Properties

Label 39T23
Order \(936\)
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 $t$ :  $23$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,17,10,30,15,20)(2,16,12,28,13,19)(3,18,11,29,14,21)(4,7,39,27,23,31)(5,9,37,26,22,33)(6,8,38,25,24,32)(34,35), (1,27,38,4,7,29,19,35,24,16,13,32)(2,25,39,5,8,28,20,36,23,17,14,33)(3,26,37,6,9,30,21,34,22,18,15,31)(10,12,11)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_4$ x 2, $C_2^2$
6:  $S_3$, $C_6$ x 3
8:  $C_4\times C_2$
12:  $D_{6}$, $C_{12}$ x 2, $C_6\times C_2$
18:  $S_3\times C_3$
24:  $S_3 \times C_4$, 24T2
36:  $C_6\times S_3$
72:  24T65
156:  $F_{13}$
312:  26T10

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 13: $F_{13}$

Low degree siblings

There are no siblings 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 $ $12$ $13$ $( 1,28,16, 4,33,19, 9,34,23,10,39,26,15)( 2,30,17, 5,31,20, 7,35,22,12,37,27, 13)( 3,29,18, 6,32,21, 8,36,24,11,38,25,14)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $13$ $3$ $( 4,28,10)( 5,30,12)( 6,29,11)( 7,17,20)( 8,18,21)( 9,16,19)(13,31,37) (14,32,38)(15,33,39)(22,35,27)(23,34,26)(24,36,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1 $ $13$ $3$ $( 4,10,28)( 5,12,30)( 6,11,29)( 7,20,17)( 8,21,18)( 9,19,16)(13,37,31) (14,38,32)(15,39,33)(22,27,35)(23,26,34)(24,25,36)$
$ 39 $ $24$ $39$ $( 1,38,35,33,29,27,23,21,17,15,11, 7, 4, 3,37,34,32,30,26,24,20,16,14,12, 9, 6, 2,39,36,31,28,25,22,19,18,13,10, 8, 5)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,11,12)(13,15,14)(16,18,17)(19,21,20) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $26$ $3$ $( 1,14, 5)( 2,15, 6)( 3,13, 4)( 7,28,24)( 8,30,23)( 9,29,22)(10,18,31) (11,17,33)(12,16,32)(19,21,20)(25,35,39)(26,36,37)(27,34,38)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $26$ $3$ $( 1,32, 5)( 2,33, 6)( 3,31, 4)( 7,10,21)( 8,12,19)( 9,11,20)(13,28,36) (14,30,34)(15,29,35)(16,38,22)(17,39,24)(18,37,23)(25,27,26)$
$ 6, 6, 6, 6, 6, 6, 2, 1 $ $39$ $6$ $( 2, 3)( 4,33,28,39,10,15)( 5,32,30,38,12,14)( 6,31,29,37,11,13) ( 7,24,17,36,20,25)( 8,22,18,35,21,27)( 9,23,16,34,19,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ $39$ $2$ $( 2, 3)( 4,39)( 5,38)( 6,37)( 7,36)( 8,35)( 9,34)(10,33)(11,31)(12,32)(13,29) (14,30)(15,28)(16,26)(17,25)(18,27)(19,23)(20,24)(21,22)$
$ 6, 6, 6, 6, 6, 6, 2, 1 $ $39$ $6$ $( 2, 3)( 4,15,10,39,28,33)( 5,14,12,38,30,32)( 6,13,11,37,29,31) ( 7,25,20,36,17,24)( 8,27,21,35,18,22)( 9,26,19,34,16,23)$
