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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 $n\leq 47$.

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$4T1
C2XC2$C_2^2$4T2
V4$C_2^2$4T2
D2$C_2^2$4T2
D4$D_{4}$4T3, 8T4
A4$A_4$4T4, 6T4, 12T4
S4$S_4$4T5, 6T7, 6T8, 8T14, 12T8, 12T9, 24T10
C5$C_5$5T1
D5$D_{5}$5T2, 10T2
F5$F_5$5T3, 10T4, 20T5
A5$A_5$5T4, 6T12, 10T7, 12T33, 15T5, 20T15, 30T9
PSL(2,5)$A_5$5T4, 6T12, 10T7, 12T33, 15T5, 20T15, 30T9
PGL(2,5)$S_5$5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62
S5$S_5$5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62
C6$C_6$6T1
D6$D_{6}$6T3, 12T3
C3XS3$S_3\times C_3$6T5, 9T4, 18T3
S3XC3$S_3\times C_3$6T5, 9T4, 18T3
S3XS3$S_3^2$6T9, 9T8, 12T16, 18T9, 18T11, 36T13
PSL(2,9)$A_6$6T15, 10T26, 15T20, 20T89, 30T88, 36T555, 40T304, 45T49
A6$A_6$6T15, 10T26, 15T20, 20T89, 30T88, 36T555, 40T304, 45T49
S6$S_6$6T16, 10T32, 12T183, 15T28, 20T145, 20T149, 30T164, 30T166, 30T176, 36T1252, 40T589, 40T592, 45T96
C7$C_7$7T1
D7$D_{7}$7T2, 14T2
F7$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)$7T5, 8T37, 14T10, 21T14, 24T284, 28T32, 42T37, 42T38
A7$A_7$7T6, 15T47, 21T33, 35T28, 42T294, 42T299
S7$S_7$7T7, 14T46, 21T38, 30T565, 35T31, 42T411, 42T412, 42T413, 42T418
C8$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}$8T6, 16T13
SL(2,3)$\SL(2,3)$8T12, 24T7
GL(2,3)$\textrm{GL(2,3)}$8T23, 16T66, 24T22
PGL(2,7)$\PGL(2,7)$8T43, 14T16, 16T713, 21T20, 24T707, 28T42, 28T46, 42T81, 42T82, 42T83
A8$A_8$8T49, 15T72, 28T433, 35T36
S8$S_8$8T50, 16T1838, 28T502, 30T1153, 35T44
C9$C_9$9T1
C3XC3$C_3^2$9T2
D9$D_{9}$9T3, 18T5
M9$C_3^2:Q_8$9T14, 12T47, 18T35, 24T82, 36T55
PSL(2,8)$\PSL(2,8)$9T27, 28T70, 36T712
A9$A_9$9T33, 36T23796
S9$S_9$9T34, 18T887, 36T28590
C10$C_{10}$10T1
D10$D_{10}$10T3, 20T4
PGL(2,9)$\PGL(2,9)$10T30, 12T182, 20T146, 30T171, 36T1254, 40T590, 45T110
M10$M_{10}$10T31, 12T181, 20T148, 20T150, 30T162, 36T1253, 40T591, 45T109
A10$A_{10}$10T44, 45T1982
S10$S_{10}$10T45, 20T1007, 45T2246
C11$C_{11}$11T1
D11$D_{11}$11T2, 22T2
F11$F_{11}$11T4, 22T4
PSL(2,11)$\PSL(2,11)$11T5, 12T179
M11$M_{11}$11T6, 12T272, 22T22
A11$A_{11}$11T7
S11$S_{11}$11T8, 22T45
C12$C_{12}$12T1
C2XC6$C_6\times C_2$12T2
C6XC2$C_6\times C_2$12T2
D12$D_{12}$12T12, 24T13
A12$A_{12}$12T300
S12$S_{12}$12T301, 24T24748
C13$C_{13}$13T1
D13$D_{13}$13T2, 26T2
F13$F_{13}$13T6, 26T8, 39T11
A13$A_{13}$13T8
S13$S_{13}$13T9, 26T83
C14$C_{14}$14T1
D14$D_{14}$14T3, 28T4
PGL(2,13)$\PGL(2,13)$14T39, 28T201, 42T284
A14A1414T62
S14S1414T63, 28T1755
C15$C_{15}$15T1
D15$D_{15}$15T2, 30T3
A15A1515T103
S15S1515T104, 30T5467
C16$C_{16}$16T1
Q16$Q_{16}$16T14
D16$D_{16}$16T56, 32T31
A16$A_{16}$16T1953
S16$S_{16}$16T1954, 32T2801205
C17$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}$18T1
D18$D_{18}$18T13
PGL(2,17)$\PGL(2,17)$18T468
A18$A_{18}$18T982
S18$S_{18}$18T983
C19$C_{19}$19T1
D19$D_{19}$19T2
A19$A_{19}$19T7
S19$S_{19}$19T8
C20$C_{20}$20T1
D20$D_{20}$20T10, 40T12
PGL(2,19)t20n36220T362, 40T5409
A20t20n111620T1116
S20t20n111720T1117, 40T315649
C21$C_{21}$21T1
D21$D_{21}$21T5, 42T5
A21A2121T163
S21S2121T164, 42T9215
C22$C_{22}$22T1
D22$D_{22}$22T3, 44T4
A22t22n5822T58
S22t22n5922T59, 44T2028
C23$C_{23}$23T1
D23$D_{23}$23T2, 46T2
F23$C_{23}:C_{11}$23T3
M23$M_{23}$23T5
A23$A_{23}$23T6
S23$S_{23}$23T7, 46T44
C24$C_{24}$24T1
D24$D_{24}$24T34
C25$C_{25}$25T1
D25$D_{25}$25T4
A25$A_{25}$25T210
S25$S_{25}$25T211
C26$C_{26}$26T1
D26$D_{26}$26T3
A26$A_{26}$26T95
S26$S_{26}$26T96
A24$A_{24}$24T24999
S24$S_{24}$24T25000
C27$C_{27}$27T1
D27$D_{27}$27T8
A27$A_{27}$27T2391
S27$S_{27}$27T2392
C28$C_{28}$28T1
D28$D_{28}$28T10
A28$A_{28}$28T1853
S28$S_{28}$28T1854
C29$C_{29}$29T1
D29$D_{29}$29T2
A29$A_{29}$29T7
S29$S_{29}$29T8
C30$C_{30}$30T1
D30$D_{30}$30T14
A30$A_{30}$30T5711
S30$S_{30}$30T5712
C31$C_{31}$31T1
D31$D_{31}$31T2
A31$A_{31}$31T11
S31$S_{31}$31T12
C32$C_{32}$32T33
D32$D_{32}$32T374
C33$C_{33}$33T1
D33$D_{33}$33T3
A33$A_{33}$33T161
S33$S_{33}$33T162
C34$C_{34}$34T1
D34$D_{34}$34T3
A34$A_{34}$34T114
S34$S_{34}$34T115
C35$C_{35}$35T1
D35$D_{35}$35T4
A35$A_{35}$35T406
S35$S_{35}$35T407
C36$C_{36}$36T1
D36$D_{36}$36T47
C37$C_{37}$37T1
D37$D_{37}$37T2
A37$A_{37}$37T10
S37$S_{37}$37T11
C38$C_{38}$38T1
D38$D_{38}$38T3
A38$A_{38}$38T75
S38$S_{38}$38T76
C39$C_{39}$39T1
D39$D_{39}$39T4
A39$A_{39}$39T305
S39$S_{39}$39T306
C40$C_{40}$40T1
D40$D_{40}$40T46
C41$C_{41}$41T1
D41$D_{41}$41T2
A41$A_{41}$41T9
S41$S_{41}$41T10
C42$C_{42}$42T1
D42$D_{42}$42T11
A42$A_{42}$42T9490
S42$S_{42}$42T9491
C43$C_{43}$43T1
D43$D_{43}$43T2
A43$A_{43}$43T9
S43$S_{43}$43T10
C44$C_{44}$44T1
D44$D_{44}$44T9
A44$A_{44}$44T2112
S44$S_{44}$44T2113
C45$C_{45}$45T1
D45$D_{45}$45T4
C46$C_{46}$46T1
D46$D_{46}$46T3
A46$A_{46}$46T55
S46$S_{46}$46T56
A45$A_{45}$45T10922
S45$S_{45}$45T10923
C47$C_{47}$47T1
D47$D_{47}$47T2
A47$A_{47}$47T5
S47$S_{47}$47T6
A36$A_{36}$36T121278
S36$S_{36}$36T121279
A40$A_{40}$40T315841
S40$S_{40}$40T315842
A32$A_{32}$32T2801323
S32$S_{32}$32T2801324
Authors:
Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2019-09-06 18:36:54
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