# Properties

 Label 39T17 Degree $39$ Order $468$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times D_{13}:C_3$

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## Group action invariants

 Degree $n$: $39$ Transitive number $t$: $17$ Group: $S_3\times D_{13}:C_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,37,30,2,39,28)(3,38,29)(4,7,18)(5,9,16,6,8,17)(10,27,33)(11,26,31,12,25,32)(13,36,20,15,34,19)(14,35,21)(23,24), (1,27,2,25,3,26)(4,24,5,22,6,23)(7,20,8,21,9,19)(10,17,11,18,12,16)(13,15,14)(28,39,29,37,30,38)(31,35,32,36,33,34)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$3$:  $C_3$
$4$:  $C_2^2$
$6$:  $S_3$, $C_6$ x 3
$12$:  $D_{6}$, $C_6\times C_2$
$18$:  $S_3\times C_3$
$36$:  $C_6\times S_3$
$78$:  $C_{13}:C_6$
$156$:  26T9

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 13: $C_{13}:C_6$

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

## Group invariants

 Order: $468=2^{2} \cdot 3^{2} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [468, 31]
 Character table: not available.