the group $\mathrm{C}_{20}$
- GAP SmallGroup id
- $[20, 2]$
- order
- $20$ (cyclic)
- structure
- $\mathrm{C}_{20}$
- presentation
- $\langle g_{1}, g_{2}, g_{3} \mid g_{1}^{2}g_{3}^{-1},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{3}^{-1}g_{1}^{-1}g_{3}g_{1},\; g_{2}^{5},\; g_{3}^{-1}g_{2}^{-1}g_{3}g_{2},\; g_{3}^{2} \rangle$
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 | 4 | 5 | 2 | 20 | 4 | 5 | 10 | 20 | 20 | 5 | 10 | 20 | 20 | 5 | 10 | 20 | 20 | 10 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $-\zeta_{4}$ | $1$ | $-1$ | $-\zeta_{4}$ | $\zeta_{4}$ | $1$ | $-1$ | $-\zeta_{4}$ | $\zeta_{4}$ | $1$ | $-1$ | $-\zeta_{4}$ | $\zeta_{4}$ | $1$ | $-1$ | $-\zeta_{4}$ | $\zeta_{4}$ | $-1$ | $\zeta_{4}$ |
| $\chi_{ 3 }$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $-1$ |
| $\chi_{ 4 }$ | $1$ | $\zeta_{4}$ | $1$ | $-1$ | $\zeta_{4}$ | $-\zeta_{4}$ | $1$ | $-1$ | $\zeta_{4}$ | $-\zeta_{4}$ | $1$ | $-1$ | $\zeta_{4}$ | $-\zeta_{4}$ | $1$ | $-1$ | $\zeta_{4}$ | $-\zeta_{4}$ | $-1$ | $-\zeta_{4}$ |
| $\chi_{ 5 }$ | $1$ | $1$ | $\zeta_{5}$ | $1$ | $\zeta_{5}$ | $1$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $\chi_{ 6 }$ | $1$ | $-\zeta_{4}$ | $\zeta_{5}$ | $-1$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $\zeta_{4}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}$ | $\zeta_{20}^{3}$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ | $\zeta_{20}^{7}$ | $-\zeta_{20}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{3}$ | $-\zeta_{20}$ | $-\zeta_{20}^{7}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{20}$ |
| $\chi_{ 7 }$ | $1$ | $-1$ | $\zeta_{5}$ | $1$ | $-\zeta_{5}$ | $-1$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
| $\chi_{ 8 }$ | $1$ | $\zeta_{4}$ | $\zeta_{5}$ | $-1$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $-\zeta_{4}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}$ | $-\zeta_{20}^{3}$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ | $-\zeta_{20}^{7}$ | $\zeta_{20}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{3}$ | $\zeta_{20}$ | $\zeta_{20}^{7}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{20}$ |
| $\chi_{ 9 }$ | $1$ | $1$ | $\zeta_{5}^{2}$ | $1$ | $\zeta_{5}^{2}$ | $1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{3}$ |
| $\chi_{ 10 }$ | $1$ | $-\zeta_{4}$ | $\zeta_{5}^{2}$ | $-1$ | $\zeta_{20}^{3}$ | $\zeta_{4}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{2}$ | $-\zeta_{20}$ | $-\zeta_{20}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $\zeta_{20}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}$ | $\zeta_{20}^{7}$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $-\zeta_{5}^{3}$ | $-\zeta_{20}^{7}$ |
| $\chi_{ 11 }$ | $1$ | $-1$ | $\zeta_{5}^{2}$ | $1$ | $-\zeta_{5}^{2}$ | $-1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}^{2}$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3}$ |
| $\chi_{ 12 }$ | $1$ | $\zeta_{4}$ | $\zeta_{5}^{2}$ | $-1$ | $-\zeta_{20}^{3}$ | $-\zeta_{4}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{2}$ | $\zeta_{20}$ | $\zeta_{20}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $-\zeta_{20}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}$ | $-\zeta_{20}^{7}$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $-\zeta_{5}^{3}$ | $\zeta_{20}^{7}$ |
| $\chi_{ 13 }$ | $1$ | $1$ | $\zeta_{5}^{3}$ | $1$ | $\zeta_{5}^{3}$ | $1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{2}$ |
| $\chi_{ 14 }$ | $1$ | $-\zeta_{4}$ | $\zeta_{5}^{3}$ | $-1$ | $\zeta_{20}^{7}$ | $\zeta_{4}$ | $\zeta_{5}$ | $-\zeta_{5}^{3}$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $-\zeta_{20}^{7}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}$ | $-\zeta_{20}$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{20}^{3}$ | $\zeta_{20}$ | $-\zeta_{5}^{2}$ | $-\zeta_{20}^{3}$ |
| $\chi_{ 15 }$ | $1$ | $-1$ | $\zeta_{5}^{3}$ | $1$ | $-\zeta_{5}^{3}$ | $-1$ | $\zeta_{5}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}$ | $-\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{5}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{2}$ |
| $\chi_{ 16 }$ | $1$ | $\zeta_{4}$ | $\zeta_{5}^{3}$ | $-1$ | $-\zeta_{20}^{7}$ | $-\zeta_{4}$ | $\zeta_{5}$ | $-\zeta_{5}^{3}$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $\zeta_{20}^{7}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}$ | $\zeta_{20}$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{20}^{3}$ | $-\zeta_{20}$ | $-\zeta_{5}^{2}$ | $\zeta_{20}^{3}$ |
| $\chi_{ 17 }$ | $1$ | $1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $\zeta_{5}$ | $\zeta_{5}$ |
| $\chi_{ 18 }$ | $1$ | $-\zeta_{4}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-1$ | $-\zeta_{20}$ | $\zeta_{4}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{20}^{7}$ | $\zeta_{20}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ | $\zeta_{20}^{3}$ | $-\zeta_{20}^{7}$ | $\zeta_{5}$ | $-\zeta_{5}^{2}$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ | $-\zeta_{20}^{3}$ | $-\zeta_{5}$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ |
| $\chi_{ 19 }$ | $1$ | $-1$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $1$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-1$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $\zeta_{5}^{2}$ | $\zeta_{5}^{3}$ | $-\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ | $\zeta_{5}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}$ | $-\zeta_{5}^{2}$ | $\zeta_{5}$ | $-\zeta_{5}$ |
| $\chi_{ 20 }$ | $1$ | $\zeta_{4}$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ | $-1$ | $\zeta_{20}$ | $-\zeta_{4}$ | $\zeta_{5}^{3}$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ | $-\zeta_{20}^{7}$ | $-\zeta_{20}$ | $\zeta_{5}^{2}$ | $-\zeta_{5}^{3}$ | $-\zeta_{20}^{3}$ | $\zeta_{20}^{7}$ | $\zeta_{5}$ | $-\zeta_{5}^{2}$ | $\zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}$ | $\zeta_{20}^{3}$ | $-\zeta_{5}$ | $-\zeta_{20}^{7} + \zeta_{20}^{5} - \zeta_{20}^{3} + \zeta_{20}$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, i, 1, 1)$$\operatorname{diag}(1, 1, \zeta_{5}, \zeta_{5}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 20$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, i, -1, -1)$$\operatorname{diag}(1, 1, \zeta_{5}, \zeta_{5}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 20$, moduli $1$