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Each Galois group is identified by its degree and an index called its T-number. This specifies both the abstract group and a faithful transitive permutation of that group. One may search for a group in the form $n$T$t$ where $n$ is the degree and $t$ is the T-number. The search results will then only show fields where the Galois group matches the requested permutation representation.

For familiar groups one can use short names from the table below. An abstract group may have more than one representation as a Galois group. Correspondingly, the familiar symbol for a group may represent several Galois groups. When searching for fields using the name of an abstract group, e.g., S3, the results may contain fields with different degrees and permutation representations. Some of these combinations are shown below for $n\leq 23$.

AliasGroup\(n\)T\(t\)
S1Trivial1T1
C1Trivial1T1
A1Trivial1T1
A2Trivial1T1
S2$C_2$2T1
C2$C_2$2T1
D1$C_2$2T1
C3$C_3$3T1
A3$C_3$3T1
S3$S_3$3T2, 6T2
D3$S_3$3T2, 6T2
C4$C_4$4T1
C2XC2$V_4$4T2
V4$V_4$4T2
D2$V_4$4T2
D4$D_4$4T3, 8T4
A4$A_4$4T4, 6T4, 12T4
S4$S_4$4T5, 6T7, 6T8, 8T14, 12T8, 12T9
C5$C_5$5T1
D5$D_5$5T2, 10T2
F5$F_5$5T3, 10T4
A5$A_5$5T4, 6T12, 10T7, 12T33
PSL(2,5)$A_5$5T4, 6T12, 10T7, 12T33
PGL(2,5)$S_5$5T5, 6T14, 10T12, 10T13, 12T74
S5$S_5$5T5, 6T14, 10T12, 10T13, 12T74
C6$C_6$6T1
D6$S_3\times C_2$6T3, 12T3
C3XS3$S_3\times C_3$6T5, 9T4
S3XC3$S_3\times C_3$6T5, 9T4
S3XS3$S_3^2$6T9, 9T8, 12T16
PSL(2,9)$A_6$6T15, 10T26
A6$A_6$6T15, 10T26
S6$S_6$6T16, 10T32, 12T183, 12T183
C7$C_7$7T1
D7$D_7$7T2, 14T2
F7$F_7$7T4, 14T4, 21T4
PSL(2,7)$\GL(3,2)$7T5, 8T37
GL(3,2)$\GL(3,2)$7T5, 8T37
A7$A_7$7T6
S7$S_7$7T7
C8$C_8$8T1
C4XC2$C_4\times C_2$8T2
C2XC4$C_4\times C_2$8T2
C2XC2XC2$C_2^3$8T3
Q8$Q_8$8T5
D8$D_8$8T6, 16T7
SL(2,3)$\SL(2,3)$8T12
GL(2,3)$\textrm{GL(2,3)}$8T23
PGL(2,7)$\PGL(2,7)$8T43
A8$A_8$8T49
S8$S_8$8T50
C9$C_9$9T1
C3XC3$C_3^2$9T2
D9$D_9$9T3
M9$C_3^2:Q_8$9T14, 12T47
PSL(2,8)$\PSL(2,8)$9T27
A9$A_9$9T33
S9$S_9$9T34
C10$C_{10}$10T1
D10$D_{10}$10T3
PGL(2,9)$\PGL(2,9)$10T30, 12T182
M10$M_{10}$10T31, 12T181
A10$A_{10}$10T44
S10$S_{10}$10T45
C11$C_{11}$11T1
D11$D_{11}$11T2, 22T2
F11$F_{11}$11T4, 22T4
PSL(2,11)$\PSL(2,11)$11T5, 12T272
M11$M_{11}$11T6
A11$A_{11}$11T7
S11$S_{11}$11T8
C12$C_{12}$12T1
C2XC6$C_6\times C_2$12T2
C6XC2$C_6\times C_2$12T2
A12$A_{12}$12T300
S12$S_{12}$12T301
C13$C_{13}$13T1
D13$D_{13}$13T2
F13$F_{13}$13T6
A13$A_{13}$13T8
S13$S_{13}$13T9
C14$C_{14}$14T1
D14$D_{14}$14T3
PGL(2,13)$\PGL(2,13)$14T39
A1414T6214T62
S1414T6314T63
C15$C_{15}$15T1
A1515T10315T103
S1515T10415T104
C16$C_{16}$16T1
Q16$Q_{16}$16T14
A16$A_{16}$16T1953
S16$S_{16}$16T1954
C17$C_{17}$17T1
D17$D_{17}$17T2
F17$F_{17}$17T5
PSL(2,17)$\PSL(2,16)$17T6
A17$A_{17}$17T9
S17$S_{17}$17T10
C18$C_{18}$18T1
PGL(2,17)$\PGL(2,17)$18T468
A18$A_{18}$18T982
S18$S_{18}$18T983
C19$C_{19}$19T1
A19$A_{19}$19T7
S19$S_{19}$19T8
C20$C_{20}$20T1
PGL(2,19)$\PGL(2,19)$20T362
A2020T111620T1116
S2020T111720T1117
C21$C_{21}$21T1
A2121T16321T163
S2121T16421T164
C2222T122T1
D2222T322T3
A2222T5822T58
S2222T5922T59
C23$C_{23}$23T1
D23$D_{23}$23T2
F23$C_{23}:C_{11}$23T3
M23$M_{23}$23T5
A23$A_{23}$23T6
S23$S_{23}$23T7
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Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2019-04-25 10:10:27
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