the group $\mathrm{G}(3,4,2)$
- GAP SmallGroup id
- $[12, 1]$
- order
- $12$
- structure
- $\mathrm{C}_{3} \rtimes \mathrm{C}_{4}$
- presentation
- $\langle g_{1}, g_{2}, g_{3} \mid g_{1}^{2}g_{2}^{-1},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{3}^{-1}g_{1}^{-1}g_{3}g_{1}g_{3}^{-1},\; g_{2}^{2},\; g_{3}^{-1}g_{2}^{-1}g_{3}g_{2},\; g_{3}^{3} \rangle$
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 | 4 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|---|
| size | 1 | 3 | 1 | 2 | 3 | 2 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $\zeta_{4}$ | $-1$ | $1$ | $-\zeta_{4}$ | $-1$ |
| $\chi_{ 3 }$ | $1$ | $-1$ | $1$ | $1$ | $-1$ | $1$ |
| $\chi_{ 4 }$ | $1$ | $-\zeta_{4}$ | $-1$ | $1$ | $\zeta_{4}$ | $-1$ |
| $\chi_{ 5 }$ | $2$ | $0$ | $2$ | $-1$ | $0$ | $-1$ |
| $\chi_{ 6 }$ | $2$ | $0$ | $-2$ | $-1$ | $0$ | $1$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, 1, i, -i)$$\operatorname{diag}(1, 1, -1, -1)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 2$, $\operatorname{ord} \omega_X = 1$, moduli $5$
- dimension 4, representation $\operatorname{diag}(1, 1, i, -1)$$\operatorname{diag}(1, 1, 1, -1)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 4$, moduli $2$
- dimension 4, representation $\operatorname{diag}(1, i, i, -i)$$\operatorname{diag}(1, -1, -1, -1)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 4$, moduli $2$
- dimension 4, representation $\operatorname{diag}(1, i, -1, -i)$$\operatorname{diag}(1, 1, -1, -1)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 2$, moduli $3$
- dimension 4, representation $\operatorname{diag}(1, i, -1, -1)$$\operatorname{diag}(1, 1, 1, -1)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 0$, $\operatorname{ord} \omega_X = 4$, moduli $2$