# Properties

 Label 39T4 Degree $39$ Order $78$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{39}$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$26$:  $D_{13}$

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

## Group invariants

 Order: $78=2 \cdot 3 \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [78, 5]
 Character table: not available.