Show commands:
Magma
magma: G := TransitiveGroup(12, 167);
Group action invariants
| Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $167$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_3\wr D_4$ | ||
| CHM label: | $[3^{4}]D(4)=3wrD(4)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,7)(3,9)(5,11), (4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $8$: $D_{4}$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $24$: $(C_6\times C_2):C_2$, $D_4 \times C_3$ $36$: $C_6\times S_3$ $72$: $C_3^2:D_4$, 12T42 $216$: 12T116, 12T121 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: None
Low degree siblings
12T167, 18T189 x 2, 24T1519 x 2, 24T1536, 36T1079 x 2, 36T1080 x 2, 36T1081 x 2, 36T1158, 36T1163, 36T1165, 36T1170, 36T1180, 36T1191 x 2, 36T1200 x 2, 36T1231 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 54 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $648=2^{3} \cdot 3^{4}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 648.719 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);