Properties

Label 39T31
Order \(1521\)
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 $t$ :  $31$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,33)(2,16,32)(3,25,35)(4,21,28)(5,26,34)(6,18,36)(7,23,39)(8,20,37)(9,19,29)(10,17,27)(11,22,31)(12,24,30)(13,15,38), (1,7,10)(2,4,11)(3,9,6)(5,8,12)(14,26,18)(15,24,25)(16,20,21)(17,19,22)(27,33,39)(28,31,32)(29,36,35)(30,34,37)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 13: None

Low degree siblings

39T31 x 3

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

Group invariants

Order:  $1521=3^{2} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1521, 9]
Character table: Data not available.