Properties

Label 39T40
Order \(3042\)
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 $t$ :  $40$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,38,10,31,8,32)(2,30,13,29,12,28)(3,36,9,27,7,34)(4,39,11,35,5,33)(6,37)(14,17,23,19,26,16)(15,21,24,20,22,18), (1,14,33,7,19,37,3,16,28,5,15,36,13,21,30,9,17,35,2,18,34,6,25,27,4,24,32,8,26,38,11,22,29,12,20,39,10,23,31)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $S_3$, $C_6$
18:  $S_3\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 13: None

Low degree siblings

26T27, 39T39

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $13$ $( 1, 6, 7, 4, 3, 8, 5,11,13,12, 9,10, 2)(27,38,39,33,28,30,34,32,29,31,37,36, 35)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $13$ $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(27,39,28,34,29,37,35,38,33,30,32,31, 36)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $13$ $( 1, 3,13, 2, 4,11,10, 7, 5, 9, 6, 8,12)(27,28,29,35,33,32,36,39,34,37,38,30, 31)$
$ 13, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $9$ $13$ $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(27,29,33,36,34,38,31,28,35,32,39,37, 30)$
$ 13, 13, 13 $ $6$ $13$ $( 1, 6, 7, 4, 3, 8, 5,11,13,12, 9,10, 2)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,37,32,28,38,36,29,30,39,35,31,34,33)$
$ 13, 13, 13 $ $18$ $13$ $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,36,31,32,30,33,38,35,37,29,34,28,39)$
$ 13, 13, 13 $ $18$ $13$ $( 1,13, 4,10, 5, 6,12, 3, 2,11, 7, 9, 8)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,28,29,35,33,32,36,39,34,37,38,30,31)$
$ 13, 13, 13 $ $6$ $13$ $( 1, 4, 5,12, 2, 7, 8,13,10, 6, 3,11, 9)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,35,36,37,31,29,32,34,30,28,33,39,38)$
$ 13, 13, 13 $ $18$ $13$ $( 1, 5, 2, 8,10, 3, 9, 4,12, 7,13, 6,11)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,39,28,34,29,37,35,38,33,30,32,31,36)$
$ 13, 13, 13 $ $18$ $13$ $( 1, 2,10, 9,12,13,11, 5, 8, 3, 4, 7, 6)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,29,33,36,34,38,31,28,35,32,39,37,30)$
$ 13, 13, 13 $ $18$ $13$ $( 1,10,12,11, 8, 4, 6, 2, 9,13, 5, 3, 7)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,32,38,29,39,31,33,37,28,36,30,35,34)$
$ 13, 13, 13 $ $18$ $13$ $( 1, 8, 9, 7,11, 2, 3,12, 6, 5,10, 4,13)(14,25,19,24,16,26,15,22,21,20,17,23, 18)(27,38,39,33,28,30,34,32,29,31,37,36,35)$
$ 13, 13, 13 $ $6$ $13$ $( 1, 7, 3, 5,13, 9, 2, 6, 4, 8,11,12,10)(14,19,16,15,21,17,18,25,24,26,22,20, 23)(27,32,38,29,39,31,33,37,28,36,30,35,34)$
$ 13, 13, 13 $ $6$ $13$ $( 1, 5, 2, 8,10, 3, 9, 4,12, 7,13, 6,11)(14,19,16,15,21,17,18,25,24,26,22,20, 23)(27,36,31,32,30,33,38,35,37,29,34,28,39)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $169$ $3$ $( 2, 9, 3)( 4,12, 6)( 5, 8, 7)(10,11,13)(15,26,19)(16,18,17)(20,25,24) (21,23,22)(28,35,37)(29,36,32)(30,39,34)(31,38,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $169$ $3$ $( 2, 3, 9)( 4, 6,12)( 5, 7, 8)(10,13,11)(15,19,26)(16,17,18)(20,24,25) (21,22,23)(28,37,35)(29,32,36)(30,34,39)(31,33,38)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $338$ $3$ $( 1,39,17)( 2,33,22)( 3,36,20)( 4,35,15)( 5,31,19)( 6,38,14)( 7,27,24) ( 8,37,18)( 9,30,25)(10,28,16)(11,29,26)(12,34,23)(13,32,21)$
