Properties

Label 39T41
Order \(3159\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

Degree $n$ :  $39$
Transitive number $t$ :  $41$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,31,8,24,37,15,29,4,21,35,12,25,2,17,32,9,22,38,13,30,5,19,36,10,26,3,18,33,7,23,39,14,28,6,20,34,11,27), (1,31,5,2,32,6,3,33,4)(7,11,20)(8,12,21)(9,10,19)(13,29,35,14,30,36,15,28,34)(16,39,24,18,38,23,17,37,22)
$|\Aut(F/K)|$:  $3$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
39:  $C_{13}:C_3$
117:  39T7
1053:  27T292

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 13: $C_{13}:C_3$

Low degree siblings

39T41 x 5

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,15,14)(16,17,18)(19,20,21)(28,29,30) (31,33,32)(37,39,38)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,14,15)(16,18,17)(19,21,20)(28,30,29) (31,32,33)(37,38,39)$
$ 13, 13, 13 $ $81$ $13$ $( 1, 8,15,21,25,32,38, 5,10,18,23,28,34)( 2, 9,13,19,26,33,39, 6,11,16,24,29, 35)( 3, 7,14,20,27,31,37, 4,12,17,22,30,36)$
$ 13, 13, 13 $ $81$ $13$ $( 1,15,25,38,10,23,34, 8,21,32, 5,18,28)( 2,13,26,39,11,24,35, 9,19,33, 6,16, 29)( 3,14,27,37,12,22,36, 7,20,31, 4,17,30)$
$ 13, 13, 13 $ $81$ $13$ $( 1,25,10,34,21, 5,28,15,38,23, 8,32,18)( 2,26,11,35,19, 6,29,13,39,24, 9,33, 16)( 3,27,12,36,20, 4,30,14,37,22, 7,31,17)$
$ 13, 13, 13 $ $81$ $13$ $( 1,10,21,28,38, 8,18,25,34, 5,15,23,32)( 2,11,19,29,39, 9,16,26,35, 6,13,24, 33)( 3,12,20,30,37, 7,17,27,36, 4,14,22,31)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $(10,12,11)(19,20,21)(22,23,24)(28,30,29)(31,32,33)(34,36,35)(37,38,39)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(22,23,24)(28,29,30) (31,33,32)(34,36,35)(37,39,38)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$
$ 39 $ $81$ $39$ $( 1, 8,15,21,26,33,37, 5,10,17,22,28,36, 2, 9,13,19,27,31,38, 6,11,18,23,29, 34, 3, 7,14,20,25,32,39, 4,12,16,24,30,35)$
$ 39 $ $81$ $39$ $( 1,15,25,38,11,23,35, 8,21,33, 4,17,30, 2,13,26,39,12,24,36, 9,19,31, 5,18, 28, 3,14,27,37,10,22,34, 7,20,32, 6,16,29)$
$ 39 $ $81$ $39$ $( 1,25,10,36,19, 4,30,13,39,22, 8,32,16, 2,26,11,34,20, 5,28,14,37,23, 9,33, 17, 3,27,12,35,21, 6,29,15,38,24, 7,31,18)$
$ 39 $ $81$ $39$ $( 1,10,20,28,37, 8,18,25,34, 4,14,22,32, 2,11,21,29,38, 9,16,26,35, 5,15,23, 33, 3,12,19,30,39, 7,17,27,36, 6,13,24,31)$
$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $(10,11,12)(19,21,20)(22,24,23)(28,29,30)(31,33,32)(34,35,36)(37,39,38)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $13$ $3$ $( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(22,24,23)(28,30,29) (31,32,33)(34,35,36)(37,38,39)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)$
$ 39 $ $81$ $39$ $( 1, 8,15,21,27,31,39, 5,10,16,24,28,35, 3, 7,14,20,26,33,38, 4,12,18,23,30, 34, 2, 9,13,19,25,32,37, 6,11,17,22,29,36)$
$ 39 $ $81$ $39$ $( 1,15,25,38,12,23,36, 8,21,31, 6,16,29, 3,14,27,37,11,22,35, 7,20,33, 5,18, 28, 2,13,26,39,10,24,34, 9,19,32, 4,17,30)$
$ 39 $ $81$ $39$ $( 1,25,10,35,20, 6,29,14,37,24, 8,32,17, 3,27,12,34,19, 5,28,13,39,23, 7,31, 16, 2,26,11,36,21, 4,30,15,38,22, 9,33,18)$
$ 39 $ $81$ $39$ $( 1,10,19,28,39, 8,18,25,34, 6,13,24,32, 3,12,21,30,38, 7,17,27,36, 5,15,23, 31, 2,11,20,29,37, 9,16,26,35, 4,14,22,33)$
