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Properties

Label 13T4
Degree $13$
Order $52$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_{13}:C_4$

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Group action invariants

Degree $n$:  $13$
Transitive number $t$:  $4$
Group:  $C_{13}:C_4$
CHM label:  $F_{52}(13)=13:4$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$\card{\Aut(F/K)}$:  $1$
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13), (1,5,12,8)(2,10,11,3)(4,7,9,6)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

26T4

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

Group invariants

Order:  $52=2^{2} \cdot 13$
Cyclic:  no
Abelian:  no
Solvable:  yes
Label:  52.3
Character table:    
     2  2  2  2  2   .   .   .
    13  1  .  .  .   1   1   1

       1a 4a 4b 2a 13a 13b 13c
    2P 1a 2a 2a 1a 13b 13c 13a
    3P 1a 4b 4a 2a 13b 13c 13a
    5P 1a 4a 4b 2a 13a 13b 13c
    7P 1a 4b 4a 2a 13c 13a 13b
   11P 1a 4b 4a 2a 13b 13c 13a
   13P 1a 4a 4b 2a  1a  1a  1a

X.1     1  1  1  1   1   1   1
X.2     1 -1 -1  1   1   1   1
X.3     1  A -A -1   1   1   1
X.4     1 -A  A -1   1   1   1
X.5     4  .  .  .   B   C   D
X.6     4  .  .  .   C   D   B
X.7     4  .  .  .   D   B   C

A = -E(4)
  = -Sqrt(-1) = -i
B = E(13)^2+E(13)^3+E(13)^10+E(13)^11
C = E(13)^4+E(13)^6+E(13)^7+E(13)^9
D = E(13)+E(13)^5+E(13)^8+E(13)^12