Properties

Label 38T31
Order \(19494\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $38$
Transitive number $t$ :  $31$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,36,17,28,15,29)(2,26,5,34,7,33)(3,35,12,21,18,37)(4,25,19,27,10,22)(6,24,14,20,13,30)(8,23,9,32,16,38)(11,31), (1,18,5,6,3,12,4,9,13)(2,15,14,17,8,16,11,7,19)(20,21,25,22,29,38,36,28,34)(23,33,35,24,37,32,31,27,30)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$, $D_{9}$
54:  18T19

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 TypeSizeOrderRepresentative
$ 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 $ $27$ $19$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$
$ 19, 19 $ $27$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$
$ 19, 19 $ $27$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$
$ 19, 19 $ $27$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$
$ 19, 19 $ $27$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$
$ 19, 19 $ $27$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $19$ $(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38)$
$ 19, 19 $ $54$ $19$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$
$ 19, 19 $ $54$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,31,23,34,26,37, 29,21,32,24,35,27,38,30,22,33,25,36,28)$
$ 19, 19 $ $54$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $19$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$
$ 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, 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, 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 $ $342$ $57$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,31,32)(21,38,24) (22,26,35)(23,33,27)(25,28,30)(29,37,36)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $342$ $57$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,23,25)(21,30,36) (22,37,28)(24,32,31)(26,27,34)(33,38,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $722$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
$ 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 $ $342$ $57$ $( 1,12,19)( 2, 4, 7)( 3,15,14)( 5,18, 9)( 6,10,16)( 8,13,11)(20,31,23,34,26, 37,29,21,32,24,35,27,38,30,22,33,25,36,28)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $342$ $57$ $( 1, 4,18)( 2,15, 6)( 3, 7,13)( 5,10, 8)( 9,16,17)(11,19,12)(20,23,26,29,32, 35,38,22,25,28,31,34,37,21,24,27,30,33,36)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 6, 6, 6, 6, 6, 6, 2 $ $3249$ $6$ $( 1,36,17,28,15,29)( 2,26, 5,34, 7,33)( 3,35,12,21,18,37)( 4,25,19,27,10,22) ( 6,24,14,20,13,30)( 8,23, 9,32,16,38)(11,31)$
$ 38 $ $513$ $38$ $( 1,37,15,26,10,34, 5,23,19,31,14,20, 9,28, 4,36,18,25,13,33, 8,22, 3,30,17, 38,12,27, 7,35, 2,24,16,32,11,21, 6,29)$
$ 38 $ $513$ $38$ $( 1,29,12,38, 4,28,15,37, 7,27,18,36,10,26, 2,35,13,25, 5,34,16,24, 8,33,19, 23,11,32, 3,22,14,31, 6,21,17,30, 9,20)$
$ 38 $ $513$ $38$ $( 1,21, 9,31,17,22, 6,32,14,23, 3,33,11,24,19,34, 8,25,16,35, 5,26,13,36, 2, 27,10,37,18,28, 7,38,15,29, 4,20,12,30)$
$ 38 $ $513$ $38$ $( 1,30,10,27,19,24, 9,21,18,37, 8,34,17,31, 7,28,16,25, 6,22,15,38, 5,35,14, 32, 4,29,13,26, 3,23,12,20, 2,36,11,33)$
$ 38 $ $513$ $38$ $( 1,28,14,30, 8,32, 2,34,15,36, 9,38, 3,21,16,23,10,25, 4,27,17,29,11,31, 5, 33,18,35,12,37, 6,20,19,22,13,24, 7,26)$
$ 38 $ $513$ $38$ $( 1,26,18,33,16,21,14,28,12,35,10,23, 8,30, 6,37, 4,25, 2,32,19,20,17,27,15, 34,13,22,11,29, 9,36, 7,24, 5,31, 3,38)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $171$ $2$ $( 1,25)( 2,31)( 3,37)( 4,24)( 5,30)( 6,36)( 7,23)( 8,29)( 9,35)(10,22)(11,28) (12,34)(13,21)(14,27)(15,33)(16,20)(17,26)(18,32)(19,38)$
$ 6, 6, 6, 6, 6, 6, 2 $ $3249$ $6$ $( 1,25, 9,38, 2,29)( 3,33,12,31,16,28)( 4,37)( 5,22,15,24,11,27) ( 6,26, 7,30,18,36)( 8,34,10,23,13,35)(14,20,19,21,17,32)$

Group invariants

Order:  $19494=2 \cdot 3^{3} \cdot 19^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.