Each Galois group is identified by a code "nTt" where \(n\) is its degree and \(t\) and its T-number. One may specify a group by this symbol.
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. These combinations for degree $\leq 11$ are shown below.
| 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 |