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Artin representations were computed with Magma by Tim Dokchitser and John Jones.

Bounds for completeness of computations come from [MR:3669395, 10.1007/s40993-017-0079-5, arXiv:1610.01228] . In each case, the LMFDB contains all faithful representations for a given group and irreducible representation with conductor less than or equal to the stated bound. Representations are distinguished by their dimension, and the smallest containing permutation representation.

GroupDimContainerBound
$C_2$12T110000
$C_3$13T111180
$C_4$14T1796
$C_5$15T1752
$C_6$16T1577
$S_3$23T2177662241
$C_7$17T1483
$C_8$18T1249
$D_4$24T222500
$Q_8$28T5215444
$C_9$19T1387
$D_{5}$25T240000
$A_{4}$34T43375000
$D_{6}$26T222500
$D_{7}$27T240000
$S_{3}C_3$26T52828
$F_{5}$47T31600000000
$S_{4}$34T522497
$S_{4}$36T8635168
$A_{4}C_2$36T622497
$C_{3}^2:C_4$46T103374494
$S_{3}^2$36T97998219
$A_{5}$312T3366627
$A_{5}$45T4613778
$A_{5}$56T1252222435
$S_{5}$45T57226
$S_{5}$410T12613778
$S_{5}$56T1452222435
$S_{5}$510T1352222435
$S_{5}$620T354439177045