the group $\mathrm{S}_3 \times \mathrm{C}_3 \times \mathrm{C}_3$
- GAP SmallGroup id
- $[54, 12]$
- order
- $54$
- structure
- $\mathrm{C}_{3} \times \mathrm{C}_{3} \times \mathrm{S}_3$
- presentation
- $\langle g_{1}, g_{2}, g_{3}, g_{4} \mid g_{1}^{2},\; g_{2}^{-1}g_{1}^{-1}g_{2}g_{1},\; g_{3}^{-1}g_{1}^{-1}g_{3}g_{1},\; g_{4}^{-1}g_{1}^{-1}g_{4}g_{1}g_{4}^{-1},\; g_{2}^{3},\; g_{3}^{-1}g_{2}^{-1}g_{3}g_{2},\; g_{4}^{-1}g_{2}^{-1}g_{4}g_{2},\; g_{3}^{3},\; g_{4}^{-1}g_{3}^{-1}g_{4}g_{3},\; g_{4}^{3} \rangle$
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 | 2 | 3 | 3 | 3 | 6 | 6 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 3 | 3 | 3 | 3 | 3 | 6 | 6 | 3 | 3 | 3 | 6 | 3 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| size | 1 | 3 | 1 | 1 | 2 | 3 | 3 | 1 | 1 | 2 | 1 | 2 | 3 | 3 | 3 | 1 | 2 | 1 | 2 | 2 | 3 | 3 | 1 | 2 | 2 | 3 | 2 |
| $\chi_{ 1 }$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |
| $\chi_{ 2 }$ | $1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $-1$ | $-1$ | $1$ | $1$ | $1$ | $-1$ | $1$ |
| $\chi_{ 3 }$ | $1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 4 }$ | $1$ | $-1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3}$ | $-\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3} + 1$ | $\zeta_{3} + 1$ | $\zeta_{3} + 1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $-1$ | $-1$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 5 }$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 6 }$ | $1$ | $-1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3} + 1$ | $\zeta_{3} + 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3}$ | $-\zeta_{3}$ | $-\zeta_{3}$ | $1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-1$ | $-1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3} + 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 7 }$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 8 }$ | $1$ | $-1$ | $\zeta_{3}$ | $1$ | $1$ | $-\zeta_{3}$ | $-1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $-1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 9 }$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $1$ |
| $\chi_{ 10 }$ | $1$ | $-1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3}$ | $-1$ | $\zeta_{3} + 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-1$ | $1$ |
| $\chi_{ 11 }$ | $1$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 12 }$ | $1$ | $-1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $-1$ | $\zeta_{3} + 1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-1$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 13 }$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 14 }$ | $1$ | $-1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $\zeta_{3} + 1$ | $-1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $-1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3}$ | $\zeta_{3}$ |
| $\chi_{ 15 }$ | $1$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 16 }$ | $1$ | $-1$ | $1$ | $\zeta_{3}$ | $1$ | $-1$ | $-\zeta_{3}$ | $1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $\zeta_{3}$ | $1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3} - 1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3} + 1$ | $-\zeta_{3} - 1$ |
| $\chi_{ 17 }$ | $1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $1$ |
| $\chi_{ 18 }$ | $1$ | $-1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $1$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $\zeta_{3} + 1$ | $-1$ | $-\zeta_{3}$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-\zeta_{3} - 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $1$ | $\zeta_{3}$ | $-\zeta_{3} - 1$ | $-1$ | $1$ |
