Properties

Label 38T41
Order \(50616\)
n \(38\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No

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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 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $666$ $2$ $( 1,18)( 2,16)( 3,17)( 4, 7)( 5,38)( 6,35)( 8,31)( 9,37)(10,20)(11,26)(12,15) (13,22)(14,19)(21,33)(23,25)(24,36)(27,28)(29,30)(32,34)$
$ 19, 19 $ $1332$ $19$ $( 1,12, 6,19,10,17,29,32,24, 8,26,21,23,27,16,37, 5,22, 4)( 2, 9,38,13, 7,18, 15,35,14,20, 3,30,34,36,31,11,33,25,28)$
$ 19, 19 $ $1332$ $19$ $( 1,21,19,16,29,22, 8,12,23,10,37,32, 4,26, 6,27,17, 5,24)( 2,30,13,31,15,25, 20, 9,34, 7,11,35,28, 3,38,36,18,33,14)$
$ 19, 19 $ $1332$ $19$ $( 1,32,16, 6, 8, 5,10,21, 4,29,27,12,24,37,19,26,22,17,23)( 2,35,31,38,20,33, 7,30,28,15,36, 9,14,11,13, 3,25,18,34)$
$ 19, 19 $ $1332$ $19$ $( 1,17,26,37,12,29,21, 5, 6,32,23,22,19,24,27, 4,10, 8,16)( 2,18, 3,11, 9,15, 30,33,38,35,34,25,13,14,36,28, 7,20,31)$
$ 19, 19 $ $1332$ $19$ $( 1,22,37,27,21, 8,32,17,19,12, 4, 5,16,23,26,24,29,10, 6)( 2,25,11,36,30,20, 35,18,13, 9,28,33,31,34, 3,14,15, 7,38)$
$ 19, 19 $ $1332$ $19$ $( 1, 5,27,26,32,10,12,22,16,21,24,17, 6, 4,37,23, 8,29,19)( 2,33,36, 3,35, 7, 9,25,31,30,14,18,38,28,11,34,20,15,13)$
$ 19, 19 $ $1332$ $19$ $( 1,29,23, 4,17,21,22,10,26, 5,19, 8,37, 6,24,16,12,32,27)( 2,15,34,28,18,30, 25, 7, 3,33,13,20,11,38,14,31, 9,35,36)$
$ 19, 19 $ $1332$ $19$ $( 1, 8, 4,24,22,32, 5,29,37,17,16,10,27,19,23, 6,21,12,26)( 2,20,28,14,25,35, 33,15,11,18,31, 7,36,13,34,38,30, 9, 3)$
$ 19, 19 $ $1332$ $19$ $( 1,10,24,23, 5,12,17, 8,27,22, 6,29,26,16, 4,19,32,21,37)( 2, 7,14,34,33, 9, 18,20,36,25,38,15, 3,31,28,13,35,30,11)$
$ 38 $ $1332$ $38$ $( 1,11,12,33, 6,25,19,28,10, 2,17, 9,29,38,32,13,24, 7, 8,18,26,15,21,35,23, 14,27,20,16, 3,37,30, 5,34,22,36, 4,31)$
$ 38 $ $1332$ $38$ $( 1,15, 6,14,10, 3,29,34,24,31,26,33,23,28,16, 9, 5,13, 4,18,12,35,19,20,17, 30,32,36, 8,11,21,25,27, 2,37,38,22, 7)$
$ 38 $ $1332$ $38$ $( 1,35,10,30,24,11,23, 2, 5, 7,12,14,17,34, 8,33,27, 9,22,18, 6,20,29,36,26, 25,16,38, 4,15,19, 3,32,31,21,28,37,13)$
$ 38 $ $1332$ $38$ $( 1,20,24,25, 5,15,17,31,27,13, 6,30,26, 2, 4,14,32,33,37,18,10,36,23,38,12, 3, 8,28,22,35,29,11,16, 7,19,34,21, 9)$
$ 38 $ $1332$ $38$ $( 1,36, 5, 3,27,35,26, 7,32, 9,10,25,12,31,22,30,16,14,21,18,24,38,17,28, 6, 11, 4,34,37,20,23,15, 8,13,29, 2,19,33)$
$ 38 $ $1332$ $38$ $( 1,38,27,11,32,20,12,13,16,33,24, 3, 6, 7,37,25, 8,30,19,18, 5,28,26,34,10, 15,22, 2,21,36,17,35, 4, 9,23,31,29,14)$
$ 38 $ $1332$ $38$ $( 1,28,32,15,16,36, 6, 9, 8,14, 5,11,10,13,21, 3, 4,25,29,18,27,34,12, 2,24, 35,37,31,19,38,26,20,22,33,17, 7,23,30)$
$ 38 $ $1332$ $38$ $( 1,34,16,35, 8,38,10,33, 4,30,27,15,24, 9,19,11,22, 3,23,18,32, 2, 6,31, 5, 20,21, 7,29,28,12,36,37,14,26,13,17,25)$
