Group action invariants
Degree $n$: | $26$ | |
Transitive number $t$: | $9$ | |
Group: | $C_2\times D_{13}:C_3$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,6,20,4,26,12)(2,5,19,3,25,11)(7,14,21,23,17,10)(8,13,22,24,18,9)(15,16), (1,13,3,7,21,5)(2,14,4,8,22,6)(9,16,24,26,20,11)(10,15,23,25,19,12) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $C_6\times C_2$ $78$: $C_{13}:C_6$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 13: $C_{13}:C_6$
Low degree siblings
26T9Siblings 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 $ | $13$ | $3$ | $( 3, 7,20)( 4, 8,19)( 5,13,11)( 6,14,12)( 9,26,21)(10,25,22)(15,18,23) (16,17,24)$ |
$ 6, 6, 6, 6, 1, 1 $ | $13$ | $6$ | $( 3, 9, 7,26,20,21)( 4,10, 8,25,19,22)( 5,17,13,24,11,16)( 6,18,14,23,12,15)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ | $13$ | $3$ | $( 3,20, 7)( 4,19, 8)( 5,11,13)( 6,12,14)( 9,21,26)(10,22,25)(15,23,18) (16,24,17)$ |
$ 6, 6, 6, 6, 1, 1 $ | $13$ | $6$ | $( 3,21,20,26, 7, 9)( 4,22,19,25, 8,10)( 5,16,11,24,13,17)( 6,15,12,23,14,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $13$ | $2$ | $( 3,26)( 4,25)( 5,24)( 6,23)( 7,21)( 8,22)( 9,20)(10,19)(11,17)(12,18)(13,16) (14,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)$ |
$ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3, 8,20, 4, 7,19)( 5,14,11, 6,13,12)( 9,25,21,10,26,22) (15,17,23,16,18,24)$ |
$ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3,10, 7,25,20,22)( 4, 9, 8,26,19,21)( 5,18,13,23,11,15) ( 6,17,14,24,12,16)$ |
$ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3,19, 7, 4,20, 8)( 5,12,13, 6,11,14)( 9,22,26,10,21,25) (15,24,18,16,23,17)$ |
$ 6, 6, 6, 6, 2 $ | $13$ | $6$ | $( 1, 2)( 3,22,20,25, 7,10)( 4,21,19,26, 8, 9)( 5,15,11,23,13,18) ( 6,16,12,24,14,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $13$ | $2$ | $( 1, 2)( 3,25)( 4,26)( 5,23)( 6,24)( 7,22)( 8,21)( 9,19)(10,20)(11,18)(12,17) (13,15)(14,16)$ |
$ 13, 13 $ | $6$ | $13$ | $( 1, 3, 5, 7, 9,11,13,16,17,20,21,24,26)( 2, 4, 6, 8,10,12,14,15,18,19,22,23, 25)$ |
$ 26 $ | $6$ | $26$ | $( 1, 4, 5, 8, 9,12,13,15,17,19,21,23,26, 2, 3, 6, 7,10,11,14,16,18,20,22,24,25 )$ |
$ 13, 13 $ | $6$ | $13$ | $( 1, 5, 9,13,17,21,26, 3, 7,11,16,20,24)( 2, 6,10,14,18,22,25, 4, 8,12,15,19, 23)$ |
$ 26 $ | $6$ | $26$ | $( 1, 6, 9,14,17,22,26, 4, 7,12,16,19,24, 2, 5,10,13,18,21,25, 3, 8,11,15,20,23 )$ |
Group invariants
Order: | $156=2^{2} \cdot 3 \cdot 13$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [156, 8] |
Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 1 1 1 1 3 1 1 1 1 1 1 1 1 1 1 1 1 . . . . 13 1 . . . . . 1 . . . . . 1 1 1 1 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 13a 26a 13b 26b 2P 1a 3b 3a 3a 3b 1a 1a 3b 3a 3a 3b 1a 13b 13b 13a 13a 3P 1a 1a 2a 1a 2a 2a 2b 2b 2c 2b 2c 2c 13a 26a 13b 26b 5P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 13b 26b 13a 26a 7P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 13b 26b 13a 26a 11P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 13b 26b 13a 26a 13P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 1a 2b 1a 2b 17P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 13a 26a 13b 26b 19P 1a 3a 6a 3b 6b 2a 2b 6c 6d 6e 6f 2c 13b 26b 13a 26a 23P 1a 3b 6b 3a 6a 2a 2b 6e 6f 6c 6d 2c 13a 26a 13b 26b X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 X.3 1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 -1 1 -1 X.5 1 A -/A /A -A -1 -1 -A /A -/A A 1 1 -1 1 -1 X.6 1 /A -A A -/A -1 -1 -/A A -A /A 1 1 -1 1 -1 X.7 1 A -/A /A -A -1 1 A -/A /A -A -1 1 1 1 1 X.8 1 /A -A A -/A -1 1 /A -A A -/A -1 1 1 1 1 X.9 1 A /A /A A 1 -1 -A -/A -/A -A -1 1 -1 1 -1 X.10 1 /A A A /A 1 -1 -/A -A -A -/A -1 1 -1 1 -1 X.11 1 A /A /A A 1 1 A /A /A A 1 1 1 1 1 X.12 1 /A A A /A 1 1 /A A A /A 1 1 1 1 1 X.13 6 . . . . . -6 . . . . . B -B *B -*B X.14 6 . . . . . -6 . . . . . *B -*B B -B X.15 6 . . . . . 6 . . . . . B B *B *B X.16 6 . . . . . 6 . . . . . *B *B 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 |