the group $\mathrm{C}_{14}$
- GAP SmallGroup id
- $[14, 2]$
- order
- $14$ (cyclic)
- structure
- $\mathrm{C}_{14}$
- presentation
- $\langle g_{1}, g_{2} \mid g_{1}^{2},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{2}^{7} \rangle$
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 | 2 | 7 | 14 | 7 | 14 | 7 | 14 | 7 | 14 | 7 | 14 | 7 | 14 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ | $1$ | $-1$ |
| $\chi_{ 3 }$ | $1$ | $1$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{4}$ | $\zeta_{7}$ | $\zeta_{7}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{3}$ |
| $\chi_{ 4 }$ | $1$ | $-1$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{4}$ | $\zeta_{7}$ | $-\zeta_{7}$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5}$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{2}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{3}$ |
| $\chi_{ 5 }$ | $1$ | $1$ | $\zeta_{7}$ | $\zeta_{7}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
| $\chi_{ 6 }$ | $1$ | $-1$ | $\zeta_{7}$ | $-\zeta_{7}$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{2}$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{3}$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{4}$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ |
| $\chi_{ 7 }$ | $1$ | $1$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{3}$ | $\zeta_{7}$ | $\zeta_{7}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{2}$ |
| $\chi_{ 8 }$ | $1$ | $-1$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5}$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{3}$ | $\zeta_{7}$ | $-\zeta_{7}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{4}$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{2}$ |
| $\chi_{ 9 }$ | $1$ | $1$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}$ | $\zeta_{7}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{5}$ |
| $\chi_{ 10 }$ | $1$ | $-1$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{2}$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{4}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ | $\zeta_{7}$ | $-\zeta_{7}$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{3}$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5}$ |
| $\chi_{ 11 }$ | $1$ | $1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{2}$ | $\zeta_{7}$ | $\zeta_{7}$ |
| $\chi_{ 12 }$ | $1$ | $-1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5}$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{4}$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{3}$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{2}$ | $\zeta_{7}$ | $-\zeta_{7}$ |
| $\chi_{ 13 }$ | $1$ | $1$ | $\zeta_{7}^{3}$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{2}$ | $\zeta_{7}^{5}$ | $\zeta_{7}^{5}$ | $\zeta_{7}$ | $\zeta_{7}$ | $\zeta_{7}^{4}$ | $\zeta_{7}^{4}$ |
| $\chi_{ 14 }$ | $1$ | $-1$ | $\zeta_{7}^{3}$ | $-\zeta_{7}^{3}$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ | $\zeta_{7}^{5} + \zeta_{7}^{4} + \zeta_{7}^{3} + \zeta_{7}^{2} + \zeta_{7} + 1$ | $\zeta_{7}^{2}$ | $-\zeta_{7}^{2}$ | $\zeta_{7}^{5}$ | $-\zeta_{7}^{5}$ | $\zeta_{7}$ | $-\zeta_{7}$ | $\zeta_{7}^{4}$ | $-\zeta_{7}^{4}$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, -1, -1, -1)$$\operatorname{diag}(1, \zeta_{7}, \zeta_{7}^{2}, \zeta_{7}^{3})$ with $q = 1$, $\operatorname{ord} \omega_X = 14$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, -1, -1, -1)$$\operatorname{diag}(1, \zeta_{7}, \zeta_{7}^{2}, \zeta_{7}^{4})$ with $q = 1$, $\operatorname{ord} \omega_X = 2$, moduli $1$