Properties

Label 38T3
Order \(76\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{38}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
38:  $D_{19}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: $D_{19}$

Low degree siblings

38T3

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

Group invariants

Order:  $76=2^{2} \cdot 19$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [76, 3]
Character table: Data not available.