the group $\mathrm{C}_{2} \times \mathrm{C}_{2}$
- GAP SmallGroup id
- $[4, 2]$
- order
- $4$
- structure
- $\mathrm{C}_{2} \times \mathrm{C}_{2}$
- presentation
- $\langle g_{1}, g_{2} \mid g_{1}^{2},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{2}^{2} \rangle$
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 | 2 | 2 | 2 |
|---|---|---|---|---|
| size | 1 | 1 | 1 | 1 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $-1$ | $1$ | $-1$ |
| $\chi_{ 3 }$ | $1$ | $1$ | $-1$ | $-1$ |
| $\chi_{ 4 }$ | $1$ | $-1$ | $-1$ | $1$ |
hyperelliptic quotients
- dimension 3, representation $\operatorname{diag}(1, -1, 1)$$\operatorname{diag}(1, 1, -1)$ with $q = 1$, $\operatorname{ord} \omega_X = 2$, moduli $3$
- dimension 3, representation $\operatorname{diag}(-1, 1, -1)$$\operatorname{diag}(1, -1, -1)$ with $q = 0$, $\operatorname{ord} \omega_X = 1$, moduli $3$
- dimension 4, representation $\operatorname{diag}(1, 1, -1, 1)$$\operatorname{diag}(1, 1, 1, -1)$ with $q = 2$, $\operatorname{ord} \omega_X = 2$, moduli $6$
- dimension 4, representation $\operatorname{diag}(1, -1, -1, 1)$$\operatorname{diag}(1, 1, 1, -1)$ with $q = 1$, $\operatorname{ord} \omega_X = 2$, moduli $6$
- dimension 4, representation $\operatorname{diag}(1, -1, 1, -1)$$\operatorname{diag}(1, 1, -1, -1)$ with $q = 1$, $\operatorname{ord} \omega_X = 1$, moduli $4$
- dimension 4, representation $\operatorname{diag}(-1, -1, 1, -1)$$\operatorname{diag}(1, 1, -1, -1)$ with $q = 0$, $\operatorname{ord} \omega_X = 2$, moduli $6$