$ 38 $ $1332$ $38$ $( 1, 2, 8,20, 4,28,24,14,22,25,32,35, 5,33,29,15,37,11,17,18,16,31,10, 7,27, 36,19,13,23,34, 6,38,21,30,12, 9,26, 3)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $703$ $2$ $( 1,35)( 2,36)( 3,27)( 4,17)( 5,14)( 6, 8)( 7,22)( 9,16)(10,34)(11,37)(13,19) (15,30)(18,24)(20,33)(21,28)(23,32)(26,38)(29,31)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $1406$ $3$ $( 1,10, 5)( 2,23, 3)( 4,24,15)( 6,37, 9)( 7,19,20)( 8,11,16)(13,33,22) (14,35,34)(17,18,30)(21,26,29)(27,36,32)(28,38,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1 $ $1406$ $4$ $( 1,28,35,21)( 2, 9,36,16)( 3,37,27,11)( 4, 7,17,22)( 5,31,14,29)( 6,32, 8,23) (10,38,34,26)(13,24,19,18)(15,20,30,33)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $1406$ $6$ $( 1,14,10,35, 5,34)( 2,27,23,36, 3,32)( 4,30,24,17,15,18)( 6,16,37, 8, 9,11) ( 7,33,19,22,20,13)(21,31,26,28,29,38)$
$ 9, 9, 9, 9, 1, 1 $ $1406$ $9$ $( 1,24,16,10,15, 8, 5, 4,11)( 2,38,20,23,31, 7, 3,28,19)( 6,14,17,37,35,18, 9, 34,30)(13,36,26,33,32,29,22,27,21)$
$ 9, 9, 9, 9, 1, 1 $ $1406$ $9$ $( 1,16,15, 5,11,24,10, 8, 4)( 2,20,31, 3,19,38,23, 7,28)( 6,17,35, 9,30,14,37, 18,34)(13,26,32,22,21,36,33,29,27)$
$ 9, 9, 9, 9, 1, 1 $ $1406$ $9$ $( 1,15,11,10, 4,16, 5,24, 8)( 2,31,19,23,28,20, 3,38, 7)( 6,35,30,37,34,17, 9, 14,18)(13,32,21,33,27,26,22,36,29)$
$ 12, 12, 12, 1, 1 $ $1406$ $12$ $( 1,26,14,28,10,29,35,38, 5,21,34,31)( 2, 8,27, 9,23,11,36, 6, 3,16,32,37) ( 4,13,30, 7,24,33,17,19,15,22,18,20)$
$ 12, 12, 12, 1, 1 $ $1406$ $12$ $( 1,38,14,21,10,31,35,26, 5,28,34,29)( 2, 6,27,16,23,37,36, 8, 3, 9,32,11) ( 4,19,30,22,24,20,17,13,15, 7,18,33)$
$ 18, 18, 1, 1 $ $1406$ $18$ $( 1, 6,24,14,16,17,10,37,15,35, 8,18, 5, 9, 4,34,11,30)( 2,22,38,27,20,21,23, 13,31,36, 7,26, 3,33,28,32,19,29)$
$ 18, 18, 1, 1 $ $1406$ $18$ $( 1,18,16,34,15, 6, 5,17,11,35,24, 9,10,30, 8,14, 4,37)( 2,26,20,32,31,22, 3, 21,19,36,38,33,23,29, 7,27,28,13)$
$ 18, 18, 1, 1 $ $1406$ $18$ $( 1, 9,15,14,11,18,10, 6, 4,35,16,30, 5,37,24,34, 8,17)( 2,33,31,27,19,26,23, 22,28,36,20,29, 3,13,38,32, 7,21)$
$ 36, 1, 1 $ $1406$ $36$ $( 1, 7, 6,26,24, 3,14,33,16,28,17,32,10,19,37,29,15, 2,35,22, 8,38,18,27, 5, 20, 9,21, 4,23,34,13,11,31,30,36)$
$ 36, 1, 1 $ $1406$ $36$ $( 1,22, 6,38,24,27,14,20,16,21,17,23,10,13,37,31,15,36,35, 7, 8,26,18, 3, 5, 33, 9,28, 4,32,34,19,11,29,30, 2)$
$ 36, 1, 1 $ $1406$ $36$ $( 1,23,18,29,16, 7,34,27,15,28, 6,13, 5, 2,17,26,11,20,35,32,24,31, 9,22,10, 3,30,21, 8,19,14,36, 4,38,37,33)$
$ 36, 1, 1 $ $1406$ $36$ $( 1,32,18,31,16,22,34, 3,15,21, 6,19, 5,36,17,38,11,33,35,23,24,29, 9, 7,10, 27,30,28, 8,13,14, 2, 4,26,37,20)$
$ 36, 1, 1 $ $1406$ $36$ $( 1,19, 9,26,15,23,14,22,11,28,18,36,10,20, 6,29, 4, 3,35,13,16,38,30,32, 5, 7,37,21,24, 2,34,33, 8,31,17,27)$
$ 36, 1, 1 $ $1406$ $36$ $( 1,13, 9,38,15,32,14, 7,11,21,18, 2,10,33, 6,31, 4,27,35,19,16,26,30,23, 5, 22,37,28,24,36,34,20, 8,29,17, 3)$
$ 37, 1 $ $1368$ $37$ $( 1, 6,13, 4,35,29,16,17,37,10,36,21,25,24,26,18,20,19,23, 8,34, 7,27,28,15, 9, 2,31,38, 5,33,14,12,22,32,30,11)$

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.