# Properties

 Label 39T8 Degree $39$ Order $156$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times D_{13}$

## Group action invariants

 Degree $n$: $39$ Transitive number $t$: $8$ Group: $S_3\times D_{13}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,32)(2,31)(3,33)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,23)(11,22)(12,24)(13,19)(14,21)(15,20)(16,18)(34,38)(35,37)(36,39), (1,29,2,30,3,28)(4,25,5,26,6,27)(7,22,8,23,9,24)(10,19,11,20,12,21)(13,16,14,17,15,18)(31,37,32,38,33,39)(34,36,35)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$26$:  $D_{13}$
$52$:  $D_{26}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 13: $D_{13}$

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

## Group invariants

 Order: $156=2^{2} \cdot 3 \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [156, 11]
 Character table: not available.