$ 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 $ $3$ $2$ $( 2, 3)( 5, 6)( 7, 8)(11,12)(13,14)(17,18)(20,21)(22,24)(25,27)(29,30)(31,32) (35,36)(37,38)$
$ 26, 13 $ $36$ $26$ $( 1,28,16, 4,33,19, 9,34,23,10,39,26,15)( 2,29,17, 6,31,21, 7,36,22,11,37,25, 13, 3,30,18, 5,32,20, 8,35,24,12,38,27,14)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ $39$ $6$ $( 2, 3)( 4,28,10)( 5,29,12, 6,30,11)( 7,18,20, 8,17,21)( 9,16,19) (13,32,37,14,31,38)(15,33,39)(22,36,27,24,35,25)(23,34,26)$
$ 6, 6, 6, 6, 3, 3, 3, 3, 2, 1 $ $39$ $6$ $( 2, 3)( 4,10,28)( 5,11,30, 6,12,29)( 7,21,17, 8,20,18)( 9,19,16) (13,38,31,14,37,32)(15,39,33)(22,25,35,24,27,36)(23,26,34)$
$ 6, 6, 6, 6, 6, 6, 1, 1, 1 $ $13$ $6$ $( 4,33,28,39,10,15)( 5,31,30,37,12,13)( 6,32,29,38,11,14)( 7,22,17,35,20,27) ( 8,24,18,36,21,25)( 9,23,16,34,19,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $13$ $2$ $( 4,39)( 5,37)( 6,38)( 7,35)( 8,36)( 9,34)(10,33)(11,32)(12,31)(13,30)(14,29) (15,28)(16,26)(17,27)(18,25)(19,23)(20,22)(21,24)$
$ 6, 6, 6, 6, 6, 6, 1, 1, 1 $ $13$ $6$ $( 4,15,10,39,28,33)( 5,13,12,37,30,31)( 6,14,11,38,29,32)( 7,27,20,35,17,22) ( 8,25,21,36,18,24)( 9,26,19,34,16,23)$
$ 6, 6, 6, 6, 6, 6, 3 $ $26$ $6$ $( 1,38, 7,19,24,13)( 2,39, 8,20,23,14)( 3,37, 9,21,22,15)( 4,29,35,16,32,27) ( 5,28,36,17,33,25)( 6,30,34,18,31,26)(10,11,12)$
$ 6, 6, 6, 6, 6, 6, 3 $ $26$ $6$ $( 1,14, 2,15, 3,13)( 4,11, 5,10, 6,12)( 7, 9, 8)(16,38,17,39,18,37) (19,36,20,34,21,35)(22,33,24,31,23,32)(25,30,26,29,27,28)$
$ 6, 6, 6, 6, 6, 6, 3 $ $26$ $6$ $( 1,32,35, 9,18,13)( 2,33,36, 7,16,14)( 3,31,34, 8,17,15)( 4, 6, 5) (10,29,22,39,21,27)(11,30,23,38,20,26)(12,28,24,37,19,25)$
$ 12, 12, 12, 1, 1, 1 $ $13$ $12$ $( 4,23,33,16,28,34,39,19,10,26,15, 9)( 5,22,31,17,30,35,37,20,12,27,13, 7) ( 6,24,32,18,29,36,38,21,11,25,14, 8)$
$ 12, 12, 12, 1, 1, 1 $ $13$ $12$ $( 4,34,15,16,10,23,39, 9,28,26,33,19)( 5,35,13,17,12,22,37, 7,30,27,31,20) ( 6,36,14,18,11,24,38, 8,29,25,32,21)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $13$ $4$ $( 4,26,39,16)( 5,27,37,17)( 6,25,38,18)( 7,12,35,31)( 8,11,36,32)( 9,10,34,33) (13,20,30,22)(14,21,29,24)(15,19,28,23)$
$ 12, 12, 12, 3 $ $26$ $12$ $( 1,38,17,26,11,22,28,32,13, 4,21, 7)( 2,39,18,27,10,24,30,33,14, 5,19, 8) ( 3,37,16,25,12,23,29,31,15, 6,20, 9)(34,36,35)$
