Properties

Label 26T6
Order \(78\)
n \(26\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{13}:C_3$

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Group action invariants

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

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

Degree 2: $C_2$

Degree 13: $C_{13}:C_6$

Low degree siblings

13T5, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $13$ $3$ $( 3, 7,19)( 4, 8,20)( 5,14,12)( 6,13,11)( 9,25,21)(10,26,22)(15,17,23) (16,18,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $13$ $3$ $( 3,19, 7)( 4,20, 8)( 5,12,14)( 6,11,13)( 9,21,25)(10,22,26)(15,23,17) (16,24,18)$
$ 6, 6, 6, 6, 2 $ $13$ $6$ $( 1, 2)( 3, 9, 7,25,19,21)( 4,10, 8,26,20,22)( 5,17,14,23,12,15) ( 6,18,13,24,11,16)$
$ 6, 6, 6, 6, 2 $ $13$ $6$ $( 1, 2)( 3,21,19,25, 7, 9)( 4,22,20,26, 8,10)( 5,15,12,23,14,17) ( 6,16,11,24,13,18)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $13$ $2$ $( 1, 2)( 3,25)( 4,26)( 5,23)( 6,24)( 7,21)( 8,22)( 9,19)(10,20)(11,18)(12,17) (13,16)(14,15)$
$ 13, 13 $ $6$ $13$ $( 1, 4, 6, 8, 9,11,13,15,17,20,21,23,25)( 2, 3, 5, 7,10,12,14,16,18,19,22,24, 26)$
$ 13, 13 $ $6$ $13$ $( 1, 6, 9,13,17,21,25, 4, 8,11,15,20,23)( 2, 5,10,14,18,22,26, 3, 7,12,16,19, 24)$

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