Group action invariants
| Degree $n$: | $26$ | |
| Transitive number $t$: | $41$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,21,7,25,2,26,4,23,11,19,3,18)(5,15,8,17,12,24,13,16,10,14,6,20)(9,22), (1,18,9,21,2,20,13,16,5,26,12,14)(3,22,4,24,8,19,11,25,10,23,6,15)(7,17) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $4$: $C_4$ $6$: $S_3$, $C_6$ $12$: $C_{12}$, $C_3 : C_4$ $18$: $S_3\times C_3$ $36$: $C_3\times (C_3 : C_4)$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T45 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 $ | $36$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ |
| $ 13, 13 $ | $36$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,18,22,26,17,21,25,16,20,24,15,19,23)$ |
| $ 13, 13 $ | $36$ | $13$ | $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ |
| $ 13, 13 $ | $36$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,22,17,25,20,15,23,18,26,21,16,24,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $169$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $169$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $169$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $169$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $(15,23,17)(16,19,20)(18,24,26)(21,25,22)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $156$ | $39$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,26,17)(15,22,20)(16,18,23) (21,24,25)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $156$ | $39$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,25,20)(15,21,23)(16,17,26) (18,22,19)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $156$ | $39$ | $( 1, 9, 7)( 2,12, 3)( 4, 5, 8)( 6,11,13)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $156$ | $39$ | $( 1, 4,13)( 2, 7, 9)( 3,10, 5)( 8,12,11)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ |
| $ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ | $338$ | $6$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,18,17,26,23,24)(16,22,20,25,19, 21)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 12, 12, 2 $ | $507$ | $12$ | $( 1,21, 7,25, 2,26, 4,23,11,19, 3,18)( 5,15, 8,17,12,24,13,16,10,14, 6,20) ( 9,22)$ |
| $ 12, 12, 2 $ | $507$ | $12$ | $( 1,25,10,14, 7,22, 8,15,12,26, 2,18)( 3,24, 5,23,13,19, 6,16, 4,17, 9,21) (11,20)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $507$ | $4$ | $( 1,22,12,18)( 2,24,11,16)( 3,26,10,14)( 4,15, 9,25)( 5,17, 8,23)( 6,19, 7,21) (13,20)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $507$ | $4$ | $( 1,19, 8,18)( 2,17, 7,20)( 3,15, 6,22)( 4,26, 5,24)( 9,16,13,21)(10,14,12,23) (11,25)$ |
| $ 12, 12, 2 $ | $507$ | $12$ | $( 1,20,10,14, 9,19,12,17, 3,23, 4,18)( 2,15, 7,16, 5,26,11,22, 6,21, 8,24) (13,25)$ |
| $ 12, 12, 2 $ | $507$ | $12$ | $( 1,16,11,21,12,15, 3,17, 6,25, 5,18)( 2,23)( 4,24,10,14, 8,26,13,22, 7,19, 9, 20)$ |
Group invariants
| Order: | $6084=2^{2} \cdot 3^{2} \cdot 13^{2}$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | not available |
| Character table: not available. |