19⋊C6 |
| ρ22 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ3ζ1917+ζ3ζ1916+ζ3ζ1914+ζ3ζ195+ζ3ζ193+ζ3ζ192 | ζ3ζ1915+ζ3ζ1913+ζ3ζ1910+ζ3ζ199+ζ3ζ196+ζ3ζ194 | ζ3ζ1918+ζ3ζ1912+ζ3ζ1911+ζ3ζ198+ζ3ζ197+ζ3ζ19 | ζ32ζ1918+ζ32ζ1912+ζ32ζ1911+ζ32ζ198+ζ32ζ197+ζ32ζ19 | ζ32ζ1917+ζ32ζ1916+ζ32ζ1914+ζ32ζ195+ζ32ζ193+ζ32ζ192 | ζ32ζ1915+ζ32ζ1913+ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194 | complex faithful |
| ρ23 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ3ζ1915+ζ3ζ1913+ζ3ζ1910+ζ3ζ199+ζ3ζ196+ζ3ζ194 | ζ3ζ1918+ζ3ζ1912+ζ3ζ1911+ζ3ζ198+ζ3ζ197+ζ3ζ19 | ζ3ζ1917+ζ3ζ1916+ζ3ζ1914+ζ3ζ195+ζ3ζ193+ζ3ζ192 | ζ32ζ1917+ζ32ζ1916+ζ32ζ1914+ζ32ζ195+ζ32ζ193+ζ32ζ192 | ζ32ζ1915+ζ32ζ1913+ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194 | ζ32ζ1918+ζ32ζ1912+ζ32ζ1911+ζ32ζ198+ζ32ζ197+ζ32ζ19 | complex faithful |
| ρ24 | 6 | 0 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ3ζ1918+ζ3ζ1912+ζ3ζ1911+ζ3ζ198+ζ3ζ197+ζ3ζ19 | ζ3ζ1917+ζ3ζ1916+ζ3ζ1914+ζ3ζ195+ζ3ζ193+ζ3ζ192 | ζ3ζ1915+ζ3ζ1913+ζ3ζ1910+ζ3ζ199+ζ3ζ196+ζ3ζ194 | ζ32ζ1915+ζ32ζ1913+ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194 | ζ32ζ1918+ζ32ζ1912+ζ32ζ1911+ζ32ζ198+ζ32ζ197+ζ32ζ19 | ζ32ζ1917+ζ32ζ1916+ζ32ζ1914+ζ32ζ195+ζ32ζ193+ζ32ζ192 | complex faithful |
| ρ25 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ32ζ1915+ζ32ζ1913+ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194 | ζ32ζ1918+ζ32ζ1912+ζ32ζ1911+ζ32ζ198+ζ32ζ197+ζ32ζ19 | ζ32ζ1917+ζ32ζ1916+ζ32ζ1914+ζ32ζ195+ζ32ζ193+ζ32ζ192 | ζ3ζ1917+ζ3ζ1916+ζ3ζ1914+ζ3ζ195+ζ3ζ193+ζ3ζ192 | ζ3ζ1915+ζ3ζ1913+ζ3ζ1910+ζ3ζ199+ζ3ζ196+ζ3ζ194 | ζ3ζ1918+ζ3ζ1912+ζ3ζ1911+ζ3ζ198+ζ3ζ197+ζ3ζ19 | complex faithful |
| ρ26 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ32ζ1917+ζ32ζ1916+ζ32ζ1914+ζ32ζ195+ζ32ζ193+ζ32ζ192 | ζ32ζ1915+ζ32ζ1913+ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194 | ζ32ζ1918+ζ32ζ1912+ζ32ζ1911+ζ32ζ198+ζ32ζ197+ζ32ζ19 | ζ3ζ1918+ζ3ζ1912+ζ3ζ1911+ζ3ζ198+ζ3ζ197+ζ3ζ19 | ζ3ζ1917+ζ3ζ1916+ζ3ζ1914+ζ3ζ195+ζ3ζ193+ζ3ζ192 | ζ3ζ1915+ζ3ζ1913+ζ3ζ1910+ζ3ζ199+ζ3ζ196+ζ3ζ194 | complex faithful |
| ρ27 | 6 | 0 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1917+ζ1916+ζ1914+ζ195+ζ193+ζ192 | ζ1915+ζ1913+ζ1910+ζ199+ζ196+ζ194 | ζ1918+ζ1912+ζ1911+ζ198+ζ197+ζ19 | ζ32ζ1918+ζ32ζ1912+ζ32ζ1911+ζ32ζ198+ζ32ζ197+ζ32ζ19 | ζ32ζ1917+ζ32ζ1916+ζ32ζ1914+ζ32ζ195+ζ32ζ193+ζ32ζ192 | ζ32ζ1915+ζ32ζ1913+ζ32ζ1910+ζ32ζ199+ζ32ζ196+ζ32ζ194 | ζ3ζ1915+ζ3ζ1913+ζ3ζ1910+ζ3ζ199+ζ3ζ196+ζ3ζ194 | ζ3ζ1918+ζ3ζ1912+ζ3ζ1911+ζ3ζ198+ζ3ζ197+ζ3ζ19 | ζ3ζ1917+ζ3ζ1916+ζ3ζ1914+ζ3ζ195+ζ3ζ193+ζ3ζ192 | complex faithful |
►On 171 points
Generators in S171
(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)
(1 60 161 39 98 142 20 79 123)(2 106 168 19 90 116 21 68 130 38 109 135 40 87 149 57 71 154)(3 95 118 56 82 147 22 114 137 18 101 166 41 76 156 37 63 128)(4 84 125 36 74 121 23 103 144 55 93 140 42 65 163 17 112 159)(5 73 132 16 66 152 24 92 151 35 85 171 43 111 170 54 104 133)(6 62 139 53 58 126 25 81 158 15 77 145 44 100 120 34 96 164)(7 108 146 33 107 157 26 70 165 52 69 119 45 89 127 14 88 138)(8 97 153 13 99 131 27 59 115 32 61 150 46 78 134 51 80 169)(9 86 160 50 91 162 28 105 122 12 110 124 47 67 141 31 72 143)(10 75 167 30 83 136 29 94 129 49 102 155 48 113 148 11 64 117)