| $\chi_{ 19 }$ | $2$ | $0$ | $2$ | $2$ | $-1$ | $0$ | $0$ | $2$ | $2$ | $-1$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $2$ | $-1$ | $2$ | $-1$ | $-1$ | $0$ | $0$ | $2$ | $-1$ | $-1$ | $0$ | $-1$ |
| $\chi_{ 20 }$ | $2$ | $0$ | $2\zeta_{3}$ | $2$ | $-1$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $2\zeta_{3}$ | $-\zeta_{3}$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $\zeta_{3} + 1$ | $2\zeta_{3}$ | $-\zeta_{3}$ | $-1$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $0$ | $\zeta_{3} + 1$ |
| $\chi_{ 21 }$ | $2$ | $0$ | $-2\zeta_{3} - 2$ | $2$ | $-1$ | $0$ | $0$ | $2\zeta_{3}$ | $-2\zeta_{3} - 2$ | $\zeta_{3} + 1$ | $2$ | $-1$ | $0$ | $0$ | $0$ | $2\zeta_{3}$ | $-\zeta_{3}$ | $-2\zeta_{3} - 2$ | $\zeta_{3} + 1$ | $-1$ | $0$ | $0$ | $2\zeta_{3}$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $-\zeta_{3}$ |
| $\chi_{ 22 }$ | $2$ | $0$ | $2$ | $2\zeta_{3}$ | $-1$ | $0$ | $0$ | $2$ | $2\zeta_{3}$ | $-1$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $0$ | $0$ | $0$ | $2\zeta_{3}$ | $-1$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $\zeta_{3} + 1$ |
| $\chi_{ 23 }$ | $2$ | $0$ | $2\zeta_{3}$ | $2\zeta_{3}$ | $-1$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $0$ | $0$ | $0$ | $2$ | $\zeta_{3} + 1$ | $2$ | $\zeta_{3} + 1$ | $\zeta_{3} + 1$ | $0$ | $0$ | $2\zeta_{3}$ | $-1$ | $-1$ | $0$ | $-\zeta_{3}$ |
| $\chi_{ 24 }$ | $2$ | $0$ | $-2\zeta_{3} - 2$ | $2\zeta_{3}$ | $-1$ | $0$ | $0$ | $2\zeta_{3}$ | $2$ | $\zeta_{3} + 1$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $0$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $-\zeta_{3}$ | $2\zeta_{3}$ | $-1$ | $\zeta_{3} + 1$ | $0$ | $0$ | $2$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $0$ | $-1$ |
| $\chi_{ 25 }$ | $2$ | $0$ | $2$ | $-2\zeta_{3} - 2$ | $-1$ | $0$ | $0$ | $2$ | $-2\zeta_{3} - 2$ | $-1$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $-1$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $0$ | $0$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $-\zeta_{3}$ | $0$ | $-\zeta_{3}$ |
| $\chi_{ 26 }$ | $2$ | $0$ | $2\zeta_{3}$ | $-2\zeta_{3} - 2$ | $-1$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $2$ | $-\zeta_{3}$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $0$ | $0$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $-2\zeta_{3} - 2$ | $-1$ | $-\zeta_{3}$ | $0$ | $0$ | $2$ | $-\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $-1$ |
| $\chi_{ 27 }$ | $2$ | $0$ | $-2\zeta_{3} - 2$ | $-2\zeta_{3} - 2$ | $-1$ | $0$ | $0$ | $2\zeta_{3}$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $2\zeta_{3}$ | $\zeta_{3} + 1$ | $0$ | $0$ | $0$ | $2$ | $-\zeta_{3}$ | $2$ | $-\zeta_{3}$ | $-\zeta_{3}$ | $0$ | $0$ | $-2\zeta_{3} - 2$ | $-1$ | $-1$ | $0$ | $\zeta_{3} + 1$ |
hyperelliptic quotients
- dimension 4, representation $\operatorname{diag}(1, 1, 1, -1)$$\operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3})$$\operatorname{diag}(1, 1, 1, \zeta_{3})$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 6$, moduli $1$
- dimension 4, representation $\operatorname{diag}(1, 1, -1, -1)$$\operatorname{diag}(1, \zeta_{3}, \zeta_{3}, \zeta_{3})$$\operatorname{diag}(1, 1, 1, \zeta_{3})$$\operatorname{diag}(1, 1, \zeta_{3}, \zeta_{3}^{2})$ with $q = 1$, $\operatorname{ord} \omega_X = 3$, moduli $1$