Group action invariants
| Degree $n$ : | $26$ | |
| Transitive number $t$ : | $50$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,23,7,25,5,20,10,26,4,24,6,16)(2,19,11,22,8,21,9,17,13,14,3,15)(12,18), (1,19,4,25,8,20,9,22,6,16,2,21)(3,23,11,26,13,17,7,18,12,15,10,24)(5,14) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $S_3$, $C_6$ x 3 8: $Q_8$ 12: $D_{6}$, $C_6\times C_2$ 18: $S_3\times C_3$ 24: 24T4, 24T5 36: $C_6\times S_3$ 72: 24T64 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $13$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| $ 13, 13 $ | $72$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ |
| $ 13, 13 $ | $72$ | $13$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18, 16)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $169$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $169$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $169$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 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 $ | $169$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $(15,17,23)(16,20,19)(18,26,24)(21,22,25)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $312$ | $39$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)(15,23,17)(16,19,20)(18,24,26) (21,25,22)$ |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $26$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $312$ | $39$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(14,26,25,24,23,22,21,20,19,18,17,16, 15)$ |
| $ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)$ |
| $ 6, 6, 2, 2, 2, 2, 2, 2, 1, 1 $ | $338$ | $6$ | $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,24,23,26,17,18)(16,21,19,25,20, 22)$ |
| $ 6, 6, 6, 6, 1, 1 $ | $338$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 12, 12, 2 $ | $1014$ | $12$ | $( 1,23, 7,25, 5,20,10,26, 4,24, 6,16)( 2,19,11,22, 8,21, 9,17,13,14, 3,15) (12,18)$ |
| $ 12, 12, 2 $ | $1014$ | $12$ | $( 1,15, 4,18, 8,22, 9,23, 6,20, 2,16)( 3,17,11,25,13,14, 7,21,12,26,10,24) ( 5,19)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,17, 5,16)( 2,20, 4,26)( 3,23)( 6,19,13,14)( 7,22,12,24)( 8,25,11,21) ( 9,15,10,18)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| $ 12, 4, 4, 4, 1, 1 $ | $338$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 12, 4, 4, 4, 1, 1 $ | $338$ | $12$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,16,18,22,17,20,26,25,23,19,24,21)$ |
| $ 12, 4, 4, 4, 1, 1 $ | $338$ | $12$ | $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,25,18,19,17,21,26,16,23,22,24,20)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 3, 5, 9, 4, 7,13,12,10, 6,11, 8)(15,21,24,19,23,25,26,20,17,22,18,16)$ |
| $ 12, 4, 4, 4, 1, 1 $ | $338$ | $12$ | $( 2, 7,11, 9,10, 3,13, 8, 4, 6, 5,12)(15,22,26,19)(16,17,25,24)(18,20,23,21)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2,12, 5, 6, 4, 8,13, 3,10, 9,11, 7)(15,16,18,22,17,20,26,25,23,19,24,21)$ |
| $ 12, 12, 1, 1 $ | $338$ | $12$ | $( 2, 8,11, 6,10,12,13, 7, 4, 9, 5, 3)(15,20,24,22,23,16,26,21,17,19,18,25)$ |
| $ 4, 4, 4, 4, 4, 4, 1, 1 $ | $338$ | $4$ | $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,19,26,22)(16,24,25,17)(18,21,23,20)$ |
| $ 12, 12, 2 $ | $1014$ | $12$ | $( 1,23)( 2,18, 5,16, 4,21,13,15,10,17,11,25)( 3,26, 9,22, 7,19,12,20, 6,24, 8, 14)$ |
| $ 12, 12, 2 $ | $1014$ | $12$ | $( 1,15, 2,26,12,19, 8,14, 7,16,10,23)( 3,24, 9,25, 4,22, 6,18,13,17, 5,20) (11,21)$ |
| $ 4, 4, 4, 4, 4, 4, 2 $ | $1014$ | $4$ | $( 1,17,13,23)( 2,24,12,16)( 3,18,11,22)( 4,25,10,15)( 5,19, 9,21)( 6,26, 8,14) ( 7,20)$ |
Group invariants
| Order: | $12168=2^{3} \cdot 3^{2} \cdot 13^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |