Group action invariants
| Degree $n$ : | $38$ | |
| Transitive number $t$ : | $30$ | |
| Parity: | $-1$ | |
| Primitive: | No | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| Generators: | (1,6)(2,5)(3,4)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(20,30)(21,29)(22,28)(23,27)(24,26)(31,38)(32,37)(33,36)(34,35), (1,29,9,26,4,35,19,27,12,32,14,36,8,24,7,22,10,28)(2,31,6,20,13,34,11,30,17,23,18,25,15,38,5,37,16,21)(3,33) | |
| $|\Aut(F/K)|$: | $1$ |
Low degree resolvents
|G/N| Galois groups for stem field(s) 2: $C_2$ x 3 3: $C_3$ 4: $C_2^2$ 6: $C_6$ x 3 9: $C_9$ 12: $C_6\times C_2$ 18: $C_{18}$ x 3 36: 36T2 342: $F_{19}$ x 2 684: 38T9 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: None
Low degree siblings
38T30 x 8Siblings 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 $ | $36$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$ |
| $ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $36$ | $19$ | $(20,27,34,22,29,36,24,31,38,26,33,21,28,35,23,30,37,25,32)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$ |
| $ 19, 19 $ | $18$ | $19$ | $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,34,29,24,38,33, 28,23,37,32,27,22,36,31,26,21,35,30,25)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,33,27,21,34,28, 22,35,29,23,36,30,24,37,31,25,38,32,26)$ |
| $ 19, 19 $ | $36$ | $19$ | $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $361$ | $2$ | $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $361$ | $3$ | $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$ |
| $ 6, 6, 6, 6, 6, 6, 1, 1 $ | $361$ | $6$ | $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$ |
| $ 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 $ | $361$ | $6$ | $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2, 4,10, 9, 6,16, 8, 3, 7,19,17,11,12,15, 5,13,18,14)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,15, 7, 9,18,11, 8, 4, 5,19, 6,14,12, 3,10,13,17,16)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2, 3, 5, 9,17,14, 8,15,10,19,18,16,12, 4, 7,13, 6,11)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,30,25,32,26,23,31, 35,37,38,29,34,27,33,36,28,24,22)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,35,36,32,29,22,31, 33,25,38,24,23,27,30,37,28,26,34)$ |
| $ 9, 9, 9, 9, 1, 1 $ | $361$ | $9$ | $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$ |
| $ 18, 18, 1, 1 $ | $361$ | $18$ | $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,33,37,32,24,34,31, 30,36,38,26,22,27,35,25,28,29,23)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,29, 9,26, 4,35,19,27,12,32,14,36, 8,24, 7,22,10,28)( 2,31, 6,20,13,34,11, 30,17,23,18,25,15,38, 5,37,16,21)( 3,33)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,30, 4,24,14,23, 3,26,17,36,13,25, 6,20, 8,35, 2,28)( 5,22,11,29,12,27, 9, 33,18,34,10,31,15,21,19,32, 7,37)(16,38)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,24,10,32,11,35, 9,29,13,22, 5,36, 2,27, 8,26,15,28)( 3,30, 6,20,19,21,12, 38, 7,23,17,34,16,31,18,37,14,25)( 4,33)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,35,12,21, 9,30,15,31, 3,29, 8,33,17,25,18,22,16,28)( 2,32,10,27,13,37, 7, 36,19,38,14,34, 5,23, 4,26, 6,20)(11,24)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,26, 3,35,13,23, 6,20, 9,24, 5,25, 4,30,18,36,12,28)( 2,21, 8,29,19,31,17, 22, 7,34,14,37,11,33,15,32,16,27)(10,38)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,33, 6,20,12,31, 4,29, 2,38,11,26,18,23,15,27,19,28)( 3,24,16,32, 5,34, 7, 25,17,37,10,21,13,36, 9,35, 8,30)(14,22)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $361$ | $6$ | $( 1,31, 5,26,11,28)( 2,25,16,36,18,24)( 3,38, 8,27, 6,20)( 4,32,19,37,13,35) ( 7,33,14,29,15,23)( 9,21,17,30,10,34)(12,22)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $361$ | $6$ | $( 1,28)( 2,34,12,37, 8,32)( 3,21, 4,27,15,36)( 5,33, 7,26,10,25) ( 6,20,18,35,17,29)( 9,38,13,24,19,22)(11,31,16,23,14,30)$ |
| $ 38 $ | $342$ | $38$ | $( 1,27,18,26,16,25,14,24,12,23,10,22, 8,21, 6,20, 4,38, 2,37,19,36,17,35,15, 34,13,33,11,32, 9,31, 7,30, 5,29, 3,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $19$ | $2$ | $( 1,26)( 2,36)( 3,27)( 4,37)( 5,28)( 6,38)( 7,29)( 8,20)( 9,30)(10,21)(11,31) (12,22)(13,32)(14,23)(15,33)(16,24)(17,34)(18,25)(19,35)$ |
| $ 38 $ | $342$ | $38$ | $( 1,32,12,36, 4,21,15,25, 7,29,18,33,10,37, 2,22,13,26, 5,30,16,34, 8,38,19, 23,11,27, 3,31,14,35, 6,20,17,24, 9,28)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $19$ | $2$ | $( 1,37)( 2,27)( 3,36)( 4,26)( 5,35)( 6,25)( 7,34)( 8,24)( 9,33)(10,23)(11,32) (12,22)(13,31)(14,21)(15,30)(16,20)(17,29)(18,38)(19,28)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $361$ | $6$ | $( 1,21,18,29, 4,28)( 2,36, 6,20,15,22)( 3,32,13,30, 7,35)( 5,24, 8,31,10,23) ( 9,27,17,33,16,37)(11,38,12,34,19,25)(14,26)$ |
| $ 6, 6, 6, 6, 6, 6, 2 $ | $361$ | $6$ | $( 1,38, 9,32, 8,28)( 2,23,16,22,19,34)( 3,27, 4,31,11,21)( 5,35,18,30,14,33) ( 6,20)( 7,24,13,29,17,26)(10,36,15,37,12,25)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,25,13,32, 4,22, 6,20,14,31, 8,37, 3,23, 2,24,17,28)( 5,21,10,35,11,34,15, 30,12,33,19,26, 9,36, 7,38,18,27)(16,29)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,34, 6,20, 7,21,11,25, 8,22,15,29, 5,38, 3,36,14,28)( 2,35,10,24, 4,37,18, 32,17,31,13,27,16,30, 9,23,19,33)(12,26)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,37, 4,23,12,30, 8,36,10,33, 9,25,19,29,14,27, 7,28)( 2,26,13,38,17,32,15, 35,16,24, 6,20,11,22,18,21, 5,31)( 3,34)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,22, 3,25, 2,33,12,29, 7,31,19,30,13,21,16,35, 5,28)( 4,36,11,37,17,27,14, 32, 6,20,10,26, 8,23, 9,34,18,38)(15,24)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,36, 5,27,10,30, 2,29,11,23, 8,25, 9,37,15,33,13,28)( 3,22,17,38, 6,20,16, 26,19,24,18,31,12,35,14,21, 7,32)( 4,34)$ |
| $ 18, 18, 2 $ | $361$ | $18$ | $( 1,23,13,31, 9,22, 4,25,12,24, 3,37, 6,20, 5,32,18,28)( 2,30,19,35, 7,27,11, 36,16,33, 8,34,17,21,14,38,15,26)(10,29)$ |
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. |