Properties

Label 38T22
Order \(6498\)
n \(38\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
9:  $C_9$
18:  $C_{18}$
171:  $C_{19}:C_{9}$
342:  $F_{19}$, 38T7

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 19: None

Low degree siblings

38T22 x 8

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 $ $1$ $1$ $()$
$ 19, 19 $ $18$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$
$ 19, 19 $ $18$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $19$ $(20,24,28,32,36,21,25,29,33,37,22,26,30,34,38,23,27,31,35)$
$ 19, 19 $ $18$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)(20,32,25,37,30,23, 35,28,21,33,26,38,31,24,36,29,22,34,27)$
$ 19, 19 $ $18$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$
$ 19, 19 $ $18$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,37,35,33,31,29, 27,25,23,21,38,36,34,32,30,28,26,24,22)$
$ 19, 19 $ $18$ $19$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)(20,23,26,29,32,35, 38,22,25,28,31,34,37,21,24,27,30,33,36)$
$ 19, 19 $ $18$ $19$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $18$ $19$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)$
$ 19, 19 $ $18$ $19$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$
$ 19, 19 $ $18$ $19$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$
$ 19, 19 $ $18$ $19$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)(20,36,33,30,27,24, 21,37,34,31,28,25,22,38,35,32,29,26,23)$
$ 19, 19 $ $9$ $19$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(20,29,38,28,37,27, 36,26,35,25,34,24,33,23,32,22,31,21,30)$
$ 19, 19 $ $18$ $19$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$
$ 19, 19 $ $18$ $19$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)(20,26,32,38,25,31, 37,24,30,36,23,29,35,22,28,34,21,27,33)$
$ 19, 19 $ $18$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)(20,28,36,25,33,22, 30,38,27,35,24,32,21,29,37,26,34,23,31)$
$ 19, 19 $ $18$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$
$ 19, 19 $ $18$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)(20,22,24,26,28,30, 32,34,36,38,21,23,25,27,29,31,33,35,37)$
$ 19, 19 $ $18$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,35,31,27,23,38, 34,30,26,22,37,33,29,25,21,36,32,28,24)$
$ 19, 19 $ $18$ $19$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)(20,24,28,32,36,21, 25,29,33,37,22,26,30,34,38,23,27,31,35)$
$ 19, 19 $ $9$ $19$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)(20,38,37,36,35,34, 33,32,31,30,29,28,27,26,25,24,23,22,21)$
$ 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, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 9, 9, 9, 9, 1, 1 $ $361$ $9$ $( 2,18, 5,12,17, 7, 8, 6,10)( 3,16, 9, 4,14,13,15,11,19)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 9, 9, 9, 9, 1, 1 $ $361$ $9$ $( 2,17,10,12, 6, 5, 8,18, 7)( 3,14,19, 4,11, 9,15,16,13)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 9, 9, 9, 9, 1, 1 $ $361$ $9$ $( 2, 6, 7,12,18,10, 8,17, 5)( 3,11,13, 4,16,19,15,14, 9)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 9, 9, 9, 9, 1, 1 $ $361$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,24,36,27,29,37,31, 26,25)(22,28,33,34,38,35,23,32,30)$
$ 9, 9, 9, 9, 1, 1 $ $361$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,26,37,27,24,25,31, 29,36)(22,32,35,34,28,30,23,38,33)$
$ 9, 9, 9, 9, 1, 1 $ $361$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,29,25,27,26,36,31, 24,37)(22,38,30,34,32,33,23,28,35)$
$ 18, 18, 2 $ $361$ $18$ $( 1,26, 8,28,13,24, 3,32, 4,35, 2,29, 6,22,17,36,14,27)( 5,38,19,23,10,34, 9, 31,11,37, 7,25,15,30,18,20,12,21)(16,33)$
$ 18, 18, 2 $ $361$ $18$ $( 1,29,18,20, 8,32,15,35,12,31,16,30,17,25, 3,38, 9,27)( 2,24, 4,33,14,21, 7, 37,10,22, 6,23, 5,28,19,34,13,26)(11,36)$
$ 18, 18, 2 $ $361$ $18$ $( 1,24,15,33,11,25, 4,30, 6,34,19,22,18,20, 2,26,12,27)( 3,28, 9,21,10,23, 7, 36,16,35, 8,38,13,29,17,37, 5,32)(14,31)$
$ 6, 6, 6, 6, 6, 6, 2 $ $361$ $6$ $( 1,27)( 2,21,12,37, 8,23)( 3,34, 4,28,15,38)( 5,22, 7,29,10,30) ( 6,35,18,20,17,26)( 9,36,13,31,19,33)(11,24,16,32,14,25)$
$ 38 $ $171$ $38$ $( 1,21, 8,34,15,28, 3,22,10,35,17,29, 5,23,12,36,19,30, 7,24,14,37, 2,31, 9, 25,16,38, 4,32,11,26,18,20, 6,33,13,27)$
$ 38 $ $171$ $38$ $( 1,29,11,34, 2,20,12,25, 3,30,13,35, 4,21,14,26, 5,31,15,36, 6,22,16,27, 7, 32,17,37, 8,23,18,28, 9,33,19,38,10,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $19$ $2$ $( 1,34)( 2,25)( 3,35)( 4,26)( 5,36)( 6,27)( 7,37)( 8,28)( 9,38)(10,29)(11,20) (12,30)(13,21)(14,31)(15,22)(16,32)(17,23)(18,33)(19,24)$
$ 6, 6, 6, 6, 6, 6, 2 $ $361$ $6$ $( 1,31,13,21, 2,27)( 3,23, 8,22, 5,34)( 4,38,15,32,16,28)( 6,30,10,33,19,35) ( 7,26,17,24,11,29)( 9,37,12,25,14,36)(18,20)$
$ 18, 18, 2 $ $361$ $18$ $( 1,25,19,32,13,36,15,22, 8,33, 4,23,18,20, 7,21,17,27)( 2,37, 6,28,11,31, 3, 30,12,24, 9,26,10,38,16,34,14,29)( 5,35)$
$ 18, 18, 2 $ $361$ $18$ $( 1,37,18,20,10,28,16,22, 2,36, 3,35, 7,31, 4,34,11,27)( 5,33,15,23,17,21, 6, 32,19,38,14,24,13,25, 9,29,12,26)( 8,30)$
$ 18, 18, 2 $ $361$ $18$ $( 1,36, 4,22,12,29, 8,35,10,32, 9,24,19,28,14,26, 7,27)( 2,25,13,37,17,31,15, 34,16,23, 6,38,11,21,18,20, 5,30)( 3,33)$

Group invariants

Order:  $6498=2 \cdot 3^{2} \cdot 19^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.