Properties

Label 39T26
Order \(1014\)
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 $t$ :  $26$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,31,11,39,6,29,13,38,7,32,12,27,4,34,9,35,3,30,10,36,8,28,2,37,5,33)(14,26,17,19,22,18,16,20,25,15,23,24,21), (1,3,13,2,4,11,10,7,5,9,6,8,12)(14,34,25,37,19,38,24,30,16,31,26,27,15,28,22,29,21,35,20,33,17,32,23,36,18,39)
$|\Aut(F/K)|$:  $13$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
6:  $S_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 13: None

Low degree siblings

26T14, 39T27

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

Group invariants

Order:  $1014=2 \cdot 3 \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1014, 7]
Character table: Data not available.