# Properties

 Label 39T19 Degree $39$ Order $468$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3\times F_{13}$

## Group action invariants

 Degree $n$: $39$ Transitive number $t$: $19$ Group: $C_3\times F_{13}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $3$ Generators: (1,4,29,26,3,6,30,25,2,5,28,27)(7,14,23,17,9,13,24,16,8,15,22,18)(10,37,21,33,11,39,20,32,12,38,19,31)(34,36,35), (1,12,24,26,18,6)(2,11,23,27,16,4)(3,10,22,25,17,5)(7,32,39,20,34,28)(8,33,37,21,35,30)(9,31,38,19,36,29)(13,15,14)

## 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
$156$:  $F_{13}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 13: $F_{13}$

## Low degree siblings

39T19 x 2

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

## Group invariants

 Order: $468=2^{2} \cdot 3^{2} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [468, 29]
 Character table: not available.