Show commands:
Magma
magma: G := TransitiveGroup(18, 472);
Group action invariants
| Degree $n$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $472$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_6\wr C_3$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (3,5)(4,6)(7,11,10)(8,12,9)(13,14)(15,17)(16,18), (1,5)(2,6)(7,12,10,8,11,9)(13,14)(15,17)(16,18), (1,9,13)(2,10,14)(3,12,15,5,8,18)(4,11,16,6,7,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 $12$: $A_4$ x 5, $C_6\times C_2$ $24$: $A_4\times C_2$ x 15 $48$: $C_2^2 \times A_4$ x 5, $C_2^4:C_3$ $96$: 12T56 x 3 $192$: 12T90 $648$: $S_3 \wr C_3 $ $1296$: 18T283 $2592$: 18T399 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 6: $A_4\times C_2$
Degree 9: $S_3 \wr C_3 $
Low degree siblings
18T472 x 23, 36T5699 x 24, 36T5711 x 24, 36T5737 x 12, 36T5755 x 12, 36T5762 x 12, 36T5781 x 12, 36T5878 x 48, 36T5879 x 48, 36T5884 x 24, 36T5885 x 24, 36T5893 x 24, 36T5894 x 24, 36T6165 x 24, 36T6166 x 24, 36T6167 x 48, 36T6170 x 24, 36T6171 x 24, 36T6172 x 48Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
There are 88 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $5184=2^{6} \cdot 3^{4}$ | 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: | 5184.bt | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);