Properties

 Label 39T44 Degree $39$ Order $6084$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

Group action invariants

 Degree $n$: $39$ Transitive number $t$: $44$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,26,32,5,15,36,11,18,29,7,16,38)(2,20,33,13,19,31,10,24,28,12,25,30)(3,14,34,8,23,39,9,17,27,4,21,35)(6,22,37), (1,20,32,6,22,30)(2,23,29,5,19,33)(3,26,39,4,16,36)(7,25,27,13,17,35)(8,15,37,12,14,38)(9,18,34,11,24,28)(10,21,31)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$4$:  $C_4$
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$12$:  $C_{12}$ x 4
$18$:  $C_6 \times C_3$
$36$:  36T3
$156$:  $F_{13}$ x 2
$468$:  39T19 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 13: None

Low degree siblings

39T44 x 3

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

Group invariants

 Order: $6084=2^{2} \cdot 3^{2} \cdot 13^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: not available
 Character table: not available.