the group $\mathrm{C}_{18}$
- GAP SmallGroup id
- $[18, 2]$
- order
- $18$ (cyclic)
- structure
- $\mathrm{C}_{18}$
- presentation
- $\langle g_{1}, g_{2}, g_{3} \mid g_{1}^{2},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{3}^{-1}g_{1}^{-1}g_{3}g_{1},\; g_{2}^{3}g_{3}^{-1},\; g_{3}^{-1}g_{2}^{-1}g_{3}g_{2},\; g_{3}^{3} \rangle$
character table
Columns are the conjugacy classes: order is the order of a class representative, size the number of elements in the class. Rows $\chi_i$ are the irreducible characters.
| order | 1 | 2 | 9 | 3 | 18 | 6 | 9 | 9 | 3 | 18 | 18 | 6 | 9 | 9 | 18 | 18 | 9 | 18 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $-1$ |
| $\chi_{ 3 }$ | $1$ | $1$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{3}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{3}$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{2}$ |
| $\chi_{ 4 }$ | $1$ | $-1$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{3}$ | $\zeta_{9}^{4} + \zeta_{9}$ | $-\zeta_{3}$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{5}$ | $-\zeta_{9}$ | $\zeta_{3} + 1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{5} + \zeta_{9}^{2}$ | $-\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $-\zeta_{9}^{2}$ |
| $\chi_{ 5 }$ | $1$ | $1$ | $\zeta_{9}^{5}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{5}$ | $-\zeta_{3} - 1$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{3}$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{3}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{4}$ |
| $\chi_{ 6 }$ | $1$ | $-1$ | $\zeta_{9}^{5}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{5}$ | $\zeta_{3} + 1$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{3}$ | $-\zeta_{9}$ | $-\zeta_{9}^{2}$ | $-\zeta_{3}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4} + \zeta_{9}$ | $\zeta_{9}^{5} + \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{4}$ |
| $\chi_{ 7 }$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 8 }$ | $1$ | $-1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3}$ | $-1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $-1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3} + 1$ |
| $\chi_{ 9 }$ | $1$ | $1$ | $\zeta_{9}$ | $\zeta_{3}$ | $\zeta_{9}$ | $\zeta_{3}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $\chi_{ 10 }$ | $1$ | $-1$ | $\zeta_{9}$ | $\zeta_{3}$ | $-\zeta_{9}$ | $-\zeta_{3}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{2}$ | $-\zeta_{9}^{4}$ | $\zeta_{3} + 1$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5}$ | $\zeta_{9}^{4} + \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{5} + \zeta_{9}^{2}$ |
| $\chi_{ 11 }$ | $1$ | $1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{3}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{3}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}$ |
| $\chi_{ 12 }$ | $1$ | $-1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{5} + \zeta_{9}^{2}$ | $\zeta_{3} + 1$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{3}$ | $\zeta_{9}^{4} + \zeta_{9}$ | $-\zeta_{9}^{5}$ | $-\zeta_{3}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $-\zeta_{9}^{4}$ | $-\zeta_{9}^{2}$ | $\zeta_{9}$ | $-\zeta_{9}$ |
| $\chi_{ 13 }$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 14 }$ | $1$ | $-1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3} + 1$ | $-1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $-1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $\zeta_{3}$ | $-\zeta_{3}$ |
| $\chi_{ 15 }$ | $1$ | $1$ | $\zeta_{9}^{4}$ | $\zeta_{3}$ | $\zeta_{9}^{4}$ | $\zeta_{3}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{9}^{5}$ |
| $\chi_{ 16 }$ | $1$ | $-1$ | $\zeta_{9}^{4}$ | $\zeta_{3}$ | $-\zeta_{9}^{4}$ | $-\zeta_{3}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{5} + \zeta_{9}^{2}$ | $\zeta_{9}^{4} + \zeta_{9}$ | $\zeta_{3} + 1$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $-\zeta_{9}^{2}$ | $-\zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{5}$ |
| $\chi_{ 17 }$ | $1$ | $1$ | $\zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{3}$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{3}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $\chi_{ 18 }$ | $1$ | $-1$ | $\zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{2}$ | $\zeta_{3} + 1$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{3}$ | $-\zeta_{9}^{4}$ | $\zeta_{9}^{5} + \zeta_{9}^{2}$ | $-\zeta_{3}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}$ | $-\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{4} + \zeta_{9}$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, -1, -1, -1)$$\operatorname{diag}(1, \zeta_{9}, \zeta_{9}^{2}, \zeta_{9}^{4})$ with $q = 1$, $\operatorname{ord} \omega_X = 18$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, -1, -1, -1)$$\operatorname{diag}(1, \zeta_{9}, \zeta_{9}^{4}, \zeta_{9}^{7})$ with $q = 1$, $\operatorname{ord} \omega_X = 6$, moduli $1$