Show commands:
Magma
magma: G := TransitiveGroup(18, 189);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $189$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_3\wr D_4$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,12,5,16,3,11,4,18,2,10,6,17)(7,15,8,13,9,14), (1,5,3,4,2,6)(7,8,9)(10,15,17)(11,13,18)(12,14,16) | magma: Generators(G);
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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 6: $S_3\times C_3$, $C_3^2:D_4$
Degree 9: None
Low degree siblings
12T167 x 2, 18T189, 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
The 54 conjugacy class representatives for $C_3\wr D_4$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $648=2^{3} \cdot 3^{4}$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 648.719 | magma: IdentifyGroup(G);
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| Character table: | 54 x 54 character table |
magma: CharacterTable(G);