Distribution of groups in curves of genus 6

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