the group $\mathrm{C}_{4} \times \mathrm{A}_4$
- GAP SmallGroup id
- $[48, 31]$
- order
- $48$
- structure
- $\mathrm{C}_{4} \times \mathrm{A}_4$
- presentation
- $\langle g_{1}, g_{2}, g_{3}, g_{4}, g_{5} \mid g_{1}^{2}g_{3}^{-1},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{3}^{-1}g_{1}^{-1}g_{3}g_{1},\; g_{4}^{-1}g_{1}^{-1}g_{4}g_{1},\; g_{5}^{-1}g_{1}^{-1}g_{5}g_{1},\; g_{2}^{3},\; g_{3}^{-1}g_{2}^{-1}g_{3}g_{2},\; g_{4}^{-1}g_{2}^{-1}g_{4}g_{2}g_{5}^{-1}g_{4}^{-1},\; g_{5}^{-1}g_{2}^{-1}g_{5}g_{2}g_{4}^{-1},\; g_{3}^{2},\; g_{4}^{-1}g_{3}^{-1}g_{4}g_{3},\; g_{5}^{-1}g_{3}^{-1}g_{5}g_{3},\; g_{4}^{2},\; g_{5}^{-1}g_{4}^{-1}g_{5}g_{4},\; g_{5}^{2} \rangle$
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 | 3 | 2 | 2 | 12 | 4 | 4 | 3 | 6 | 2 | 12 | 12 | 4 | 6 | 12 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 1 | 4 | 1 | 3 | 4 | 1 | 3 | 4 | 4 | 3 | 4 | 4 | 3 | 4 | 4 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 3 }$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 4 }$ | $1$ | $\zeta_{4}$ | $1$ | $-1$ | $1$ | $\zeta_{4}$ | $-\zeta_{4}$ | $\zeta_{4}$ | $1$ | $-1$ | $-1$ | $\zeta_{4}$ | $-\zeta_{4}$ | $-\zeta_{4}$ | $-1$ | $-\zeta_{4}$ |
| $\chi_{ 5 }$ | $1$ | $\zeta_{4}$ | $-\zeta_{3} - 1$ | $-1$ | $1$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $-\zeta_{4}$ | $\zeta_{4}$ | $\zeta_{3}$ | $\zeta_{3} + 1$ | $-1$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{4}$ | $-\zeta_{3}$ | $\zeta_{12}$ |
| $\chi_{ 6 }$ | $1$ | $\zeta_{4}$ | $\zeta_{3}$ | $-1$ | $1$ | $-\zeta_{12}$ | $-\zeta_{4}$ | $\zeta_{4}$ | $-\zeta_{3} - 1$ | $-\zeta_{3}$ | $-1$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{4}$ | $\zeta_{3} + 1$ | $\zeta_{12}^{3} - \zeta_{12}$ |
| $\chi_{ 7 }$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $-1$ |
| $\chi_{ 8 }$ | $1$ | $-1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3} + 1$ | $-1$ | $-1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $-1$ | $\zeta_{3}$ | $-\zeta_{3}$ |
| $\chi_{ 9 }$ | $1$ | $-1$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3}$ | $-1$ | $-1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $-1$ | $-\zeta_{3} - 1$ | $\zeta_{3} + 1$ |
| $\chi_{ 10 }$ | $1$ | $-\zeta_{4}$ | $1$ | $-1$ | $1$ | $-\zeta_{4}$ | $\zeta_{4}$ | $-\zeta_{4}$ | $1$ | $-1$ | $-1$ | $-\zeta_{4}$ | $\zeta_{4}$ | $\zeta_{4}$ | $-1$ | $\zeta_{4}$ |
| $\chi_{ 11 }$ | $1$ | $-\zeta_{4}$ | $-\zeta_{3} - 1$ | $-1$ | $1$ | $\zeta_{12}^{3} - \zeta_{12}$ | $\zeta_{4}$ | $-\zeta_{4}$ | $\zeta_{3}$ | $\zeta_{3} + 1$ | $-1$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{4}$ | $-\zeta_{3}$ | $-\zeta_{12}$ |
| $\chi_{ 12 }$ | $1$ | $-\zeta_{4}$ | $\zeta_{3}$ | $-1$ | $1$ | $\zeta_{12}$ | $\zeta_{4}$ | $-\zeta_{4}$ | $-\zeta_{3} - 1$ | $-\zeta_{3}$ | $-1$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{4}$ | $\zeta_{3} + 1$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
| $\chi_{ 13 }$ | $3$ | $3$ | $0$ | $3$ | $-1$ | $0$ | $3$ | $-1$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $-1$ | $0$ | $0$ |
| $\chi_{ 14 }$ | $3$ | $-3$ | $0$ | $3$ | $-1$ | $0$ | $-3$ | $1$ | $0$ | $0$ | $-1$ | $0$ | $0$ | $1$ | $0$ | $0$ |
| $\chi_{ 15 }$ | $3$ | $3\zeta_{4}$ | $0$ | $-3$ | $-1$ | $0$ | $-3\zeta_{4}$ | $-\zeta_{4}$ | $0$ | $0$ | $1$ | $0$ | $0$ | $\zeta_{4}$ | $0$ | $0$ |
| $\chi_{ 16 }$ | $3$ | $-3\zeta_{4}$ | $0$ | $-3$ | $-1$ | $0$ | $3\zeta_{4}$ | $\zeta_{4}$ | $0$ | $0$ | $1$ | $0$ | $0$ | $-\zeta_{4}$ | $0$ | $0$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, i, i, i)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$$\operatorname{diag}(1, -1, -1, -1)$$\operatorname{diag}(1, 1, -1, -1)$$\operatorname{diag}(1, 1, -1, -1)$ with $q = 1$, $\operatorname{ord} \omega_X = 4$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, 1, 1, i)$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$$\operatorname{diag}(1, 1, 1, -1)$$\operatorname{diag}(1, 1, -1, -1)$$\operatorname{diag}(1, 1, -1, -1)$ with $q = 0$, $\operatorname{ord} \omega_X = 4$, moduli $1$