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