↑

Properties

Label 39T8
Degree $39$
Order $156$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_{13}$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(39, 8);
 

Group action invariants

Degree $n$:  $39$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_{13}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,32)(2,31)(3,33)(4,30)(5,29)(6,28)(7,27)(8,26)(9,25)(10,23)(11,22)(12,24)(13,19)(14,21)(15,20)(16,18)(34,38)(35,37)(36,39), (1,29,2,30,3,28)(4,25,5,26,6,27)(7,22,8,23,9,24)(10,19,11,20,12,21)(13,16,14,17,15,18)(31,37,32,38,33,39)(34,36,35)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$26$:  $D_{13}$
$52$:  $D_{26}$

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

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $156=2^{2} \cdot 3 \cdot 13$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  156.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 6A 13A1 13A2 13A3 13A4 13A5 13A6 26A1 26A3 26A5 26A7 26A9 26A11 39A1 39A2 39A4 39A5 39A7 39A10
Size 1 3 13 39 2 26 2 2 2 2 2 2 6 6 6 6 6 6 4 4 4 4 4 4
2 P 1A 1A 1A 1A 3A 3A 13A2 13A5 13A6 13A4 13A1 13A3 13A3 13A6 13A5 13A1 13A4 13A2 39A10 39A7 39A4 39A1 39A2 39A5
3 P 1A 2A 2B 2C 1A 2B 13A3 13A1 13A4 13A6 13A5 13A2 26A9 26A5 26A11 26A3 26A1 26A7 13A6 13A1 13A5 13A2 13A4 13A3
13 P 1A 2A 2B 2C 3A 6A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 3A 3A 3A 3A 3A 3A
Type
156.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
156.11.1b R 1 −1 −1 1 1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
156.11.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 1 1
156.11.1d R 1 1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
156.11.2a R 2 0 2 0 −1 −1 2 2 2 2 2 2 0 0 0 0 0 0 −1 −1 −1 −1 −1 −1
156.11.2b R 2 0 −2 0 −1 1 2 2 2 2 2 2 0 0 0 0 0 0 −1 −1 −1 −1 −1 −1
156.11.2c1 R 2 2 0 0 2 0 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−3+ζ133 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−2+ζ132 ζ13−5+ζ135 ζ13−1+ζ13 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−4+ζ134 ζ13−2+ζ132
156.11.2c2 R 2 2 0 0 2 0 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−4+ζ134 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−6+ζ136 ζ13−2+ζ132 ζ13−3+ζ133 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−1+ζ13 ζ13−6+ζ136
156.11.2c3 R 2 2 0 0 2 0 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−2+ζ132 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−3+ζ133 ζ13−1+ζ13 ζ13−5+ζ135 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−6+ζ136 ζ13−3+ζ133
156.11.2c4 R 2 2 0 0 2 0 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−5+ζ135 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−1+ζ13 ζ13−4+ζ134 ζ13−6+ζ136 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−2+ζ132 ζ13−1+ζ13
156.11.2c5 R 2 2 0 0 2 0 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−1+ζ13 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−5+ζ135 ζ13−6+ζ136 ζ13−4+ζ134 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−3+ζ133 ζ13−5+ζ135
156.11.2c6 R 2 2 0 0 2 0 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−6+ζ136 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−4+ζ134 ζ13−3+ζ133 ζ13−2+ζ132 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−5+ζ135 ζ13−4+ζ134
156.11.2d1 R 2 −2 0 0 2 0 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−5+ζ135 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−3+ζ133 −ζ13−3−ζ133 −ζ13−4−ζ134 −ζ13−2−ζ132 −ζ13−5−ζ135 −ζ13−1−ζ13 −ζ13−6−ζ136 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−4+ζ134 ζ13−2+ζ132
156.11.2d2 R 2 −2 0 0 2 0 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−2+ζ132 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−4+ζ134 −ζ13−4−ζ134 −ζ13−1−ζ13 −ζ13−6−ζ136 −ζ13−2−ζ132 −ζ13−3−ζ133 −ζ13−5−ζ135 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−1+ζ13 ζ13−6+ζ136
156.11.2d3 R 2 −2 0 0 2 0 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−1+ζ13 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−2+ζ132 −ζ13−2−ζ132 −ζ13−6−ζ136 −ζ13−3−ζ133 −ζ13−1−ζ13 −ζ13−5−ζ135 −ζ13−4−ζ134 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−6+ζ136 ζ13−3+ζ133
156.11.2d4 R 2 −2 0 0 2 0 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−4+ζ134 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−5+ζ135 −ζ13−5−ζ135 −ζ13−2−ζ132 −ζ13−1−ζ13 −ζ13−4−ζ134 −ζ13−6−ζ136 −ζ13−3−ζ133 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−2+ζ132 ζ13−1+ζ13
156.11.2d5 R 2 −2 0 0 2 0 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−6+ζ136 ζ13−5+ζ135 ζ13−3+ζ133 ζ13−1+ζ13 −ζ13−1−ζ13 −ζ13−3−ζ133 −ζ13−5−ζ135 −ζ13−6−ζ136 −ζ13−4−ζ134 −ζ13−2−ζ132 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−4+ζ134 ζ13−3+ζ133 ζ13−5+ζ135
156.11.2d6 R 2 −2 0 0 2 0 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−3+ζ133 ζ13−4+ζ134 ζ13−5+ζ135 ζ13−6+ζ136 −ζ13−6−ζ136 −ζ13−5−ζ135 −ζ13−4−ζ134 −ζ13−3−ζ133 −ζ13−2−ζ132 −ζ13−1−ζ13 ζ13−3+ζ133 ζ13−6+ζ136 ζ13−1+ζ13 ζ13−2+ζ132 ζ13−5+ζ135 ζ13−4+ζ134
156.11.4a1 R 4 0 0 0 −2 0 2ζ13−6+2ζ136 2ζ13−1+2ζ13 2ζ13−5+2ζ135 2ζ13−2+2ζ132 2ζ13−4+2ζ134 2ζ13−3+2ζ133 0 0 0 0 0 0 −ζ13−5−ζ135 −ζ13−3−ζ133 −ζ13−6−ζ136 −ζ13−1−ζ13 −ζ13−4−ζ134 −ζ13−2−ζ132
156.11.4a2 R 4 0 0 0 −2 0 2ζ13−5+2ζ135 2ζ13−3+2ζ133 2ζ13−2+2ζ132 2ζ13−6+2ζ136 2ζ13−1+2ζ13 2ζ13−4+2ζ134 0 0 0 0 0 0 −ζ13−2−ζ132 −ζ13−4−ζ134 −ζ13−5−ζ135 −ζ13−3−ζ133 −ζ13−1−ζ13 −ζ13−6−ζ136
156.11.4a3 R 4 0 0 0 −2 0 2ζ13−4+2ζ134 2ζ13−5+2ζ135 2ζ13−1+2ζ13 2ζ13−3+2ζ133 2ζ13−6+2ζ136 2ζ13−2+2ζ132 0 0 0 0 0 0 −ζ13−1−ζ13 −ζ13−2−ζ132 −ζ13−4−ζ134 −ζ13−5−ζ135 −ζ13−6−ζ136 −ζ13−3−ζ133
156.11.4a4 R 4 0 0 0 −2 0 2ζ13−3+2ζ133 2ζ13−6+2ζ136 2ζ13−4+2ζ134 2ζ13−1+2ζ13 2ζ13−2+2ζ132 2ζ13−5+2ζ135 0 0 0 0 0 0 −ζ13−4−ζ134 −ζ13−5−ζ135 −ζ13−3−ζ133 −ζ13−6−ζ136 −ζ13−2−ζ132 −ζ13−1−ζ13
156.11.4a5 R 4 0 0 0 −2 0 2ζ13−2+2ζ132 2ζ13−4+2ζ134 2ζ13−6+2ζ136 2ζ13−5+2ζ135 2ζ13−3+2ζ133 2ζ13−1+2ζ13 0 0 0 0 0 0 −ζ13−6−ζ136 −ζ13−1−ζ13 −ζ13−2−ζ132 −ζ13−4−ζ134 −ζ13−3−ζ133 −ζ13−5−ζ135
156.11.4a6 R 4 0 0 0 −2 0 2ζ13−1+2ζ13 2ζ13−2+2ζ132 2ζ13−3+2ζ133 2ζ13−4+2ζ134 2ζ13−5+2ζ135 2ζ13−6+2ζ136 0 0 0 0 0 0 −ζ13−3−ζ133 −ζ13−6−ζ136 −ζ13−1−ζ13 −ζ13−2−ζ132 −ζ13−5−ζ135 −ζ13−4−ζ134

magma: CharacterTable(G);