the group $\mathrm{Heis}(3)$
- GAP SmallGroup id
- $[27, 3]$
- order
- $27$
- structure
- $(\mathrm{C}_{3} \times \mathrm{C}_{3}) \rtimes \mathrm{C}_{3}$
- presentation
- $\langle g_{1}, g_{2}, g_{3} \mid g_{1}^{3},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1}g_{3}^{-1},\; g_{3}^{-1}g_{1}^{-1}g_{3}g_{1},\; g_{2}^{3},\; g_{3}^{-1}g_{2}^{-1}g_{3}g_{2},\; g_{3}^{3} \rangle$
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 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 3 | 1 | 3 | 3 | 3 | 1 | 3 | 3 | 3 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ |
| $\chi_{ 3 }$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ |
| $\chi_{ 4 }$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 5 }$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $1$ | $\zeta_{3}$ |
| $\chi_{ 6 }$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ |
| $\chi_{ 7 }$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 8 }$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ |
| $\chi_{ 9 }$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $1$ | $1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 10 }$ | $3$ | $0$ | $0$ | $3\zeta_{3}$ | $0$ | $0$ | $0$ | $-3\zeta_{3} - 3$ | $0$ | $0$ | $0$ |
| $\chi_{ 11 }$ | $3$ | $0$ | $0$ | $-3\zeta_{3} - 3$ | $0$ | $0$ | $0$ | $3\zeta_{3}$ | $0$ | $0$ | $0$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$$\operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3})$ with $q = 1$, $\operatorname{ord} \omega_X = 1$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3}^{2})$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$$\operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3})$ with $q = 0$, $\operatorname{ord} \omega_X = 3$, moduli $0$