Group action invariants
Degree $n$: | $26$ | |
Transitive number $t$: | $6$ | |
Group: | $D_{13}:C_3$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
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) |
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, 39T6Siblings 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, 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 |