# Properties

 Label 39T12 Degree $39$ Order $234$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times C_{13}:C_3$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$39$:  $C_{13}:C_3$
$78$:  26T5

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $S_3$

Degree 13: $C_{13}:C_3$

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

## Group invariants

 Order: $234=2 \cdot 3^{2} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [234, 8]
 Character table: not available.