Group action invariants
| Degree $n$: | $26$ | |
| Transitive number $t$: | $30$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,17,5,24)(2,22,4,19)(3,14)(6,16,13,25)(7,21,12,20)(8,26,11,15)(9,18,10,23), (1,17,6,24,11,18,3,25,8,19,13,26,5,20,10,14,2,21,7,15,12,22,4,16,9,23) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 6: $S_3$ 8: $C_4\times C_2$ 12: $D_{6}$ 24: $S_3 \times C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T42 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Cycle Type | Size | Order | Representative |
| $ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ |
| $ 13, 13 $ | $24$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $13$ | $(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $12$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ |
| $ 13, 13 $ | $12$ | $13$ | $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ |
| $ 13, 13 $ | $12$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $507$ | $4$ | $( 1,17, 5,24)( 2,22, 4,19)( 3,14)( 6,16,13,25)( 7,21,12,20)( 8,26,11,15) ( 9,18,10,23)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $507$ | $4$ | $( 1,24)( 2,19,13,16)( 3,14,12,21)( 4,22,11,26)( 5,17,10,18)( 6,25, 9,23) ( 7,20, 8,15)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,25,18,19,17,21,26,16,23,22,24,20)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,16,18,22,17,20,26,25,23,19,24,21)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 26 $ | $156$ | $26$ | $( 1,17,12,25,10,20, 8,15, 6,23, 4,18, 2,26,13,21,11,16, 9,24, 7,19, 5,14, 3,22 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $39$ | $2$ | $( 1,25)( 2,21)( 3,17)( 4,26)( 5,22)( 6,18)( 7,14)( 8,23)( 9,19)(10,15)(11,24) (12,20)(13,16)$ |
| $ 26 $ | $156$ | $26$ | $( 1,23, 7,25,13,14, 6,16,12,18, 5,20,11,22, 4,24,10,26, 3,15, 9,17, 2,19, 8,21 )$ |
| $ 26 $ | $156$ | $26$ | $( 1,16, 2,25, 3,21, 4,17, 5,26, 6,22, 7,18, 8,14, 9,23,10,19,11,15,12,24,13,20 )$ |
| $ 26 $ | $156$ | $26$ | $( 1,24, 8,26, 2,15, 9,17, 3,19,10,21, 4,23,11,25, 5,14,12,16, 6,18,13,20, 7,22 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $39$ | $2$ | $( 1,19)( 2,23)( 3,14)( 4,18)( 5,22)( 6,26)( 7,17)( 8,21)( 9,25)(10,16)(11,20) (12,24)(13,15)$ |
| $ 26 $ | $156$ | $26$ | $( 1,17, 6,24,11,18, 3,25, 8,19,13,26, 5,20,10,14, 2,21, 7,15,12,22, 4,16, 9,23 )$ |
| $ 26 $ | $156$ | $26$ | $( 1,23, 4,22, 7,21,10,20,13,19, 3,18, 6,17, 9,16,12,15, 2,14, 5,26, 8,25,11,24 )$ |
Group invariants
| Order: | $4056=2^{3} \cdot 3 \cdot 13^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |