Properties

Label 38T38
Order \(38988\)
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 $t$ :  $38$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,35,14,25,4,21,19,27,6,37,16,22)(2,24,3,32,11,20,18,38,17,30,9,23)(5,29,8,34,13,36,15,33,12,28,7,26)(10,31), (1,8,5,9,10,15,2,13,11)(3,18,17,12,6,14,16,7,19)(21,29,25,27,26,36,31,24,37)(22,38,30,34,32,33,23,28,35)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $S_3$, $C_6$
12:  $C_{12}$, $C_3 : C_4$
18:  $S_3\times C_3$, $D_{9}$
36:  $C_3\times (C_3 : C_4)$, 36T9
54:  18T19
108:  36T68

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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 $ $1$ $1$ $()$
$ 19, 19 $ $108$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,25,30,35,21,26, 31,36,22,27,32,37,23,28,33,38,24,29,34)$
$ 19, 19 $ $108$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)(20,30,21,31,22,32, 23,33,24,34,25,35,26,36,27,37,28,38,29)$
$ 19, 19 $ $108$ $19$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)(20,21,22,23,24,25, 26,27,28,29,30,31,32,33,34,35,36,37,38)$
$ 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $36$ $19$ $(20,31,23,34,26,37,29,21,32,24,35,27,38,30,22,33,25,36,28)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,27,31) (22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $361$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $361$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $361$ $2$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,38)(22,37) (23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $361$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,32,31,38,27,28) (22,25,23,37,34,36)(24,30,26,35,29,33)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $38$ $3$ $(21,27,31)(22,34,23)(24,29,26)(25,36,37)(28,38,32)(30,33,35)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $684$ $57$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)(20,25,22)(21,32,33) (23,27,36)(24,34,28)(26,29,31)(30,38,37)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $722$ $3$ $( 2, 8,12)( 3,15, 4)( 5,10, 7)( 6,17,18)( 9,19,13)(11,14,16)(21,31,27) (22,23,34)(24,26,29)(25,37,36)(28,32,38)(30,35,33)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $38$ $3$ $( 2,12, 8)( 3, 4,15)( 5, 7,10)( 6,18,17)( 9,13,19)(11,16,14)$
$ 19, 3, 3, 3, 3, 3, 3, 1 $ $684$ $57$ $( 1,11, 7)( 2, 3,14)( 4, 6, 9)( 5,17,16)( 8,12,18)(10,15,13)(20,25,30,35,21, 26,31,36,22,27,32,37,23,28,33,38,24,29,34)$
$ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $722$ $6$ $( 2, 9, 8,19,12,13)( 3,17,15,18, 4, 6)( 5,14,10,16, 7,11)(21,38)(22,37)(23,36) (24,35)(25,34)(26,33)(27,32)(28,31)(29,30)$
$ 6, 6, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $722$ $6$ $( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11) (21,32,31,38,27,28)(22,25,23,37,34,36)(24,30,26,35,29,33)$
$ 6, 6, 6, 6, 6, 6, 1, 1 $ $722$ $6$ $( 2,13,12,19, 8, 9)( 3, 6, 4,18,15,17)( 5,11, 7,16,10,14)(21,28,27,38,31,32) (22,36,34,37,23,25)(24,33,29,35,26,30)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 5,17, 8,10,18,12, 7, 6)( 3, 9,14,15,19,16, 4,13,11)(21,37,24,31,36,26,27, 25,29)(22,35,28,23,33,32,34,30,38)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2,10, 6, 8, 7,17,12, 5,18)( 3,19,11,15,13,14, 4, 9,16)(21,25,26,31,37,29,27, 36,24)(22,30,32,23,35,38,34,33,28)$
$ 9, 9, 9, 9, 1, 1 $ $722$ $9$ $( 2, 7,18, 8, 5, 6,12,10,17)( 3,13,16,15, 9,11, 4,19,14)(21,36,29,31,25,24,27, 37,26)(22,33,38,23,30,28,34,35,32)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,14,18,13, 5,15,12,11,17,19, 7, 3, 8,16, 6, 9,10, 4)(21,23,29,28,25,35,27, 22,26,38,36,30,31,34,24,32,37,33)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,16,17,13,10, 3,12,14, 6,19, 5, 4, 8,11,18, 9, 7,15)(21,22,24,28,36,33,27, 34,29,38,37,35,31,23,26,32,25,30)$
$ 18, 18, 1, 1 $ $722$ $18$ $( 2,11, 6,13, 7, 4,12,16,18,19,10,15, 8,14,17, 9, 5, 3)(21,34,26,28,37,30,27, 23,24,38,25,33,31,22,29,32,36,35)$
$ 12, 12, 12, 2 $ $3249$ $12$ $( 1,35,14,25, 4,21,19,27, 6,37,16,22)( 2,24, 3,32,11,20,18,38,17,30, 9,23) ( 5,29, 8,34,13,36,15,33,12,28, 7,26)(10,31)$
$ 12, 12, 12, 2 $ $3249$ $12$ $( 1,30,19,31, 7,24,15,35,16,34, 9,22)( 2,29,12,38,18,32,14,36, 4,27,17,33) ( 3,28, 5,26,10,21,13,37,11,20, 6,25)( 8,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $3249$ $4$ $( 1,33, 8,22)( 2,26, 7,29)( 3,38, 6,36)( 4,31, 5,24)( 9,34,19,21)(10,27,18,28) (11,20,17,35)(12,32,16,23)(13,25,15,30)(14,37)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2 $ $3249$ $4$ $( 1,26,14,22)( 2,33,13,34)( 3,21,12,27)( 4,28,11,20)( 5,35,10,32)( 6,23, 9,25) ( 7,30, 8,37)(15,29,19,38)(16,36,18,31)(17,24)$
$ 12, 12, 12, 2 $ $3249$ $12$ $( 1,24,18,21, 2,35, 7,33, 9,36, 6,22)( 3,27,15,26,16,37, 5,30,12,31,11,20) ( 4,38)( 8,25,17,29,13,23,19,32,10,28,14,34)$
$ 12, 12, 12, 2 $ $3249$ $12$ $( 1,29, 9,37,10,38, 3,31,14,23,13,22)( 2,30)( 4,32, 7,35, 5,33,19,28,16,25,18, 27)( 6,34,12,21, 8,36,17,26,11,20,15,24)$

Group invariants

Order:  $38988=2^{2} \cdot 3^{3} \cdot 19^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.