Properties

Label 39T5
Order \(78\)
n \(39\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_{13}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
26:  $D_{13}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

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

Group invariants

Order:  $78=2 \cdot 3 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [78, 4]
Character table: Data not available.