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Distribution of groups in curves of genus 11 with quotient genus 0

Isomorphism class GAP/Magma Group Distinct generating vectors
$C_2$ [2, 1] 1
$C_3$ [3, 1] 4
$C_4$ [4, 1] 14
$C_2^2$ [4, 2] 33
$S_3$ [6, 1] 1602
$C_6$ [6, 2] 31
$C_8$ [8, 1] 27
$C_2\times C_4$ [8, 2] 159
$D_4$ [8, 3] 286
$Q_8$ [8, 4] 62
$C_2^3$ [8, 5] 525
$C_9$ [9, 1] 24
$D_5$ [10, 1] 7812
$C_{10}$ [10, 2] 24
$C_3:C_4$ [12, 1] 41
$C_{12}$ [12, 2] 29
$A_4$ [12, 3] 48
$D_6$ [12, 4] 257
$C_2\times C_6$ [12, 5] 69
$C_{15}$ [15, 1] 16
$C_{16}$ [16, 1] 20
$C_4^2$ [16, 2] 96
$C_4:C_4$ [16, 4] 24
$C_2\times C_8$ [16, 5] 64
$\OD_{16}$ [16, 6] 22
$D_8$ [16, 7] 28
$\SD_{16}$ [16, 8] 28
$Q_{16}$ [16, 9] 12
$C_2\times D_4$ [16, 11] 212
$C_2\times Q_8$ [16, 12] 20
$D_4:C_2$ [16, 13] 89
$C_{18}$ [18, 2] 6
$C_5:C_4$ [20, 1] 40
$C_{20}$ [20, 2] 8
$F_5$ [20, 3] 82
$D_{10}$ [20, 4] 660
$D_{11}$ [22, 1] 605
$C_{22}$ [22, 2] 5
$C_{23}$ [23, 1] 88
$C_{24}$ [24, 2] 12
$\SL(2,3)$ [24, 3] 12
$C_4\times S_3$ [24, 5] 20
$D_{12}$ [24, 6] 48
$C_6:C_4$ [24, 7] 20
$C_3:D_4$ [24, 8] 12
$C_2\times C_{12}$ [24, 9] 36
$C_3\times D_4$ [24, 10] 4
$S_4$ [24, 12] 32
$C_2\times A_4$ [24, 13] 21
$C_2\times D_6$ [24, 14] 144
$C_3\times D_5$ [30, 2] 12
$D_{15}$ [30, 3] 216
$C_{30}$ [30, 4] 8
$D_4:C_4$ [32, 9] 24
$C_4\wr C_2$ [32, 11] 12
$C_8.C_4$ [32, 15] 8
$C_2\times C_{16}$ [32, 16] 16
$\OD_{32}$ [32, 17] 4
$C_2\times D_8$ [32, 39] 16
$C_2\times \SD_{16}$ [32, 40] 16
$D_8:C_2$ [32, 42] 12
$D_8:C_2$ [32, 43] 8
$Q_{16}:C_2$ [32, 44] 4
$C_{33}$ [33, 1] 10
$C_5:C_8$ [40, 3] 6
$C_4\times D_5$ [40, 5] 48
$D_{20}$ [40, 6] 48
$C_5:D_4$ [40, 8] 48
$C_2\times F_5$ [40, 12] 30
$C_2\times D_{10}$ [40, 13] 144
$C_{11}:C_4$ [44, 1] 10
$C_{44}$ [44, 2] 10
$D_{22}$ [44, 3] 10
$C_{46}$ [46, 2] 22
$C_{12}:C_4$ [48, 11] 16
$C_6.D_4$ [48, 12] 8
$C_{12}:C_4$ [48, 13] 8
$C_2\times C_{24}$ [48, 23] 16
$C_3\times \OD_{16}$ [48, 24] 4
$C_2.S_4$ [48, 28] 1
$\GL(2,3)$ [48, 29] 14
$C_2\times \SL(2,3)$ [48, 32] 8
$C_2\times D_{12}$ [48, 36] 16
$D_{12}:C_2$ [48, 37] 4
$S_3\times D_4$ [48, 38] 4
$C_2\times S_4$ [48, 48] 8
$A_5$ [60, 5] 18
$C_3\times F_5$ [60, 6] 2
$C_{15}:C_4$ [60, 7] 4
$S_3\times D_5$ [60, 8] 12
$C_4.D_8$ [64, 40] 8
$\OD_{32}:C_2$ [64, 42] 8
$C_3\times D_{11}$ [66, 2] 10
$D_5:C_8$ [80, 28] 8
$C_{20}.C_4$ [80, 29] 4
$C_4\times F_5$ [80, 30] 8
$C_{20}:C_4$ [80, 31] 4
$D_4\times D_5$ [80, 39] 24
$D_{20}:C_2$ [80, 42] 24
$C_4\times D_{11}$ [88, 4] 20
$D_{12}:C_4$ [96, 28] 16
$D_{12}:C_4$ [96, 32] 8
$C_2\times \GL(2,3)$ [96, 189] 8
$\GL(2,3):C_2$ [96, 190] 2
$S_5$ [120, 34] 2
$C_2\times A_5$ [120, 35] 18
$S_3\times F_5$ [120, 36] 2
$D_{20}:C_4$ [160, 82] 8
$D_{20}:C_4$ [160, 85] 8
$C_2\times S_5$ [240, 189] 2