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