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Distribution of groups in curves of genus 2 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] 1 1 1
$C_4$ [4, 1] 1 1 1
$C_2^2$ [4, 2] 3 1 3
$C_5$ [5, 1] 4 1 4
$S_3$ [6, 1] 2 1 1
$C_6$ [6, 2] 3 2 3
$C_8$ [8, 1] 2 1 2
$D_4$ [8, 3] 1 1 1
$Q_8$ [8, 4] 1 1 1
$C_{10}$ [10, 2] 4 1 4
$C_3:C_4$ [12, 1] 1 1 1
$D_6$ [12, 4] 1 1 1
$C_2\times C_6$ [12, 5] 6 1 6
$\SD_{16}$ [16, 8] 2 1 2
$\SL(2,3)$ [24, 3] 1 1 1
$C_3:D_4$ [24, 8] 2 1 2
$\GL(2,3)$ [48, 29] 2 1 2

Distribution of groups in curves of genus 2 with quotient genus greater than 0

Isomorphism class GAP/Magma Group Distinct generating vectors Quotient Genera
$C_2$ [2, 1] 1 1