# Properties

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

# Related objects

## 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $4, 4, 4, 1$ $13$ $4$ $( 2, 6,13, 9)( 3,11,12, 4)( 5, 8,10, 7)$ $4, 4, 4, 1$ $13$ $4$ $( 2, 9,13, 6)( 3, 4,12,11)( 5, 7,10, 8)$ $2, 2, 2, 2, 2, 2, 1$ $13$ $2$ $( 2,13)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ $13$ $4$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$ $13$ $4$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$ $13$ $4$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$

## 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