Group action invariants
| Degree $n$ : | $38$ | |
| Transitive number $t$ : | $18$ | |
| Parity: | $1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,23,18,33,13,20,10,35,12,25,17,38)(2,37,11,30,5,22,9,21,19,28,6,36)(3,32,4,27,16,24,8,26,7,31,14,34)(15,29), (1,36,7,25,3,26,12,38,6,30,10,29)(2,31,19,22,14,28,11,24,13,33,18,27)(4,21,5,35,17,32,9,34,8,20,15,23)(16,37) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ 3: $C_3$ 4: $C_4$ 6: $C_6$ 12: $C_{12}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: None
Low degree siblings
38T18 x 9Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,32,25,37,30,23,35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,25,30,35,21,26,31,36,22,27,32,37,23,28,33,38,24,29,34)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,38,37,36,35,34, 33,32,31,30,29,28,27,26,25,24,23,22,21)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,30,21,31,22,32,23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $12$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $361$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,23,18,33,13,20,10,35,12,25,17,38)( 2,37,11,30, 5,22, 9,21,19,28, 6,36) ( 3,32, 4,27,16,24, 8,26, 7,31,14,34)(15,29)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,34, 4,21, 9,31,11,35, 8,29, 3,38)( 2,36,12,37,16,26,10,33,19,32,15,24) ( 5,23,17,28,18,30, 7,27,14,22,13,20)( 6,25)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $361$ | $4$ | $( 1,22,19,38)( 2,25,18,35)( 3,28,17,32)( 4,31,16,29)( 5,34,15,26)( 6,37,14,23) ( 7,21,13,20)( 8,24,12,36)( 9,27,11,33)(10,30)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ | $361$ | $4$ | $( 1,37, 7,38)( 2,34, 6,22)( 3,31, 5,25)( 4,28)( 8,35,19,21)( 9,32,18,24) (10,29,17,27)(11,26,16,30)(12,23,15,33)(13,20,14,36)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,36,19,31, 7,28,15,30,16,35, 9,38)( 2,22,12,34,18,26,14,25, 4,32,17,21) ( 3,27, 5,37,10,24,13,20,11,29, 6,23)( 8,33)$ |
| $ 12, 12, 12, 2 $ | $361$ | $12$ | $( 1,25,15,35,13,20,16,33, 2,23, 4,38)( 3,21,12,22, 8,30,14,37, 5,36, 9,28) ( 6,34,17,31,10,26,11,24,19,27, 7,32)(18,29)$ |
Group invariants
| Order: | $4332=2^{2} \cdot 3 \cdot 19^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |