Properties

Label 39T42
Order \(4056\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

Degree $n$ :  $39$
Transitive number $t$ :  $42$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,31,21,4,29,22,2,39,17,12,28,16)(3,34,26,7,27,23,13,36,25,9,30,15)(5,37,18,10,38,24,11,33,20,6,32,14)(8,35,19), (1,5,9,13,4,8,12,3,7,11,2,6,10)(14,35,24,39,21,30,18,34,15,38,25,29,22,33,19,37,16,28,26,32,23,36,20,27,17,31)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$
8:  $C_4\times C_2$
12:  $D_{6}$
24:  $S_3 \times C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 13: None

Low degree siblings

26T30, 39T42

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 $ $1$ $1$ $()$
$ 13, 13, 13 $ $12$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,25,23,21,19,17,15,26,24,22,20,18, 16)(27,34,28,35,29,36,30,37,31,38,32,39,33)$
$ 13, 13, 13 $ $12$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,23,19,15,24,20,16,25,21,17,26,22, 18)(27,28,29,30,31,32,33,34,35,36,37,38,39)$
$ 13, 13, 13 $ $12$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,19,24,16,21,26,18,23,15,20,25,17, 22)(27,29,31,33,35,37,39,28,30,32,34,36,38)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $(14,25,23,21,19,17,15,26,24,22,20,18,16)(27,33,39,32,38,31,37,30,36,29,35,28, 34)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,23,19,15,24,20,16,25,21,17,26,22, 18)$
$ 13, 13, 13 $ $24$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,21,15,22,16,23,17,24,18,25,19,26, 20)(27,34,28,35,29,36,30,37,31,38,32,39,33)$
$ 13, 13, 13 $ $24$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)(14,22,17,25,20,15,23,18,26,21,16,24, 19)(27,37,34,31,28,38,35,32,29,39,36,33,30)$
$ 13, 13, 13 $ $24$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)(14,26,25,24,23,22,21,20,19,18,17,16, 15)(27,36,32,28,37,33,29,38,34,30,39,35,31)$
$ 13, 13, 13 $ $24$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,21,15,22,16,23,17,24,18,25,19,26, 20)(27,33,39,32,38,31,37,30,36,29,35,28,34)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $12$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,19,24,16,21,26,18,23,15,20,25,17, 22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $338$ $3$ $( 1,29,17)( 2,28,21)( 3,27,25)( 4,39,16)( 5,38,20)( 6,37,24)( 7,36,15) ( 8,35,19)( 9,34,23)(10,33,14)(11,32,18)(12,31,22)(13,30,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $169$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,26)(16,25)(17,24)(18,23)(19,22) (20,21)(27,33)(28,32)(29,31)(34,39)(35,38)(36,37)$
$ 6, 6, 6, 6, 6, 6, 3 $ $338$ $6$ $( 1,29,22, 8,36,24)( 2,30,26, 7,35,20)( 3,31,17, 6,34,16)( 4,32,21, 5,33,25) ( 9,37,15,13,28,18)(10,38,19,12,27,14)(11,39,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $169$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(15,19,26,22)(16,24,25,17)(18,21,23,20) (27,28,33,32)(29,38,31,35)(34,37,39,36)$
$ 12, 12, 12, 3 $ $338$ $12$ $( 1,29,20,11,31,19,13,34,24, 3,32,25)( 2,37,16, 6,30,26,12,39,15, 8,33,18) ( 4,27,21, 9,28,14,10,36,23, 5,35,17)( 7,38,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $169$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(15,22,26,19)(16,17,25,24)(18,20,23,21) (27,32,33,28)(29,35,31,38)(34,36,39,37)$
$ 12, 12, 12, 3 $ $338$ $12$ $( 1,29,19, 7,33,21,11,27,18, 5,36,16)( 2,34,15,12,32,14,10,35,22,13,37,23) ( 3,39,24, 4,31,20, 9,30,26, 8,38,17)( 6,28,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $39$ $2$ $(14,33)(15,36)(16,39)(17,29)(18,32)(19,35)(20,38)(21,28)(22,31)(23,34)(24,37) (25,27)(26,30)$
$ 26, 13 $ $156$ $26$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)(14,27,23,28,19,29,15,30,24,31,20,32, 16,33,25,34,21,35,17,36,26,37,22,38,18,39)$
$ 26, 13 $ $156$ $26$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)(14,34,19,36,24,38,16,27,21,29,26,31, 18,33,23,35,15,37,20,39,25,28,17,30,22,32)$
$ 26, 13 $ $156$ $26$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)(14,35,24,39,21,30,18,34,15,38,25,29, 22,33,19,37,16,28,26,32,23,36,20,27,17,31)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $156$ $26$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,27,16,34,18,28,20,35,22,29,24, 36,26,30,15,37,17,31,19,38,21,32,23,39,25,33)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $156$ $26$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,39,21,31,15,36,22,28,16,33,23, 38,17,30,24,35,18,27,25,32,19,37,26,29,20,34)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $156$ $26$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,38,26,28,25,31,24,34,23,37,22, 27,21,30,20,33,19,36,18,39,17,29,16,32,15,35)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $39$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(14,30)(15,27)(16,37)(17,34)(18,31) (19,28)(20,38)(21,35)(22,32)(23,29)(24,39)(25,36)(26,33)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $507$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)(14,32,21,33)(15,34,20,31)(16,36,19,29) (17,38,18,27)(22,35,26,30)(23,37,25,28)(24,39)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $507$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)(14,28,18,33)(15,39,17,35)(16,37) (19,31,26,30)(20,29,25,32)(21,27,24,34)(22,38,23,36)$

Group invariants

Order:  $4056=2^{3} \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.