Distribution of groups in curves of genus 6 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] | 3 | 2 | 3 |
| $C_4$ | [4, 1] | 6 | 4 | 6 |
| $C_2^2$ | [4, 2] | 10 | 3 | 10 |
| $C_5$ | [5, 1] | 12 | 3 | 12 |
| $S_3$ | [6, 1] | 89 | 2 | 2 |
| $C_6$ | [6, 2] | 10 | 6 | 10 |
| $C_7$ | [7, 1] | 18 | 4 | 18 |
| $C_8$ | [8, 1] | 8 | 3 | 8 |
| $D_4$ | [8, 3] | 13 | 3 | 4 |
| $Q_8$ | [8, 4] | 1 | 1 | 1 |
| $C_9$ | [9, 1] | 9 | 2 | 9 |
| $D_5$ | [10, 1] | 312 | 1 | 1 |
| $C_{10}$ | [10, 2] | 8 | 2 | 8 |
| $C_3:C_4$ | [12, 1] | 3 | 2 | 2 |
| $C_{12}$ | [12, 2] | 7 | 3 | 7 |
| $A_4$ | [12, 3] | 9 | 1 | 1 |
| $D_6$ | [12, 4] | 21 | 3 | 4 |
| $C_2\times C_6$ | [12, 5] | 7 | 2 | 7 |
| $C_{13}$ | [13, 1] | 28 | 3 | 28 |
| $D_7$ | [14, 1] | 12 | 2 | 6 |
| $C_{14}$ | [14, 2] | 21 | 4 | 21 |
| $C_{15}$ | [15, 1] | 12 | 2 | 12 |
| $C_{16}$ | [16, 1] | 8 | 1 | 8 |
| $D_8$ | [16, 7] | 4 | 1 | 2 |
| $\SD_{16}$ | [16, 8] | 2 | 1 | 2 |
| $D_9$ | [18, 1] | 6 | 1 | 3 |
| $C_{18}$ | [18, 2] | 6 | 1 | 6 |
| $C_{20}$ | [20, 2] | 8 | 1 | 8 |
| $F_5$ | [20, 3] | 6 | 1 | 1 |
| $D_{10}$ | [20, 4] | 24 | 1 | 2 |
| $C_{21}$ | [21, 2] | 12 | 1 | 12 |
| $C_3:C_8$ | [24, 1] | 2 | 1 | 2 |
| $C_{24}$ | [24, 2] | 4 | 1 | 4 |
| $\SL(2,3)$ | [24, 3] | 1 | 1 | 1 |
| $C_3:Q_8$ | [24, 4] | 2 | 1 | 2 |
| $D_{12}$ | [24, 6] | 4 | 2 | 3 |
| $C_3:D_4$ | [24, 8] | 2 | 1 | 1 |
| $C_3\times D_4$ | [24, 10] | 2 | 1 | 2 |
| $S_4$ | [24, 12] | 3 | 1 | 1 |
| $C_5^2$ | [25, 2] | 80 | 1 | 80 |
| $C_{26}$ | [26, 2] | 12 | 1 | 12 |
| $C_7:C_4$ | [28, 1] | 3 | 1 | 3 |
| $D_{14}$ | [28, 3] | 3 | 1 | 3 |
| $C_2\times C_{14}$ | [28, 4] | 18 | 1 | 18 |
| $C_5\times S_3$ | [30, 1] | 4 | 1 | 4 |
| $C_2^2:C_9$ | [36, 3] | 3 | 1 | 3 |
| $C_{13}:C_3$ | [39, 1] | 4 | 1 | 4 |
| $C_{24}:C_2$ | [48, 6] | 4 | 1 | 4 |
| $C_3:D_8$ | [48, 15] | 4 | 1 | 4 |
| $\GL(2,3)$ | [48, 29] | 2 | 1 | 2 |
| $C_5\times D_5$ | [50, 3] | 8 | 1 | 8 |
| $C_7:D_4$ | [56, 7] | 6 | 1 | 6 |
| $A_5$ | [60, 5] | 18 | 1 | 1 |
| $C_2^2:D_9$ | [72, 15] | 3 | 1 | 3 |
| $C_5^2:C_3$ | [75, 2] | 8 | 1 | 8 |
| $S_5$ | [120, 34] | 1 | 1 | 1 |
| $C_5^2:S_3$ | [150, 5] | 4 | 1 | 4 |