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