Group action invariants
| Degree $n$ : | $38$ | |
| Transitive number $t$ : | $28$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,24,3,28,6,34)(2,26,14,31,13,29)(4,30,17,37,8,38)(5,32,9,21,15,33)(7,36,12,27,10,23)(11,25,18,20,19,22)(16,35), (1,38,6,23,3,32)(2,35,13,21,14,37)(4,29,8,36,17,28)(5,26,15,34,9,33)(7,20,10,30,12,24)(11,27,19,22,18,25)(16,31) | |
| $|\Aut(F/K)|$: | $1$ |
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 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 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 $ | $1$ | $1$ | $()$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,33,27,21,34,28,22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,27,34,22,29,36, 24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $19$ | $(20,34,29,24,38,33,28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 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 $ | $361$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $3$ | $(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$ |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $228$ | $57$ | $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,21,28)(22,35,31) (24,30,34)(25,37,26)(27,32,29)(33,36,38)$ |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $228$ | $57$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,22,36)(21,29,28) (23,24,31)(25,38,34)(27,33,37)(30,35,32)$ |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $228$ | $57$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,24,33)(21,31,25) (22,38,36)(23,26,28)(27,35,34)(29,30,37)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $38$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$ |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $228$ | $57$ | $( 1,16,10)( 2, 8,17)( 3,19, 5)( 4,11,12)( 6,14, 7)(13,15,18)(20,21,22,23,24, 25,26,27,28,29,30,31,32,33,34,35,36,37,38)$ |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $228$ | $57$ | $( 1,12,19)( 2, 4, 7)( 3,15,14)( 5,18, 9)( 6,10,16)( 8,13,11)(20,22,24,26,28, 30,32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 3, 3, 3, 3, 3, 3, 1 $ | $228$ | $57$ | $( 1, 4,18)( 2,15, 6)( 3, 7,13)( 5,10, 8)( 9,16,17)(11,19,12)(20,24,28,32,36, 21,25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $722$ | $3$ | $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $1083$ | $6$ | $( 1,24, 3,28, 6,34)( 2,26,14,31,13,29)( 4,30,17,37, 8,38)( 5,32, 9,21,15,33) ( 7,36,12,27,10,23)(11,25,18,20,19,22)(16,35)$ |
| $ 38 $ | $342$ | $38$ | $( 1,26,11,37, 2,29,12,21, 3,32,13,24, 4,35,14,27, 5,38,15,30, 6,22,16,33, 7, 25,17,36, 8,28,18,20, 9,31,19,23,10,34)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $57$ | $2$ | $( 1,27)( 2,30)( 3,33)( 4,36)( 5,20)( 6,23)( 7,26)( 8,29)( 9,32)(10,35)(11,38) (12,22)(13,25)(14,28)(15,31)(16,34)(17,37)(18,21)(19,24)$ |
| $ 38 $ | $342$ | $38$ | $( 1,30, 9,35,17,21, 6,26,14,31, 3,36,11,22,19,27, 8,32,16,37, 5,23,13,28, 2, 33,10,38,18,24, 7,29,15,34, 4,20,12,25)$ |
| $ 38 $ | $342$ | $38$ | $( 1,23, 3,29, 5,35, 7,22, 9,28,11,34,13,21,15,27,17,33,19,20, 2,26, 4,32, 6, 38, 8,25,10,31,12,37,14,24,16,30,18,36)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $1083$ | $6$ | $( 1,29, 8,32,19,34)( 2,24,15,35,11,36)( 3,38)( 4,33,10,22,14,21) ( 5,28,17,25, 6,23)( 7,37,12,31, 9,27)(13,26,16,30,18,20)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $722$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
| $ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $722$ | $6$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11) (21,32,31,38,27,28)(22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $722$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,38)(22,37)(23,36) (24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 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 $ | $361$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $1083$ | $6$ | $( 1,24, 5,25,11,36)( 2,29,16,23,18,33)( 3,34, 8,21, 6,30)( 4,20,19,38,13,27) ( 7,35,14,32,15,37)( 9,26,17,28,10,31)(12,22)$ |
| $ 38 $ | $342$ | $38$ | $( 1,26, 6,35,11,25,16,34, 2,24, 7,33,12,23,17,32, 3,22, 8,31,13,21,18,30, 4, 20, 9,29,14,38,19,28, 5,37,10,27,15,36)$ |
| $ 38 $ | $342$ | $38$ | $( 1,27,11,26, 2,25,12,24, 3,23,13,22, 4,21,14,20, 5,38,15,37, 6,36,16,35, 7, 34,17,33, 8,32,18,31, 9,30,19,29,10,28)$ |
| $ 38 $ | $342$ | $38$ | $( 1,34, 8,20,15,25, 3,30,10,35,17,21, 5,26,12,31,19,36, 7,22,14,27, 2,32, 9, 37,16,23, 4,28,11,33,18,38, 6,24,13,29)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $57$ | $2$ | $( 1,25)( 2,23)( 3,21)( 4,38)( 5,36)( 6,34)( 7,32)( 8,30)( 9,28)(10,26)(11,24) (12,22)(13,20)(14,37)(15,35)(16,33)(17,31)(18,29)(19,27)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $1083$ | $6$ | $( 1,29,11,37, 5,36)( 2,26,18,35,16,22)( 3,23, 6,33, 8,27)( 4,20,13,31,19,32) ( 7,30,15,25,14,28)( 9,24,10,21,17,38)(12,34)$ |
Group invariants
| Order: | $12996=2^{2} \cdot 3^{2} \cdot 19^{2}$ | |
| Cyclic: | No | |
| Abelian: | No | |
| Solvable: | Yes | |
| GAP id: | Data not available |
| Character table: Data not available. |