Group action invariants
| Degree $n$: | $26$ | |
| Transitive number $t$: | $40$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $1$ | |
| Generators: | (1,19,5,14,9,22,13,17,4,25,8,20,12,15,3,23,7,18,11,26,2,21,6,16,10,24), (1,19,11,25,10,14)(2,17,7,20,13,21)(3,15)(4,26,12,23,6,22)(5,24,8,18,9,16) |
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 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: None
Low degree siblings
39T46 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 $ | $18$ | $13$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $18$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,20,23,26,16,19,22,25,15,18,21, 24)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,26,25,24,23,22,21,20,19,18,17,16,15)$ |
| $ 13, 13 $ | $36$ | $13$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,16,18,20,22,24,26,15,17,19,21,23, 25)$ |
| $ 13, 13 $ | $36$ | $13$ | $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)(14,22,17,25,20,15,23,18,26,21,16,24, 19)$ |
| $ 13, 13 $ | $18$ | $13$ | $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$ |
| $ 13, 13 $ | $18$ | $13$ | $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)(14,20,26,19,25,18,24,17,23,16,22,15, 21)$ |
| $ 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $13$ | $(14,25,23,21,19,17,15,26,24,22,20,18,16)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $338$ | $3$ | $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)(15,17,23)(16,20,19)(18,26,24) (21,22,25)$ |
| $ 26 $ | $234$ | $26$ | $( 1,19, 5,14, 9,22,13,17, 4,25, 8,20,12,15, 3,23, 7,18,11,26, 2,21, 6,16,10,24 )$ |
| $ 26 $ | $234$ | $26$ | $( 1,22, 8,23, 2,24, 9,25, 3,26,10,14, 4,15,11,16, 5,17,12,18, 6,19,13,20, 7,21 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $39$ | $2$ | $( 1,15)( 2,17)( 3,19)( 4,21)( 5,23)( 6,25)( 7,14)( 8,16)( 9,18)(10,20)(11,22) (12,24)(13,26)$ |
| $ 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 $ | $156$ | $39$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,20)(15,25,16)(17,18,21) (19,24,26)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $156$ | $39$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,26)(15,20,22)(16,23,18) (21,25,24)$ |
| $ 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, 2 $ | $507$ | $6$ | $( 1,19,10,24,11,26)( 2,21,13,17, 7,18)( 3,23)( 4,25, 6,16,12,15) ( 5,14, 9,22, 8,20)$ |
| $ 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 $ | $156$ | $39$ | $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,22,16)(15,18,19)(17,23,25) (20,24,21)$ |
| $ 13, 3, 3, 3, 3, 1 $ | $156$ | $39$ | $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,17,18)(15,26,21)(16,22,24) (19,23,20)$ |
| $ 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, 2 $ | $507$ | $6$ | $( 1,19,12,15, 7,18)( 2,21, 8,20,10,24)( 3,23, 4,25,13,17)( 5,14, 9,22, 6,16) (11,26)$ |
| $ 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 $ | $169$ | $6$ | $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 6, 6, 6, 6, 2 $ | $507$ | $6$ | $( 1,19, 6,22,12,23)( 2,17)( 3,15,11,25, 5,24)( 4,26, 7,20, 8,18) ( 9,16,13,21,10,14)$ |
| $ 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 $ | $338$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,24,23,26,17,18)(16,21,19,25,20,22)$ |
| $ 26 $ | $234$ | $26$ | $( 1,19,13,21,12,23,11,25,10,14, 9,16, 8,18, 7,20, 6,22, 5,24, 4,26, 3,15, 2,17 )$ |
| $ 26 $ | $234$ | $26$ | $( 1,18, 7,19,13,20, 6,21,12,22, 5,23,11,24, 4,25,10,26, 3,14, 9,15, 2,16, 8,17 )$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $39$ | $2$ | $( 1,17)( 2,15)( 3,26)( 4,24)( 5,22)( 6,20)( 7,18)( 8,16)( 9,14)(10,25)(11,23) (12,21)(13,19)$ |
| $ 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 $ | $169$ | $6$ | $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)(15,18,17,26,23,24)(16,22,20,25,19,21)$ |
| $ 6, 6, 6, 6, 2 $ | $507$ | $6$ | $( 1,19, 8,18, 3,15)( 2,17,11,25,12,23)( 4,26)( 5,24, 7,20,13,21) ( 6,22,10,14, 9,16)$ |
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. |