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 $n\leq 47$.
| Alias | Group | \(n\)T\(t\) |
|---|---|---|
| S1 | Trivial | 1T1 |
| C1 | Trivial | 1T1 |
| A1 | Trivial | 1T1 |
| A2 | Trivial | 1T1 |
| 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 |
| A14 | A14 | 14T62 |
| S14 | S14 | 14T63, 28T1755 |
| C15 | $C_{15}$ | 15T1 |
| D15 | $D_{15}$ | 15T2, 30T3 |
| A15 | A15 | 15T103 |
| S15 | S15 | 15T104, 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) | t20n362 | 20T362, 40T5409 |
| A20 | t20n1116 | 20T1116 |
| S20 | t20n1117 | 20T1117, 40T315649 |
| C21 | $C_{21}$ | 21T1 |
| D21 | $D_{21}$ | 21T5, 42T5 |
| A21 | A21 | 21T163 |
| S21 | S21 | 21T164, 42T9215 |
| C22 | $C_{22}$ | 22T1 |
| D22 | $D_{22}$ | 22T3, 44T4 |
| A22 | t22n58 | 22T58 |
| S22 | t22n59 | 22T59, 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
Referred to by:
History:
(expand/hide all)
- nf.galois_group
- lmfdb/abvar/fq/main.py (line 283)
- lmfdb/abvar/fq/search_parsing.py (line 61)
- lmfdb/artin_representations/templates/artin-representation-index.html (line 71)
- lmfdb/local_fields/templates/lf-index.html (line 138)
- lmfdb/number_fields/templates/nf-index.html (line 92)
- lmfdb/utils/search_parsing.py (line 616)
- 2019-12-29 13:06:18 by John Jones
- 2019-12-15 22:29:07 by Alex J. Best
- 2019-12-15 22:21:03 by Alex J. Best
- 2019-09-06 18:36:54 by John Jones (Reviewed)
- 2019-04-25 10:10:27 by John Jones (Reviewed)
- 2012-07-01 07:27:56 by John Jones