Show commands:
Magma
magma: G := TransitiveGroup(18, 137);
Group action invariants
Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $137$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_3^3:D_6$ | ||
Parity: | $-1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,6,4,10,17,7)(2,5,3,9,18,8)(11,13)(12,14)(15,16), (1,15,4,11,17,14)(2,16,3,12,18,13)(5,8,9)(6,7,10) | magma: Generators(G);
|
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 $36$: $S_3^2$ $108$: $C_3^2 : D_{6} $ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 9: $((C_3^3:C_3):C_2):C_2$
Low degree siblings
9T24 x 3, 18T129 x 3, 18T136 x 3, 18T137 x 2, 27T121, 27T128 x 3, 27T129, 36T502 x 3Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $(11,14,15)(12,13,16)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $6$ | $3$ | $( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $9$ | $2$ | $( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$ |
$ 6, 6, 1, 1, 1, 1, 1, 1 $ | $18$ | $6$ | $( 5,11, 8,14, 9,15)( 6,12, 7,13,10,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5, 6)( 7, 9)( 8,10)(11,12)(13,15)(14,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $27$ | $2$ | $( 1, 2)( 3,17)( 4,18)( 5,12)( 6,11)( 7,15)( 8,16)( 9,13)(10,14)$ |
$ 6, 6, 2, 2, 2 $ | $54$ | $6$ | $( 1, 2)( 3,17)( 4,18)( 5,12, 8,16, 9,13)( 6,11, 7,15,10,14)$ |
$ 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,14,15)(12,13,16)$ |
$ 3, 3, 3, 3, 3, 3 $ | $6$ | $3$ | $( 1, 4,17)( 2, 3,18)( 5, 8, 9)( 6, 7,10)(11,15,14)(12,16,13)$ |
$ 3, 3, 2, 2, 2, 2, 2, 2 $ | $18$ | $6$ | $( 1, 4,17)( 2, 3,18)( 5,11)( 6,12)( 7,13)( 8,14)( 9,15)(10,16)$ |
$ 6, 6, 3, 3 $ | $18$ | $6$ | $( 1, 4,17)( 2, 3,18)( 5,11, 8,14, 9,15)( 6,12, 7,13,10,16)$ |
$ 6, 6, 3, 3 $ | $18$ | $6$ | $( 1, 4,17)( 2, 3,18)( 5,11, 9,15, 8,14)( 6,12,10,16, 7,13)$ |
$ 3, 3, 3, 3, 3, 3 $ | $18$ | $3$ | $( 1, 5,11)( 2, 6,12)( 3, 7,13)( 4, 8,14)( 9,15,17)(10,16,18)$ |
$ 9, 9 $ | $36$ | $9$ | $( 1, 5,11, 4, 8,14,17, 9,15)( 2, 6,12, 3, 7,13,18,10,16)$ |
$ 6, 6, 6 $ | $54$ | $6$ | $( 1, 6,11, 2, 5,12)( 3, 9,13,17, 7,15)( 4,10,14,18, 8,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $324=2^{2} \cdot 3^{4}$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 324.39 | magma: IdentifyGroup(G);
|
Character table: |
2 2 1 1 1 2 1 2 2 1 1 1 1 1 1 1 . 1 3 4 3 3 3 2 2 1 1 1 4 3 2 2 2 2 2 1 1a 3a 3b 3c 2a 6a 2b 2c 6b 3d 3e 6c 6d 6e 3f 9a 6f 2P 1a 3a 3b 3c 1a 3b 1a 1a 3c 3d 3e 3a 3e 3d 3f 9a 3f 3P 1a 1a 1a 1a 2a 2a 2b 2c 2c 1a 1a 2a 2a 2a 1a 3d 2b 5P 1a 3a 3b 3c 2a 6a 2b 2c 6b 3d 3e 6c 6d 6e 3f 9a 6f 7P 1a 3a 3b 3c 2a 6a 2b 2c 6b 3d 3e 6c 6d 6e 3f 9a 6f X.1 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 -1 X.3 1 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 -1 X.5 2 2 2 2 . . -2 . . 2 2 . . . -1 -1 1 X.6 2 2 2 2 . . 2 . . 2 2 . . . -1 -1 -1 X.7 2 -1 -1 2 -2 1 . . . 2 -1 1 1 -2 2 -1 . X.8 2 -1 -1 2 2 -1 . . . 2 -1 -1 -1 2 2 -1 . X.9 4 -2 -2 4 . . . . . 4 -2 . . . -2 1 . X.10 6 -3 3 . -2 1 . . . -3 . 1 -2 1 . . . X.11 6 -3 3 . 2 -1 . . . -3 . -1 2 -1 . . . X.12 6 . -3 . -2 1 . . . -3 3 -2 1 1 . . . X.13 6 . -3 . 2 -1 . . . -3 3 2 -1 -1 . . . X.14 6 . . -3 . . . -2 1 6 . . . . . . . X.15 6 . . -3 . . . 2 -1 6 . . . . . . . X.16 6 3 . . -2 -2 . . . -3 -3 1 1 1 . . . X.17 6 3 . . 2 2 . . . -3 -3 -1 -1 -1 . . . |
magma: CharacterTable(G);