Group action invariants
| Degree $n$ : | $26$ | |
| Transitive number $t$ : | $38$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,21,3,18,5,15,7,25,9,22,11,19,13,16,2,26,4,23,6,20,8,17,10,14,12,24), (1,13,12,11,10,9,8,7,6,5,4,3,2)(14,24,26,16)(15,19,25,21)(17,22,23,18) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 4: $C_4$ x 2, $C_2^2$ 8: $D_{4}$ x 2, $C_4\times C_2$ 16: $C_2^2:C_4$ 32: $C_4\wr C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
There are no siblings 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 $ | $16$ | $13$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,21,18,15,25,22,19,16,26,23,20, 17)$ |
| $ 13, 13 $ | $16$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ |
| $ 13, 13 $ | $16$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,16,17,18,19,20,21,22,23,24,25, 26)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $13$ | $(14,19,24,16,21,26,18,23,15,20,25,17,22)$ |
| $ 13, 13 $ | $32$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ |
| $ 13, 13 $ | $32$ | $13$ | $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)(14,21,15,22,16,23,17,24,18,25,19,26, 20)$ |
| $ 13, 13 $ | $32$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $13$ | $(14,24,21,18,15,25,22,19,16,26,23,20,17)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $13$ | $(14,21,15,22,16,23,17,24,18,25,19,26,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $169$ | $2$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 26 $ | $208$ | $26$ | $( 1,21, 3,18, 5,15, 7,25, 9,22,11,19,13,16, 2,26, 4,23, 6,20, 8,17,10,14,12,24 )$ |
| $ 26 $ | $208$ | $26$ | $( 1,18, 7,22,13,26, 6,17,12,21, 5,25,11,16, 4,20,10,24, 3,15, 9,19, 2,23, 8,14 )$ |
| $ 26 $ | $208$ | $26$ | $( 1,22, 6,21,11,20, 3,19, 8,18,13,17, 5,16,10,15, 2,14, 7,26,12,25, 4,24, 9,23 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $52$ | $2$ | $( 1,16)( 2,21)( 3,26)( 4,18)( 5,23)( 6,15)( 7,20)( 8,25)( 9,17)(10,22)(11,14) (12,19)(13,24)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $338$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $2$ | $(15,26)(16,25)(17,24)(18,23)(19,22)(20,21)$ |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $104$ | $26$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24)(15,23)(16,22)(17,21)(18,20) (25,26)$ |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $104$ | $26$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21)(15,20)(16,19)(17,18)(22,26) (23,25)$ |
| $ 13, 2, 2, 2, 2, 2, 2, 1 $ | $104$ | $26$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15)(16,26)(17,25)(18,24)(19,23) (20,22)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $676$ | $4$ | $( 1,21, 8,17)( 2,26, 7,25)( 3,18, 6,20)( 4,23, 5,15)( 9,22,13,16)(10,14,12,24) (11,19)$ |
| $ 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 $ | $169$ | $4$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $4$ | $(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| $ 13, 4, 4, 4, 1 $ | $104$ | $52$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,26,16)(15,19,25,21) (17,22,23,18)$ |
| $ 13, 4, 4, 4, 1 $ | $104$ | $52$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,25,18)(15,16,24,23) (17,19,22,20)$ |
| $ 13, 4, 4, 4, 1 $ | $104$ | $52$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,23,22)(16,18,21,19) (17,26,20,24)$ |
| $ 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $338$ | $4$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,19,26,22)(16,24,25,17) (18,21,23,20)$ |
| $ 8, 8, 8, 2 $ | $676$ | $8$ | $( 1,21, 5,15,11,19, 7,25)( 2,26,13,16,10,14,12,24)( 3,18, 8,17, 9,22, 4,23) ( 6,20)$ |
| $ 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $4$ | $(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 13, 4, 4, 4, 1 $ | $104$ | $52$ | $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,24,22,25)(15,16,21,20) (17,26,19,23)$ |
| $ 13, 4, 4, 4, 1 $ | $104$ | $52$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,21,17,23)(15,26,16,18) (19,20,25,24)$ |
| $ 13, 4, 4, 4, 1 $ | $104$ | $52$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,15,20,19)(16,25,18,22) (21,24,26,23)$ |
| $ 4, 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $338$ | $4$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,22,26,19)(16,17,25,24) (18,20,23,21)$ |
| $ 8, 8, 8, 2 $ | $676$ | $8$ | $( 1,21, 6,20, 5,15,13,16)( 2,26,11,19, 4,23, 8,17)( 3,18)( 7,25,10,14,12,24, 9,22)$ |
Group invariants
| Order: | $5408=2^{5} \cdot 13^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |