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
26T4Siblings 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 |