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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 of the form Cj Dj, Sj, Aj ($1\leq j\leq 47$), for cyclic, dihedral, symmetric, and alternating groups, or 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.

AliasGroup\(n\)T\(t\)
V4$C_2^2$4T2
C2XC2$C_2^2$4T2
F5$F_5$5T3, 10T4, 20T5
PSL(2,5)$A_5$5T4, 6T12, 10T7, 12T33, 15T5, 20T15, 30T9
PGL(2,5)$S_5$5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62
S3XC3$S_3\times C_3$6T5, 9T4, 18T3
S3XS3$S_3^2$6T9, 9T8, 12T16, 18T9, 18T11, 36T13
PSL(2,9)$A_6$6T15, 10T26, 15T20, 20T89, 30T88, 36T555, 40T304, 45T49
F7$F_7$7T4, 14T4, 21T4, 42T4
GL(3,2)$\GL(3,2)$7T5, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38
PSL(2,7)$\GL(3,2)$7T5, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38
C4XC2$C_4\times C_2$8T2
C2XC2XC2$C_2^3$8T3
Q8$Q_8$8T5
SL(2,3)$\SL(2,3)$8T12, 24T7
GL(2,3)$\textrm{GL(2,3)}$8T23, 16T66, 24T22
PGL(2,7)$\PGL(2,7)$8T43, 14T16, 16T713, 21T20, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83
C3XC3$C_3^2$9T2
M9$C_3^2:Q_8$9T14, 12T47, 18T35, 24T82, 36T55
PSL(2,8)$\PSL(2,8)$9T27, 28T70, 36T712
PGL(2,9)$\PGL(2,9)$10T30, 12T182, 20T146, 30T171, 36T1254, 40T590, 45T110
M10$M_{10}$10T31, 12T181, 20T148, 20T150, 30T162, 36T1253, 40T591, 45T109
F11$F_{11}$11T4, 22T4
PSL(2,11)$\PSL(2,11)$11T5, 12T179
M11$M_{11}$11T6, 12T272, 22T22
C6XC2$C_6\times C_2$12T2
C3:C4$C_3 : C_4$12T5
M12$M_{12}$12T295
F13$F_{13}$13T6, 26T8, 39T11
PSL(3,3)$\PSL(3,3)$13T7, 26T39, 39T43
PSL(2,13)$\PSL(2,13)$14T30, 28T120, 42T176
PGL(2,13)$\PGL(2,13)$14T39, 28T201, 42T284
Q8XC2$Q_8\times C_2$16T7
C4:C4$C_4:C_4$16T8
Q16$Q_{16}$16T14
F17$F_{17}$17T5
PSL(2,17)$\PSL(2,16)$17T6
PGL(2,17)$\PGL(2,17)$18T468
C5:C4$C_5:C_4$20T2
PGL(2,19)t20n36220T362, 40T5409
M22$M_{22}$22T38
F23$C_{23}:C_{11}$23T3
M23$M_{23}$23T5
Q8XC3$C_3\times Q_8$24T4
C3:Q8$C_3:Q_8$24T5
C3:C8$C_3:C_8$24T8
M24$M_{24}$24T24680
PSP(4,3)$\PSp(4,3)$27T993, 36T12781, 40T14344, 40T14345, 45T666
C7:C4$C_7:C_4$28T3
PSU(3,3)$\PSU(3,3)$28T323, 36T6815
Q32$Q_{32}$32T51
C5:C8$C_5:C_8$40T3

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  • Review status: reviewed
  • Last edited by John Jones on 2019-12-29 13:06:18
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