Properties

Label 38T41
Degree $38$
Order $50616$
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)
$|\Aut(F/K)|$:  $1$
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)

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

Group invariants

Order:  $50616=2^{3} \cdot 3^{2} \cdot 19 \cdot 37$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.