Properties

Label 13T5
Order \(78\)
n \(13\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{13}:C_6$

Related objects

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Group action invariants

Degree $n$ :  $13$
Transitive number $t$ :  $5$
Group :  $C_{13}:C_6$
CHM label :  $F_{78}(13)=13:6$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13), (1,4,3,12,9,10)(2,8,6,11,5,7)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T6, 39T6

Siblings are shown 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 1 $ $13$ $3$ $( 2, 4,10)( 3, 7, 6)( 5,13,11)( 8, 9,12)$
$ 6, 6, 1 $ $13$ $6$ $( 2, 5, 4,13,10,11)( 3, 9, 7,12, 6, 8)$
$ 3, 3, 3, 3, 1 $ $13$ $3$ $( 2,10, 4)( 3, 6, 7)( 5,11,13)( 8,12, 9)$
$ 6, 6, 1 $ $13$ $6$ $( 2,11,10,13, 4, 5)( 3, 8, 6,12, 7, 9)$
$ 2, 2, 2, 2, 2, 2, 1 $ $13$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$
$ 13 $ $6$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$
$ 13 $ $6$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$

Group invariants

Order:  $78=2 \cdot 3 \cdot 13$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [78, 1]
Character table:   
     2  1  1   1  1   1  1   .   .
     3  1  1   1  1   1  1   .   .
    13  1  .   .  .   .  .   1   1

       1a 3a  6a 3b  6b 2a 13a 13b
    2P 1a 3b  3a 3a  3b 1a 13b 13a
    3P 1a 1a  2a 1a  2a 2a 13a 13b
    5P 1a 3b  6b 3a  6a 2a 13b 13a
    7P 1a 3a  6a 3b  6b 2a 13b 13a
   11P 1a 3b  6b 3a  6a 2a 13b 13a
   13P 1a 3a  6a 3b  6b 2a  1a  1a

X.1     1  1   1  1   1  1   1   1
X.2     1  1  -1  1  -1 -1   1   1
X.3     1  A -/A /A  -A -1   1   1
X.4     1 /A  -A  A -/A -1   1   1
X.5     1  A  /A /A   A  1   1   1
X.6     1 /A   A  A  /A  1   1   1
X.7     6  .   .  .   .  .   B  *B
X.8     6  .   .  .   .  .  *B   B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(13)^2+E(13)^5+E(13)^6+E(13)^7+E(13)^8+E(13)^11
  = (-1-Sqrt(13))/2 = -1-b13