$ 39 $ $78$ $39$ $( 1,34,22, 9,39,16,11,36,25, 3,32,23, 6,33,21,10,35,26,13,29,19, 8,28,18, 7, 27,20, 2,31,15,12,30,24, 5,38,14, 4,37,17)$
$ 39 $ $78$ $39$ $( 1,32,23, 7,38,14, 3,29,19, 5,39,16,13,31,15, 9,33,21, 2,37,17, 6,28,18, 4, 36,25, 8,30,24,11,35,26,12,34,22,10,27,20)$
$ 39 $ $78$ $39$ $( 1,29,19,11,27,20, 6,30,24,13,37,17, 7,39,16,12,32,23, 4,35,26, 9,28,18, 3, 31,15,10,38,14, 8,34,22, 2,36,25, 5,33,21)$
$ 39 $ $78$ $39$ $( 1,35,26, 6,31,15, 7,34,22, 4,33,21, 3,27,20, 8,37,17, 5,32,23,11,28,18,13, 38,14,12,36,25, 9,29,19,10,30,24, 2,39,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $26$ $3$ $( 1,28,18)( 2,32,23)( 3,30,24)( 4,29,19)( 5,35,26)( 6,38,14)( 7,36,25) ( 8,39,16)( 9,27,20)(10,37,17)(11,31,15)(12,33,21)(13,34,22)$
$ 39 $ $78$ $39$ $( 1,30,20, 9,34,21,11,32,22, 3,29,15, 6,31,26,10,37,16,13,36,24, 8,35,19, 7, 27,25, 2,38,14,12,39,18, 5,33,23, 4,28,17)$
$ 39 $ $78$ $39$ $( 1,34,21,13,35,19, 4,30,20,10,36,24, 5,28,17, 6,37,16,12,33,23, 3,31,26, 2, 39,18,11,29,15, 7,38,14, 9,32,22, 8,27,25)$
$ 39 $ $78$ $39$ $( 1,32,22, 5,30,20, 2,33,23, 8,38,14,10,35,19, 3,37,16, 9,29,15, 4,34,21,12, 28,17, 7,39,18,13,27,25, 6,36,24,11,31,26)$
$ 39 $ $78$ $39$ $( 1,33,23, 6,32,22, 7,36,24, 4,39,18, 3,34,21, 8,37,16, 5,38,14,11,30,20,13, 31,26,12,27,25, 9,28,17,10,29,15, 2,35,19)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $26$ $3$ $( 1,37,16)( 2,34,21)( 3,27,25)( 4,31,26)( 5,29,15)( 6,38,14)( 7,30,20) ( 8,28,17)( 9,36,24)(10,39,18)(11,35,19)(12,32,22)(13,33,23)$
$ 6, 6, 6, 6, 6, 6, 2, 1 $ $507$ $6$ $( 2,12, 9, 6, 3, 4)( 5,10, 8,11, 7,13)(14,38,16,27,24,37)(15,34,20,39,26,33) (17,30,21,35,19,32)(18,36,25,28,22,31)(23,29)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $117$ $26$ $( 2, 6)( 3,12)( 4, 9)( 5,11)( 7,10)( 8,13)(14,33,26,31,17,39,19,29,22,38,18, 32,16,27,20,34,25,35,15,30,23,36,24,28,21,37)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $117$ $26$ $( 1, 6)( 2, 7)( 3, 9)( 4,10)( 5,13)( 8,12)(14,28,21,36,24,30,23,35,15,34,25, 27,20,32,16,38,18,29,22,39,19,31,17,33,26,37)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $117$ $26$ $( 1, 3)( 2, 8)( 4, 6)( 5,10)( 9,11)(12,13)(14,32,16,28,21,38,18,36,24,29,22, 30,23,39,19,35,15,31,17,34,25,33,26,27,20,37)$
$ 26, 2, 2, 2, 2, 2, 2, 1 $ $117$ $26$ $( 1,13)( 2,12)( 4, 8)( 5, 7)( 6,11)( 9,10)(14,36,24,35,15,27,20,38,18,39,19, 33,26,28,21,30,23,34,25,32,16,29,22,31,17,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $39$ $2$ $( 1,11)( 2,13)( 3, 4)( 5, 6)( 7, 8)(10,12)(14,37)(15,35)(16,32)(17,31)(18,38) (19,39)(20,27)(21,28)(22,29)(23,30)(24,36)(25,34)(26,33)$
$ 6, 6, 6, 6, 6, 6, 2, 1 $ $507$ $6$ $( 2, 4, 3, 6, 9,12)( 5,13, 7,11, 8,10)(14,31,16,27,25,37)(15,39) (17,34,24,35,26,38)(18,29,21,28,23,32)(19,36,20,30,22,33)$

Group invariants

Order:  $3042=2 \cdot 3^{2} \cdot 13^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.