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

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.