Group action invariants
| Degree $n$ : | $38$ | |
| Transitive number $t$ : | $41$ | |
| Parity: | $-1$ | |
| Primitive: | Yes | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (3,21,20,38,23,5,11,4,9,10,13,35,22,26,14,8,16,29,24,7,34,17,12,25,33,27,15,19,6,28,31,32,37,30,36,18), (1,21,2)(3,7,20)(5,13,11)(6,30,37)(8,9,36)(10,26,18)(12,14,17)(15,35,34)(16,29,33)(19,22,24)(23,31,25)(27,28,38) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 19: 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $1406$ | $3$ | $( 1,26, 5)( 2,16, 9)( 3,33,35)( 6, 7,25)( 8,19,18)(10,38,37)(11,12,31) (13,30,17)(15,21,23)(20,28,36)(22,27,24)(29,34,32)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $1406$ | $9$ | $( 1,22,31,26,27,11, 5,24,12)( 2,18,38,16, 8,37, 9,19,10)( 3,21,36,33,23,20,35, 15,28)( 6,32,13, 7,29,30,25,34,17)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $1406$ | $9$ | $( 1,31,27, 5,12,22,26,11,24)( 2,38, 8, 9,10,18,16,37,19)( 3,36,23,35,28,21,33, 20,15)( 6,13,29,25,17,32, 7,30,34)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $1406$ | $9$ | $( 1,27,12,26,24,31, 5,22,11)( 2, 8,10,16,19,38, 9,18,37)( 3,23,28,33,15,36,35, 21,20)( 6,29,17, 7,34,13,25,32,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $703$ | $2$ | $( 1,25)( 2,36)( 3,19)( 5, 7)( 6,26)( 8,35)( 9,28)(10,21)(11,13)(12,30)(15,37) (16,20)(17,31)(18,33)(22,34)(23,38)(24,29)(27,32)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $1406$ | $6$ | $( 1, 6, 5,25,26, 7)( 2,20, 9,36,16,28)( 3,18,35,19,33, 8)(10,23,37,21,38,15) (11,30,31,13,12,17)(22,32,24,34,27,29)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ | $1406$ | $4$ | $( 1,37,25,15)( 2,32,36,27)( 3,31,19,17)( 5,38, 7,23)( 6,21,26,10)( 8,30,35,12) ( 9,34,28,22)(11,18,13,33)(16,29,20,24)$ |
| $ 12, 12, 12, 1, 1 $ | $1406$ | $12$ | $( 1,38, 6,15, 5,10,25,23,26,37, 7,21)( 2,34,20,27, 9,29,36,22,16,32,28,24) ( 3,12,18,17,35,11,19,30,33,31, 8,13)$ |
| $ 12, 12, 12, 1, 1 $ | $1406$ | $12$ | $( 1,23, 6,37, 5,21,25,38,26,15, 7,10)( 2,22,20,32, 9,24,36,34,16,27,28,29) ( 3,30,18,31,35,13,19,12,33,17, 8,11)$ |
| $ 18, 18, 1, 1 $ | $1406$ | $18$ | $( 1,32,12, 6,24,17, 5,34,11,25,27,30,26,29,31, 7,22,13)( 2,35,10,20,19,23, 9, 33,37,36, 8,21,16, 3,38,28,18,15)$ |
| $ 18, 18, 1, 1 $ | $1406$ | $18$ | $( 1,30,24, 7,11,32,26,17,22,25,12,29, 5,13,27, 6,31,34)( 2,21,19,28,37,35,16, 23,18,36,10, 3, 9,15, 8,20,38,33)$ |
| $ 18, 18, 1, 1 $ | $1406$ | $18$ | $( 1,29,11, 6,22,30, 5,32,31,25,24,13,26,34,12, 7,27,17)( 2, 3,37,20,18,21, 9, 35,38,36,19,15,16,33,10,28, 8,23)$ |
| $ 36, 1, 1 $ | $1406$ | $36$ | $( 1, 3,32,38,12,28, 6,18,24,15,17, 2, 5,35,34,10,11,20,25,19,27,23,30, 9,26, 33,29,37,31,36, 7, 8,22,21,13,16)$ |
| $ 36, 1, 1 $ | $1406$ | $36$ | $( 1,19,32,23,12, 9, 6,33,24,37,17,36, 5, 8,34,21,11,16,25, 3,27,38,30,28,26, 18,29,15,31, 2, 7,35,22,10,13,20)$ |
| $ 36, 1, 1 $ | $1406$ | $36$ | $( 1,36,30,10,24, 3, 7, 9,11,15,32, 8,26,20,17,38,22,33,25, 2,12,21,29,19, 5, 28,13,37,27,35, 6,16,31,23,34,18)$ |
| $ 36, 1, 1 $ | $1406$ | $36$ | $( 1, 2,30,21,24,19, 7,28,11,37,32,35,26,16,17,23,22,18,25,36,12,10,29, 3, 5, 9,13,15,27, 8, 6,20,31,38,34,33)$ |
| $ 36, 1, 1 $ | $1406$ | $36$ | $( 1,35,29,38,11,36, 6,19,22,15,30,16, 5,33,32,10,31,28,25, 8,24,23,13, 2,26, 3,34,37,12,20, 7,18,27,21,17, 9)$ |
