# Properties

 Label 39T7 Degree $39$ Order $117$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_3\times C_{13}:C_3$

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## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$ x 4
$9$:  $C_3^2$
$39$:  $C_{13}:C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 13: $C_{13}:C_3$

## Low degree siblings

39T7 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $13$ $3$ $( 4,12,28)( 5,10,29)( 6,11,30)( 7,20,18)( 8,21,16)( 9,19,17)(13,39,32) (14,37,33)(15,38,31)(22,25,36)(23,26,34)(24,27,35)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1$ $13$ $3$ $( 4,28,12)( 5,29,10)( 6,30,11)( 7,18,20)( 8,16,21)( 9,17,19)(13,32,39) (14,33,37)(15,31,38)(22,36,25)(23,34,26)(24,35,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 2, 3)( 4,10,30)( 5,11,28)( 6,12,29)( 7,21,17)( 8,19,18)( 9,20,16) (13,37,31)(14,38,32)(15,39,33)(22,26,35)(23,27,36)(24,25,34)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 2, 3)( 4,29,11)( 5,30,12)( 6,28,10)( 7,16,19)( 8,17,20)( 9,18,21) (13,33,38)(14,31,39)(15,32,37)(22,34,27)(23,35,25)(24,36,26)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $1$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 3, 2)( 4,11,29)( 5,12,30)( 6,10,28)( 7,19,16)( 8,20,17)( 9,21,18) (13,38,33)(14,39,31)(15,37,32)(22,27,34)(23,25,35)(24,26,36)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $13$ $3$ $( 1, 3, 2)( 4,30,10)( 5,28,11)( 6,29,12)( 7,17,21)( 8,18,19)( 9,16,20) (13,31,37)(14,32,38)(15,33,39)(22,35,26)(23,36,27)(24,34,25)$ $39$ $3$ $39$ $( 1, 4, 9,10,14,17,21,23,25,29,33,36,38, 3, 6, 8,12,13,16,20,22,27,28,32,35, 37, 2, 5, 7,11,15,18,19,24,26,30,31,34,39)$ $13, 13, 13$ $3$ $13$ $( 1, 5, 8,10,15,16,21,24,27,29,31,35,38)( 2, 6, 9,11,13,17,19,22,25,30,32,36, 39)( 3, 4, 7,12,14,18,20,23,26,28,33,34,37)$ $39$ $3$ $39$ $( 1, 6, 7,10,13,18,21,22,26,29,32,34,38, 2, 4, 8,11,14,16,19,23,27,30,33,35, 39, 3, 5, 9,12,15,17,20,24,25,28,31,36,37)$ $39$ $3$ $39$ $( 1, 7,13,21,26,32,38, 4,11,16,23,30,35, 3, 9,15,20,25,31,37, 6,10,18,22,29, 34, 2, 8,14,19,27,33,39, 5,12,17,24,28,36)$ $13, 13, 13$ $3$ $13$ $( 1, 8,15,21,27,31,38, 5,10,16,24,29,35)( 2, 9,13,19,25,32,39, 6,11,17,22,30, 36)( 3, 7,14,20,26,33,37, 4,12,18,23,28,34)$ $39$ $3$ $39$ $( 1, 9,14,21,25,33,38, 6,12,16,22,28,35, 2, 7,15,19,26,31,39, 4,10,17,23,29, 36, 3, 8,13,20,27,32,37, 5,11,18,24,30,34)$ $39$ $3$ $39$ $( 1,13,26,38,11,23,35, 9,20,31, 6,18,29, 2,14,27,39,12,24,36, 7,21,32, 4,16, 30, 3,15,25,37,10,22,34, 8,19,33, 5,17,28)$ $39$ $3$ $39$ $( 1,14,25,38,12,22,35, 7,19,31, 4,17,29, 3,13,27,37,11,24,34, 9,21,33, 6,16, 28, 2,15,26,39,10,23,36, 8,20,32, 5,18,30)$ $13, 13, 13$ $3$ $13$ $( 1,15,27,38,10,24,35, 8,21,31, 5,16,29)( 2,13,25,39,11,22,36, 9,19,32, 6,17, 30)( 3,14,26,37,12,23,34, 7,20,33, 4,18,28)$ $39$ $3$ $39$ $( 1,22, 4,27, 9,28,10,32,14,35,17,37,21, 2,23, 5,25, 7,29,11,33,15,36,18,38, 19, 3,24, 6,26, 8,30,12,31,13,34,16,39,20)$ $39$ $3$ $39$ $( 1,23, 6,27, 7,30,10,33,13,35,18,39,21, 3,22, 5,26, 9,29,12,32,15,34,17,38, 20, 2,24, 4,25, 8,28,11,31,14,36,16,37,19)$ $13, 13, 13$ $3$ $13$ $( 1,24, 5,27, 8,29,10,31,15,35,16,38,21)( 2,22, 6,25, 9,30,11,32,13,36,17,39, 19)( 3,23, 4,26, 7,28,12,33,14,34,18,37,20)$

## Group invariants

 Order: $117=3^{2} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [117, 3]
 Character table: not available.