# Properties

 Label 39T15 Degree $39$ Order $312$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times D_{13}.C_2$

## Group action invariants

 Degree $n$: $39$ Transitive number $t$: $15$ Group: $S_3\times D_{13}.C_2$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,28)(2,30)(3,29)(4,27)(5,26)(6,25)(7,22)(8,24)(9,23)(10,21)(11,20)(12,19)(13,17)(14,16)(15,18)(31,39)(32,38)(33,37)(34,35), (1,23,11,29)(2,22,12,28)(3,24,10,30)(4,39,9,13)(5,38,7,15)(6,37,8,14)(16,21,36,31)(17,20,34,33)(18,19,35,32)(26,27)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_3 \times C_4$
$52$:  $C_{13}:C_4$
$104$:  26T7

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 13: $C_{13}:C_4$

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

## Group invariants

 Order: $312=2^{3} \cdot 3 \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [312, 46]
 Character table: not available.