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