# Properties

 Label 39T13 Degree $39$ Order $234$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3\times D_{13}:C_3$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$ x 4
$6$:  $C_6$ x 4
$9$:  $C_3^2$
$18$:  $C_6 \times C_3$
$78$:  $C_{13}:C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 13: $C_{13}:C_6$

## Low degree siblings

39T13 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $13$ $3$ $( 4,10,30)( 5,11,28)( 6,12,29)( 7,21,18)( 8,19,16)( 9,20,17)(13,39,31) (14,37,32)(15,38,33)(22,25,34)(23,26,35)(24,27,36)$ $6, 6, 6, 6, 6, 6, 1, 1, 1$ $13$ $6$ $( 4,13,10,39,30,31)( 5,14,11,37,28,32)( 6,15,12,38,29,33)( 7,25,21,34,18,22) ( 8,26,19,35,16,23)( 9,27,20,36,17,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $13$ $3$ $( 4,30,10)( 5,28,11)( 6,29,12)( 7,18,21)( 8,16,19)( 9,17,20)(13,31,39) (14,32,37)(15,33,38)(22,34,25)(23,35,26)(24,36,27)$ $6, 6, 6, 6, 6, 6, 1, 1, 1$ $13$ $6$ $( 4,31,30,39,10,13)( 5,32,28,37,11,14)( 6,33,29,38,12,15)( 7,22,18,34,21,25) ( 8,23,16,35,19,26)( 9,24,17,36,20,27)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1$ $13$ $2$ $( 4,39)( 5,37)( 6,38)( 7,34)( 8,35)( 9,36)(10,31)(11,32)(12,33)(13,30)(14,28) (15,29)(16,26)(17,27)(18,25)(19,23)(20,24)(21,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 2, 3)( 4,11,29)( 5,12,30)( 6,10,28)( 7,19,17)( 8,20,18)( 9,21,16) (13,37,33)(14,38,31)(15,39,32)(22,26,36)(23,27,34)(24,25,35)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 2, 3)( 4,14,12,39,28,33)( 5,15,10,37,29,31)( 6,13,11,38,30,32) ( 7,26,20,34,16,24)( 8,27,21,35,17,22)( 9,25,19,36,18,23)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 2, 3)( 4,28,12)( 5,29,10)( 6,30,11)( 7,16,20)( 8,17,21)( 9,18,19) (13,32,38)(14,33,39)(15,31,37)(22,35,27)(23,36,25)(24,34,26)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 2, 3)( 4,32,29,39,11,15)( 5,33,30,37,12,13)( 6,31,28,38,10,14) ( 7,23,17,34,19,27)( 8,24,18,35,20,25)( 9,22,16,36,21,26)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 2, 3)( 4,37, 6,39, 5,38)( 7,35, 9,34, 8,36)(10,32,12,31,11,33) (13,28,15,30,14,29)(16,27,18,26,17,25)(19,24,21,23,20,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 3, 2)( 4,12,28)( 5,10,29)( 6,11,30)( 7,20,16)( 8,21,17)( 9,19,18) (13,38,32)(14,39,33)(15,37,31)(22,27,35)(23,25,36)(24,26,34)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 3, 2)( 4,15,11,39,29,32)( 5,13,12,37,30,33)( 6,14,10,38,28,31) ( 7,27,19,34,17,23)( 8,25,20,35,18,24)( 9,26,21,36,16,22)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 3, 2)( 4,29,11)( 5,30,12)( 6,28,10)( 7,17,19)( 8,18,20)( 9,16,21) (13,33,37)(14,31,38)(15,32,39)(22,36,26)(23,34,27)(24,35,25)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 3, 2)( 4,33,28,39,12,14)( 5,31,29,37,10,15)( 6,32,30,38,11,13) ( 7,24,16,34,20,26)( 8,22,17,35,21,27)( 9,23,18,36,19,25)$ $6, 6, 6, 6, 6, 6, 3$ $13$ $6$ $( 1, 3, 2)( 4,38, 5,39, 6,37)( 7,36, 8,34, 9,35)(10,33,11,31,12,32) (13,29,14,30,15,28)(16,25,17,26,18,27)(19,22,20,23,21,24)$ $13, 13, 13$ $6$ $13$ $( 1, 4, 8,10,13,16,19,23,26,30,31,35,39)( 2, 5, 9,11,14,17,20,24,27,28,32,36, 37)( 3, 6, 7,12,15,18,21,22,25,29,33,34,38)$ $39$ $6$ $39$ $( 1, 5, 7,10,14,18,19,24,25,30,32,34,39, 2, 6, 8,11,15,16,20,22,26,28,33,35, 37, 3, 4, 9,12,13,17,21,23,27,29,31,36,38)$ $39$ $6$ $39$ $( 1, 6, 9,10,15,17,19,22,27,30,33,36,39, 3, 5, 8,12,14,16,21,24,26,29,32,35, 38, 2, 4, 7,11,13,18,20,23,25,28,31,34,37)$ $39$ $6$ $39$ $( 1, 7,14,19,25,32,39, 6,11,16,22,28,35, 3, 9,13,21,27,31,38, 5,10,18,24,30, 34, 2, 8,15,20,26,33,37, 4,12,17,23,29,36)$ $13, 13, 13$ $6$ $13$ $( 1, 8,13,19,26,31,39, 4,10,16,23,30,35)( 2, 9,14,20,27,32,37, 5,11,17,24,28, 36)( 3, 7,15,21,25,33,38, 6,12,18,22,29,34)$ $39$ $6$ $39$ $( 1, 9,15,19,27,33,39, 5,12,16,24,29,35, 2, 7,13,20,25,31,37, 6,10,17,22,30, 36, 3, 8,14,21,26,32,38, 4,11,18,23,28,34)$

## Group invariants

 Order: $234=2 \cdot 3^{2} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [234, 7]
 Character table: not available.