the group $\mathrm{C}_{9}$
- GAP SmallGroup id
- $[9, 1]$
- order
- $9$ (cyclic)
- structure
- $\mathrm{C}_{9}$
- presentation
- $\langle g_{1}, g_{2} \mid g_{1}^{3}g_{2}^{-1},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{2}^{3} \rangle$
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 | 9 | 3 | 9 | 9 | 3 | 9 | 9 | 9 |
|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $\zeta_{9}^{5}$ | $-\zeta_{3} - 1$ | $\zeta_{9}$ | $\zeta_{9}^{2}$ | $\zeta_{3}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ |
| $\chi_{ 3 }$ | $1$ | $\zeta_{9}$ | $\zeta_{3}$ | $\zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ |
| $\chi_{ 4 }$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $\chi_{ 5 }$ | $1$ | $\zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{4}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{3}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ | $-\zeta_{9}^{4} - \zeta_{9}$ |
| $\chi_{ 6 }$ | $1$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{3}$ | $\zeta_{9}^{5}$ | $\zeta_{9}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ |
| $\chi_{ 7 }$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $\chi_{ 8 }$ | $1$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{3} - 1$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $\zeta_{9}^{5}$ | $\zeta_{3}$ | $\zeta_{9}^{4}$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ |
| $\chi_{ 9 }$ | $1$ | $\zeta_{9}^{4}$ | $\zeta_{3}$ | $-\zeta_{9}^{5} - \zeta_{9}^{2}$ | $-\zeta_{9}^{4} - \zeta_{9}$ | $-\zeta_{3} - 1$ | $\zeta_{9}^{2}$ | $\zeta_{9}$ | $\zeta_{9}^{5}$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, \zeta_{9}, \zeta_{9}^{2}, \zeta_{9}^{4})$ with $q = 1$, $\operatorname{ord} \omega_X = 9$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, \zeta_{9}, \zeta_{9}^{4}, \zeta_{9}^{7})$ with $q = 1$, $\operatorname{ord} \omega_X = 3$, moduli $1$