| $ 36, 1, 1 $ | $1406$ | $36$ | $( 1, 8,29,23,11, 2, 6, 3,22,37,30,20, 5,18,32,21,31, 9,25,35,24,38,13,36,26, 19,34,15,12,16, 7,33,27,10,17,28)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,10, 7,34,16,22,26,25, 2,15,29,12,23,19, 5, 6,13, 8,11)( 3,18,17,21,27, 9, 36,33, 4,32,35,30,37,38,24,20,31,14,28)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,12,34, 5,26, 8,15,10,23,16, 6,25,11,29, 7,19,22,13, 2)( 3,30,21,24,36,14, 32,18,37,27,20,33,28,35,17,38, 9,31, 4)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,25, 5, 7,15,13,16,12,11,26,19,10, 2, 6,34,29, 8,22,23)( 3,33,24,17,32,31, 27,30,28,36,38,18, 4,20,21,35,14, 9,37)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,22,29, 6,10,26,12,13, 7,25,23, 8,34, 2,19,11,16,15, 5)( 3, 9,35,20,18,36, 30,31,17,33,37,14,21, 4,38,28,27,32,24)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1, 8, 6,19,12,15,25,22,34,10,11,13, 5,23,29, 2,26,16, 7)( 3,14,20,38,30,32, 33, 9,21,18,28,31,24,37,35, 4,36,27,17)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,13,19,29,25,16,10, 8, 5,12, 2,22, 7,11, 6,23,15,26,34)( 3,31,38,35,33,27, 18,14,24,30, 4, 9,17,28,20,37,32,36,21)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,26,23,11,22,12, 8,16,29,13,34,15, 6, 7, 2, 5,10,25,19)( 3,36,37,28, 9,30, 14,27,35,31,21,32,20,17, 4,24,18,33,38)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,15,11, 2, 8,25,13,26, 6,22, 5,16,19,34,23, 7,12,10,29)( 3,32,28, 4,14,33, 31,36,20, 9,24,27,38,21,37,17,30,18,35)$ |
| $ 19, 19 $ | $1332$ | $19$ | $( 1,16, 2,23,13,10,22,15,19, 8, 7,26,29, 5,11,34,25,12, 6)( 3,27, 4,37,31,18, 9,32,38,14,17,36,35,24,28,21,33,30,20)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $666$ | $2$ | $( 1,17)( 2,35)( 3, 8)( 4,26)( 5,31)( 6,14)( 7,27)( 9,34)(10,21)(11,18)(12,38) (13,28)(15,30)(16,36)(19,20)(22,33)(23,24)(25,32)(29,37)$ |
| $ 38 $ | $1332$ | $38$ | $( 1,30,11,35, 8,32,13, 4, 6,33, 5,36,19, 9,23,27,12,21,29,17,15,18, 2, 3,25, 28,26,14,22,31,16,20,34,24, 7,38,10,37)$ |
| $ 38 $ | $1332$ | $38$ | $( 1,18, 8,28, 6,31,19,24,12,37,15,35,25, 4,22,36,34,27,10,17,11, 3,13,14, 5, 20,23,38,29,30, 2,32,26,33,16, 9, 7,21)$ |
| $ 38 $ | $1332$ | $38$ | $( 1, 3, 6,20,12,30,25,33,34,21,11,28, 5,24,29,35,26,36, 7,17, 8,14,19,38,15, 32,22, 9,10,18,13,31,23,37, 2, 4,16,27)$ |
| $ 38 $ | $1332$ | $38$ | $( 1,14,12,32,34,18, 5,37,26,27, 8,20,15,33,10,28,23,35,16,17, 6,38,25, 9,11, 31,29, 4, 7, 3,19,30,22,21,13,24, 2,36)$ |
| $ 38 $ | $1332$ | $38$ | $( 1,38,34,31,26, 3,15,21,23,36, 6,32,11,37, 7,20,22,28, 2,17,12, 9, 5, 4, 8, 30,10,24,16,14,25,18,29,27,19,33,13,35)$ |
| $ 38 $ | $1332$ | $38$ | $( 1, 9,26,30,23,14,11,27,22,35,12,31, 8,21,16,32,29,20,13,17,34, 4,15,24, 6, 18, 7,33, 2,38, 5, 3,10,36,25,37,19,28)$ |
| $ 38 $ | $1332$ | $38$ | $( 1, 4,23,18,22,38, 8,36,29,28,34,30, 6,27, 2,31,10,32,19,17,26,24,11,33,12, 3,16,37,13, 9,15,14, 7,35, 5,21,25,20)$ |
| $ 38 $ | $1332$ | $38$ | $( 1,24,22, 3,29, 9, 6,35,10,20,26,18,12,36,13,30, 7,31,25,17,23,33, 8,37,34, 14, 2,21,19, 4,11,38,16,28,15,27, 5,32)$ |
| $ 38 $ | $1332$ | $38$ | $( 1,33,29,14,10, 4,12,28, 7,32,23, 3,34,35,19,18,16,30, 5,17,22,37, 6,21,26, 38,13,27,25,24, 8, 9, 2,20,11,36,15,31)$ |
| $ 37, 1 $ | $1368$ | $37$ | $( 1,33,20,21, 3,14, 2,25,29,28,30,22,24,23,27,12,38,11,31,32,19,13, 6,35, 4, 9,37,18,16,26,36,34,15, 5,10,17, 8)$ |
Group invariants
| Order: | $50616=2^{3} \cdot 3^{2} \cdot 19 \cdot 37$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | No | |
| GAP id: | Data not available |
| Character table: Data not available. |