Group action invariants
Degree $n$: | $18$ | |
Transitive number $t$: | $9$ | |
Group: | $S_3^2$ | |
Parity: | $-1$ | |
Primitive: | no | |
Nilpotency class: | $-1$ (not nilpotent) | |
$|\Aut(F/K)|$: | $2$ | |
Generators: | (1,5,4,8,17,9)(2,6,3,7,18,10)(11,16,14,12,15,13), (3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11), (1,2)(3,4)(5,11)(6,12)(7,13)(8,14)(9,15)(10,16)(17,18) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ x 2 $12$: $D_{6}$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$ x 2
Degree 6: $D_{6}$ x 2, $S_3^2$
Degree 9: $S_3^2$
Low degree siblings
6T9, 9T8, 12T16, 18T11 x 2Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $9$ | $2$ | $( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)(17,18)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5,10)( 6, 9)( 7, 8)(11,16)(12,15)(13,14)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 3,17, 2, 4,18)( 5,14, 9,11, 8,15)( 6,13,10,12, 7,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ |
$ 6, 6, 6 $ | $6$ | $6$ | $( 1, 5,14,18, 7,12)( 2, 6,13,17, 8,11)( 3,10,16, 4, 9,15)$ |
$ 3, 3, 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 6,15)( 2, 5,16)( 3, 8,12)( 4, 7,11)( 9,13,18)(10,14,17)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 7,14)( 2, 8,13)( 3, 9,16)( 4,10,15)( 5,12,18)( 6,11,17)$ |
Group invariants
Order: | $36=2^{2} \cdot 3^{2}$ | |
Cyclic: | no | |
Abelian: | no | |
Solvable: | yes | |
GAP id: | [36, 10] |
Character table: |
2 2 2 2 2 1 1 1 . 1 3 2 . 1 1 1 2 1 2 2 1a 2a 2b 2c 6a 3a 6b 3b 3c 2P 1a 1a 1a 1a 3a 3a 3c 3b 3c 3P 1a 2a 2b 2c 2b 1a 2c 1a 1a 5P 1a 2a 2b 2c 6a 3a 6b 3b 3c X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 1 1 X.3 1 -1 1 -1 1 1 -1 1 1 X.4 1 1 -1 -1 -1 1 -1 1 1 X.5 2 . . -2 . 2 1 -1 -1 X.6 2 . . 2 . 2 -1 -1 -1 X.7 2 . -2 . 1 -1 . -1 2 X.8 2 . 2 . -1 -1 . -1 2 X.9 4 . . . . -2 . 1 -2 |