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ $117$ $9$ $( 1,36,17, 3,35,16, 2,34,18)( 7,14,32, 8,15,33, 9,13,31)(10,24,20,11,22,21,12, 23,19)(25,30,38)(26,28,39)(27,29,37)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $117$ $3$ $( 1,36,18)( 2,34,16)( 3,35,17)( 4, 5, 6)( 7,13,33)( 8,14,31)( 9,15,32) (10,24,21)(11,22,19)(12,23,20)(25,28,38)(26,29,39)(27,30,37)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,36,16, 2,34,17, 3,35,18)( 4, 6, 5)( 7,15,31, 9,14,33, 8,13,32) (10,24,19,12,23,21,11,22,20)(25,29,38)(26,30,39)(27,28,37)$
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ $117$ $9$ $( 1,35,16, 2,36,17, 3,34,18)( 7,14,33, 9,13,32, 8,15,31)(10,22,19,12,24,21,11, 23,20)(25,29,38)(26,30,39)(27,28,37)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,35,17, 3,34,16, 2,36,18)( 4, 5, 6)( 7,13,31, 8,14,32, 9,15,33) (10,22,20,11,23,21,12,24,19)(25,30,38)(26,28,39)(27,29,37)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $117$ $3$ $( 1,35,18)( 2,36,16)( 3,34,17)( 4, 6, 5)( 7,15,32)( 8,13,33)( 9,14,31) (10,22,21)(11,23,19)(12,24,20)(25,28,38)(26,29,39)(27,30,37)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $117$ $3$ $( 1,34,18)( 2,35,16)( 3,36,17)( 7,14,31)( 8,15,32)( 9,13,33)(10,23,21) (11,24,19)(12,22,20)(25,28,38)(26,29,39)(27,30,37)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,34,16, 2,35,17, 3,36,18)( 4, 5, 6)( 7,13,32, 9,15,31, 8,14,33) (10,23,19,12,22,21,11,24,20)(25,29,38)(26,30,39)(27,28,37)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,34,17, 3,36,16, 2,35,18)( 4, 6, 5)( 7,15,33, 8,13,31, 9,14,32) (10,23,20,11,24,21,12,22,19)(25,30,38)(26,28,39)(27,29,37)$
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ $117$ $9$ $( 1,17,35, 2,18,36, 3,16,34)( 7,32,15, 9,31,14, 8,33,13)(10,20,22,12,19,24,11, 21,23)(25,38,30)(26,39,28)(27,37,29)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,18,36, 3,17,35, 2,16,34)( 4, 5, 6)( 7,31,13, 8,32,14, 9,33,15) (10,21,23,11,19,24,12,20,22)(25,37,30)(26,38,28)(27,39,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $117$ $3$ $( 1,16,34)( 2,17,35)( 3,18,36)( 4, 6, 5)( 7,33,14)( 8,31,15)( 9,32,13) (10,19,24)(11,20,22)(12,21,23)(25,39,30)(26,37,28)(27,38,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $117$ $3$ $( 1,17,34)( 2,18,35)( 3,16,36)( 7,33,13)( 8,31,14)( 9,32,15)(10,21,24) (11,19,22)(12,20,23)(25,39,30)(26,37,28)(27,38,29)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,18,35, 2,16,36, 3,17,34)( 4, 5, 6)( 7,32,14, 9,31,13, 8,33,15) (10,19,22,12,21,24,11,20,23)(25,38,30)(26,39,28)(27,37,29)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,16,36, 3,18,35, 2,17,34)( 4, 6, 5)( 7,31,15, 8,32,13, 9,33,14) (10,20,23,11,21,24,12,19,22)(25,37,30)(26,38,28)(27,39,29)$
$ 9, 9, 9, 3, 3, 3, 1, 1, 1 $ $117$ $9$ $( 1,17,36, 3,16,35, 2,18,34)( 7,31,14, 8,32,15, 9,33,13)(10,19,23,11,20,24,12, 21,22)(25,37,30)(26,38,28)(27,39,29)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $117$ $3$ $( 1,18,34)( 2,16,35)( 3,17,36)( 4, 5, 6)( 7,33,15)( 8,31,13)( 9,32,14) (10,20,24)(11,21,22)(12,19,23)(25,39,30)(26,37,28)(27,38,29)$
$ 9, 9, 9, 3, 3, 3, 3 $ $117$ $9$ $( 1,16,35, 2,17,36, 3,18,34)( 4, 6, 5)( 7,32,13, 9,31,15, 8,33,14) (10,21,22,12,20,24,11,19,23)(25,38,30)(26,39,28)(27,37,29)$

Group invariants

Order:  $3159=3^{5} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [3159, 313]
Character table: Data not available.