Show commands:
Magma
magma: G := TransitiveGroup(12, 23);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $23$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_2 \times S_4$ | ||
CHM label: | $S_{4}(6d)[x]2=[1/8.2^{6}]S(3)$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (2,8)(3,9)(4,10)(5,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ $24$: $S_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: None
Degree 6: $D_{6}$, $S_4$, $S_4\times C_2$
Low degree siblings
6T11 x 2, 8T24 x 2, 12T21, 12T22, 12T23, 12T24 x 2, 16T61, 24T46, 24T47, 24T48 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 2, 4)( 3, 5)( 8,10)( 9,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 8)( 3, 9)( 4,10)( 5,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3,12)( 4, 5)( 6, 9)( 7, 8)(10,11)$ |
$ 6, 6 $ | $8$ | $6$ | $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$ |
$ 4, 4, 2, 2 $ | $6$ | $4$ | $( 1, 2, 7, 8)( 3, 6, 9,12)( 4,11)( 5,10)$ |
$ 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$ |
$ 4, 4, 2, 2 $ | $6$ | $4$ | $( 1, 3, 7, 9)( 2, 6, 8,12)( 4,10)( 5,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $48=2^{4} \cdot 3$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 48.48 | magma: IdentifyGroup(G);
|
Character table: |
2 4 3 4 3 1 3 1 3 4 4 3 1 . . . 1 . 1 . . 1 1a 2a 2b 2c 6a 4a 3a 4b 2d 2e 2P 1a 1a 1a 1a 3a 2b 3a 2b 1a 1a 3P 1a 2a 2b 2c 2e 4a 1a 4b 2d 2e 5P 1a 2a 2b 2c 6a 4a 3a 4b 2d 2e X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 -1 1 -1 1 1 X.3 1 -1 1 1 -1 1 1 -1 -1 -1 X.4 1 1 1 -1 -1 -1 1 1 -1 -1 X.5 2 . 2 . -1 . -1 . 2 2 X.6 2 . 2 . 1 . -1 . -2 -2 X.7 3 -1 -1 -1 . 1 . 1 -1 3 X.8 3 -1 -1 1 . -1 . 1 1 -3 X.9 3 1 -1 -1 . 1 . -1 1 -3 X.10 3 1 -1 1 . -1 . -1 -1 3 |
magma: CharacterTable(G);