G:=sub<Sym(171)| (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), (1,60,161,39,98,142,20,79,123)(2,106,168,19,90,116,21,68,130,38,109,135,40,87,149,57,71,154)(3,95,118,56,82,147,22,114,137,18,101,166,41,76,156,37,63,128)(4,84,125,36,74,121,23,103,144,55,93,140,42,65,163,17,112,159)(5,73,132,16,66,152,24,92,151,35,85,171,43,111,170,54,104,133)(6,62,139,53,58,126,25,81,158,15,77,145,44,100,120,34,96,164)(7,108,146,33,107,157,26,70,165,52,69,119,45,89,127,14,88,138)(8,97,153,13,99,131,27,59,115,32,61,150,46,78,134,51,80,169)(9,86,160,50,91,162,28,105,122,12,110,124,47,67,141,31,72,143)(10,75,167,30,83,136,29,94,129,49,102,155,48,113,148,11,64,117)>;
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), (1,60,161,39,98,142,20,79,123)(2,106,168,19,90,116,21,68,130,38,109,135,40,87,149,57,71,154)(3,95,118,56,82,147,22,114,137,18,101,166,41,76,156,37,63,128)(4,84,125,36,74,121,23,103,144,55,93,140,42,65,163,17,112,159)(5,73,132,16,66,152,24,92,151,35,85,171,43,111,170,54,104,133)(6,62,139,53,58,126,25,81,158,15,77,145,44,100,120,34,96,164)(7,108,146,33,107,157,26,70,165,52,69,119,45,89,127,14,88,138)(8,97,153,13,99,131,27,59,115,32,61,150,46,78,134,51,80,169)(9,86,160,50,91,162,28,105,122,12,110,124,47,67,141,31,72,143)(10,75,167,30,83,136,29,94,129,49,102,155,48,113,148,11,64,117) );
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)], [(1,60,161,39,98,142,20,79,123),(2,106,168,19,90,116,21,68,130,38,109,135,40,87,149,57,71,154),(3,95,118,56,82,147,22,114,137,18,101,166,41,76,156,37,63,128),(4,84,125,36,74,121,23,103,144,55,93,140,42,65,163,17,112,159),(5,73,132,16,66,152,24,92,151,35,85,171,43,111,170,54,104,133),(6,62,139,53,58,126,25,81,158,15,77,145,44,100,120,34,96,164),(7,108,146,33,107,157,26,70,165,52,69,119,45,89,127,14,88,138),(8,97,153,13,99,131,27,59,115,32,61,150,46,78,134,51,80,169),(9,86,160,50,91,162,28,105,122,12,110,124,47,67,141,31,72,143),(10,75,167,30,83,136,29,94,129,49,102,155,48,113,148,11,64,117)]])
Matrix representation of C57.C6 ►in GL7(𝔽2053)
| 1855 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 131 | 1454 | 1466 | 1456 | 863 | 1453 |
| 0 | 600 | 589 | 1329 | 1910 | 1331 | 1321 |
| 0 | 732 | 1323 | 1463 | 721 | 731 | 1 |
| 0 | 2052 | 2052 | 1922 | 600 | 1320 | 2051 |
| 0 | 2 | 1465 | 854 | 1595 | 1455 | 733 |
| 0 | 1320 | 599 | 1190 | 599 | 1320 | 2052 |
| 276 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1044 | 298 | 1367 | 125 | 928 | 278 |
| 0 | 614 | 1222 | 933 | 933 | 1222 | 614 |
| 0 | 278 | 928 | 125 | 1367 | 298 | 1044 |
| 0 | 586 | 2044 | 953 | 366 | 1037 | 1238 |
| 0 | 1543 | 286 | 666 | 1773 | 1750 | 1286 |
| 0 | 0 | 1652 | 1421 | 515 | 1421 | 1652 |
G:=sub<GL(7,GF(2053))| [1855,0,0,0,0,0,0,0,131,600,732,2052,2,1320,0,1454,589,1323,2052,1465,599,0,1466,1329,1463,1922,854,1190,0,1456,1910,721,600,1595,599,0,863,1331,731,1320,1455,1320,0,1453,1321,1,2051,733,2052],[276,0,0,0,0,0,0,0,1044,614,278,586,1543,0,0,298,1222,928,2044,286,1652,0,1367,933,125,953,666,1421,0,125,933,1367,366,1773,515,0,928,1222,298,1037,1750,1421,0,278,614,1044,1238,1286,1652] >;
C57.C6 in GAP, Magma, Sage, TeX
C_{57}.C_6 % in TeX
G:=Group("C57.C6"); // GroupNames label
G:=SmallGroup(342,1);
// by ID
G=gap.SmallGroup(342,1);
# by ID
G:=PCGroup([4,-2,-3,-3,-19,29,5187,1015]);
// Polycyclic
G:=Group<a,b|a^57=1,b^6=a^19,b*a*b^-1=a^31>;
// generators/relations
Export
Subgroup lattice of C57.C6 in TeX
Character table of C57.C6 in TeX