Distribution of groups in curves of genus 4

Isomorphism class GAP/Magma Group Distinct generating vectors
$C_2$ [2,1] 1
$C_3$ [3,1] 3
$C_4$ [4,1] 3
$C_2^2$ [4,2] 6
$C_5$ [5,1] 7
$S_3$ [6,1] 44
$C_6$ [6,2] 9
$C_8$ [8,1] 2
$D_4$ [8,3] 9
$Q_8$ [8,4] 1
$C_9$ [9,1] 6
$C_3^2$ [9,2] 30
$D_5$ [10,1] 6
$C_{10}$ [10,2] 10
$C_{12}$ [12,2] 6
$A_4$ [12,3] 8
$D_6$ [12,4] 10
$C_2\times C_6$ [12,5] 8
$C_{15}$ [15,1] 8
$C_{16}$ [16,1] 4
$D_8$ [16,7] 2
$Q_{16}$ [16,9] 2
$C_{18}$ [18,2] 6
$C_3\times S_3$ [18,3] 7
$C_3:S_3$ [18,4] 12
$C_3\times C_6$ [18,5] 24
$C_5:C_4$ [20,1] 2
$F_5$ [20,3] 1
$D_{10}$ [20,4] 2
$C_2\times C_{10}$ [20,5] 12
$\SL(2,3)$ [24,3] 2
$C_3\times D_4$ [24,10] 4
$S_4$ [24,12] 4
$SD_{32}$ [32,19] 4
$C_3:S_3.C_2$ [36,9] 2
$S_3^2$ [36,10] 2
$C_3\times A_4$ [36,11] 6
$C_6\times S_3$ [36,12] 4
$C_5:D_4$ [40,8] 4
$A_5$ [60,5] 1
$S_3\wr C_2$ [72,40] 2
$C_3\times S_4$ [72,42] 2
$S_5$ [120,34] 1