$ 12, 12, 12, 3 $ $26$ $12$ $( 1,14,30,39,21,17,23,11,35,26, 6, 7)( 2,15,29,37,19,18,22,10,36,27, 4, 8) ( 3,13,28,38,20,16,24,12,34,25, 5, 9)(31,33,32)$
$ 12, 12, 12, 3 $ $26$ $12$ $( 1,32,37, 9, 3,31,39, 8, 2,33,38, 7)( 4,18,35,23, 6,17,34,24, 5,16,36,22) (10,25,30,15,11,27,28,14,12,26,29,13)(19,21,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $39$ $4$ $( 2, 3)( 4,16,39,26)( 5,18,37,25)( 6,17,38,27)( 7,32,35,11)( 8,31,36,12) ( 9,33,34,10)(13,24,30,21)(14,22,29,20)(15,23,28,19)$
$ 12, 12, 12, 2, 1 $ $39$ $12$ $( 2, 3)( 4,19,33,26,28, 9,39,23,10,16,15,34)( 5,21,31,25,30, 8,37,24,12,18,13, 36)( 6,20,32,27,29, 7,38,22,11,17,14,35)$
$ 12, 12, 12, 2, 1 $ $39$ $12$ $( 2, 3)( 4, 9,15,26,10,19,39,34,28,16,33,23)( 5, 8,13,25,12,21,37,36,30,18,31, 24)( 6, 7,14,27,11,20,38,35,29,17,32,22)$
$ 12, 12, 12, 2, 1 $ $39$ $12$ $( 2, 3)( 4,23,33,16,28,34,39,19,10,26,15, 9)( 5,24,31,18,30,36,37,21,12,25,13, 8)( 6,22,32,17,29,35,38,20,11,27,14, 7)$
$ 12, 12, 12, 2, 1 $ $39$ $12$ $( 2, 3)( 4,34,15,16,10,23,39, 9,28,26,33,19)( 5,36,13,18,12,24,37, 8,30,25,31, 21)( 6,35,14,17,11,22,38, 7,29,27,32,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 1 $ $39$ $4$ $( 2, 3)( 4,26,39,16)( 5,25,37,18)( 6,27,38,17)( 7,11,35,32)( 8,12,36,31) ( 9,10,34,33)(13,21,30,24)(14,20,29,22)(15,19,28,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 1, 1 $ $13$ $4$ $( 4,16,39,26)( 5,17,37,27)( 6,18,38,25)( 7,31,35,12)( 8,32,36,11)( 9,33,34,10) (13,22,30,20)(14,24,29,21)(15,23,28,19)$
$ 12, 12, 12, 1, 1, 1 $ $13$ $12$ $( 4,19,33,26,28, 9,39,23,10,16,15,34)( 5,20,31,27,30, 7,37,22,12,17,13,35) ( 6,21,32,25,29, 8,38,24,11,18,14,36)$
$ 12, 12, 12, 1, 1, 1 $ $13$ $12$ $( 4, 9,15,26,10,19,39,34,28,16,33,23)( 5, 7,13,27,12,20,37,35,30,17,31,22) ( 6, 8,14,25,11,21,38,36,29,18,32,24)$
$ 12, 12, 12, 3 $ $26$ $12$ $( 1,38,22,26, 3,37,23,25, 2,39,24,27)( 4,14,20,10, 6,13,19,11, 5,15,21,12) ( 7,28,18,35, 9,29,17,34, 8,30,16,36)(31,33,32)$
$ 12, 12, 12, 3 $ $26$ $12$ $( 1,14, 7,10,29,20, 4,32,37,34,18,27)( 2,15, 8,12,28,21, 5,33,38,35,16,25) ( 3,13, 9,11,30,19, 6,31,39,36,17,26)(22,23,24)$
$ 12, 12, 12, 3 $ $26$ $12$ $( 1,32,13,16,24,35,19,29, 7, 4,38,27)( 2,33,14,17,23,36,20,28, 8, 5,39,25) ( 3,31,15,18,22,34,21,30, 9, 6,37,26)(10,11,12)$

Group invariants

Order:  $936=2^{3} \cdot 3^{2} \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [936, 167]
Character table: Data not available.