Distribution of groups in curves of genus 5 with quotient genus 0
| Isomorphism class | GAP/Magma Group | Distinct generating vectors | Topologically inequivalent actions | Braid inequivalent actions |
|---|---|---|---|---|
| $C_2$ | [2, 1] | 1 | 1 | 1 |
| $C_3$ | [3, 1] | 2 | 1 | 2 |
| $C_4$ | [4, 1] | 5 | 3 | 5 |
| $C_2^2$ | [4, 2] | 12 | 3 | 12 |
| $S_3$ | [6, 1] | 18 | 1 | 1 |
| $C_6$ | [6, 2] | 6 | 4 | 6 |
| $C_8$ | [8, 1] | 6 | 2 | 6 |
| $C_2\times C_4$ | [8, 2] | 27 | 7 | 27 |
| $D_4$ | [8, 3] | 22 | 3 | 5 |
| $Q_8$ | [8, 4] | 6 | 1 | 3 |
| $C_2^3$ | [8, 5] | 77 | 3 | 77 |
| $D_5$ | [10, 1] | 50 | 1 | 2 |
| $C_{10}$ | [10, 2] | 2 | 1 | 2 |
| $C_{11}$ | [11, 1] | 20 | 2 | 20 |
| $C_3:C_4$ | [12, 1] | 2 | 1 | 2 |
| $C_{12}$ | [12, 2] | 4 | 1 | 4 |
| $A_4$ | [12, 3] | 5 | 2 | 2 |
| $D_6$ | [12, 4] | 15 | 3 | 5 |
| $C_2\times C_6$ | [12, 5] | 12 | 2 | 12 |
| $C_{15}$ | [15, 1] | 4 | 1 | 4 |
| $C_2^2:C_4$ | [16, 3] | 28 | 4 | 20 |
| $C_2\times C_8$ | [16, 5] | 8 | 1 | 8 |
| $\OD_{16}$ | [16, 6] | 2 | 1 | 2 |
| $D_8$ | [16, 7] | 8 | 1 | 1 |
| $\SD_{16}$ | [16, 8] | 8 | 1 | 1 |
| $C_2^2\times C_4$ | [16, 10] | 48 | 1 | 48 |
| $C_2\times D_4$ | [16, 11] | 22 | 3 | 11 |
| $D_4:C_2$ | [16, 13] | 6 | 1 | 3 |
| $C_2^4$ | [16, 14] | 168 | 1 | 168 |
| $C_5:C_4$ | [20, 1] | 4 | 1 | 4 |
| $C_{20}$ | [20, 2] | 4 | 1 | 4 |
| $D_{10}$ | [20, 4] | 4 | 1 | 4 |
| $C_{22}$ | [22, 2] | 10 | 1 | 10 |
| $C_6:C_4$ | [24, 7] | 4 | 1 | 4 |
| $C_3:D_4$ | [24, 8] | 4 | 1 | 2 |
| $C_2\times C_{12}$ | [24, 9] | 8 | 1 | 8 |
| $S_4$ | [24, 12] | 6 | 1 | 1 |
| $C_2\times A_4$ | [24, 13] | 6 | 3 | 4 |
| $C_2\times D_6$ | [24, 14] | 12 | 1 | 12 |
| $C_3\times D_5$ | [30, 2] | 4 | 1 | 4 |
| $C_2.C_4^2$ | [32, 2] | 16 | 1 | 16 |
| $C_2^2:C_8$ | [32, 5] | 8 | 1 | 8 |
| $C_2^3:C_4$ | [32, 6] | 2 | 1 | 2 |
| $\OD_{16}:C_2$ | [32, 7] | 4 | 1 | 4 |
| $C_2^2\wr C_2$ | [32, 27] | 24 | 1 | 12 |
| $C_4:D_4$ | [32, 28] | 8 | 1 | 4 |
| $D_8:C_2$ | [32, 43] | 4 | 1 | 1 |
| $C_4\times D_5$ | [40, 5] | 8 | 1 | 8 |
| $D_6:C_4$ | [48, 14] | 8 | 1 | 8 |
| $A_4:C_4$ | [48, 30] | 3 | 2 | 3 |
| $C_2\times S_4$ | [48, 48] | 3 | 1 | 1 |
| $C_2^2\times A_4$ | [48, 49] | 6 | 1 | 6 |
| $A_5$ | [60, 5] | 2 | 1 | 2 |
| $C_2^2.D_8$ | [64, 8] | 8 | 1 | 8 |
| $C_2\wr C_4$ | [64, 32] | 4 | 1 | 4 |
| $C_2^4:C_5$ | [80, 49] | 6 | 1 | 6 |
| $C_2^3.A_4$ | [96, 3] | 4 | 1 | 4 |
| $\GL(2,\mathbb{Z}/4)$ | [96, 195] | 2 | 1 | 2 |
| $C_2\times A_5$ | [120, 35] | 2 | 1 | 2 |
| $C_2^4:D_5$ | [160, 234] | 6 | 1 | 6 |
| $C_2^3.S_4$ | [192, 181] | 4 | 1 | 4 |