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