the group $\mathrm{C}_{15}$
- GAP SmallGroup id
- $[15, 1]$
- order
- $15$ (cyclic)
- structure
- $\mathrm{C}_{15}$
- presentation
- $\langle g_{1}, g_{2} \mid g_{1}^{3},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{2}^{5} \rangle$
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 | 5 | 3 | 15 | 5 | 15 | 15 | 5 | 15 | 15 | 5 | 15 | 15 | 15 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $\chi_{ 3 }$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $\chi_{ 4 }$ | $1$ | $1$ | $\zeta_{5}^{2}$ | $1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{3}$ |
| $\chi_{ 5 }$ | $1$ | $\zeta_{3}$ | $\zeta_{5}^{2}$ | $-\zeta_{3} - 1$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{15}$ | $\zeta_{15}^{2}$ | $\zeta_{5}$ | $\zeta_{15}^{7}$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $\zeta_{5}^{3}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $\zeta_{15}^{4}$ |
| $\chi_{ 6 }$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{3}$ | $\zeta_{15}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $\zeta_{15}^{7}$ | $\zeta_{5}$ | $\zeta_{15}^{2}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $\zeta_{5}^{3}$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $\zeta_{15}^{4}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ |
| $\chi_{ 7 }$ | $1$ | $1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}$ |
| $\chi_{ 8 }$ | $1$ | $\zeta_{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{15}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{15}^{7}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $\zeta_{5}^{2}$ | $\zeta_{15}^{4}$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $\zeta_{5}$ | $\zeta_{15}$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ |
| $\chi_{ 9 }$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{3}$ | $\zeta_{15}^{7}$ | $\zeta_{5}^{3}$ | $\zeta_{15}^{2}$ | $\zeta_{15}^{4}$ | $\zeta_{5}^{2}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $\zeta_{15}$ | $\zeta_{5}$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ |
| $\chi_{ 10 }$ | $1$ | $1$ | $\zeta_{5}$ | $1$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $\chi_{ 11 }$ | $1$ | $\zeta_{3}$ | $\zeta_{5}$ | $-\zeta_{3} - 1$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $\zeta_{5}^{2}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $\zeta_{5}^{3}$ | $\zeta_{15}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{15}^{4}$ | $\zeta_{15}^{2}$ | $\zeta_{15}^{7}$ |
| $\chi_{ 12 }$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{5}$ | $\zeta_{3}$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $\zeta_{5}^{2}$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $\zeta_{15}$ | $\zeta_{5}^{3}$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $\zeta_{15}^{4}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $\zeta_{15}^{7}$ | $\zeta_{15}^{2}$ |
| $\chi_{ 13 }$ | $1$ | $1$ | $\zeta_{5}^{3}$ | $1$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{2}$ |
| $\chi_{ 14 }$ | $1$ | $\zeta_{3}$ | $\zeta_{5}^{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $\zeta_{5}$ | $\zeta_{15}^{4}$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $\zeta_{15}^{2}$ | $\zeta_{5}^{2}$ | $\zeta_{15}^{7}$ | $-\zeta_{15}^{6} - \zeta_{15}$ | $\zeta_{15}$ |
| $\chi_{ 15 }$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{3}$ | $\zeta_{15}^{4}$ | $\zeta_{5}$ | $-\zeta_{15}^{7} + \zeta_{15}^{6} - \zeta_{15}^{4} + \zeta_{15}^{3} - \zeta_{15}^{2} + 1$ | $-\zeta_{15}^{7} + \zeta_{15}^{5} - \zeta_{15}^{4} - \zeta_{15} + 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{15}^{7} - \zeta_{15}^{5} + \zeta_{15}^{4} - \zeta_{15}^{3} + \zeta_{15} - 1$ | $\zeta_{15}^{7}$ | $\zeta_{5}^{2}$ | $\zeta_{15}^{2}$ | $\zeta_{15}$ | $-\zeta_{15}^{6} - \zeta_{15}$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, \zeta_{3}, 1, 1)$$\operatorname{diag}(1, 1, \zeta_{5}, \zeta_{5}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 15$, moduli $1$