# Properties

 Label 39T22 Degree $39$ Order $702$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

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## Group action invariants

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

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: None

Degree 13: $C_{13}$

## Low degree siblings

27T219

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

## Group invariants

 Order: $702=2 \cdot 3^{3} \cdot 13$ Cyclic: no Abelian: no Solvable: yes GAP id: [702, 47]
 Character table: not available.