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Magma
magma: G := TransitiveGroup(38, 16);
Group invariants
| Abstract group: | $C_{19}^2:D_6$ | magma: IdentifyGroup(G);
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| Order: | $4332=2^{2} \cdot 3 \cdot 19^{2}$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,9,16,15,7,19)(2,17,4,14,18,12)(3,6,11,13,10,5)(20,31,30,37,26,27)(21,24,22,36,33,35)(23,29,25,34,28,32)$, $(1,27,12,33,4,20,15,26,7,32,18,38,10,25,2,31,13,37,5,24,16,30,8,36,19,23,11,29,3,35,14,22,6,28,17,34,9,21)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $12$: $D_{6}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 19: None
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Conjugacy classes not computedmagma: ConjugacyClasses(G);
Character table
63 x 63 character tablemagma: CharacterTable(G);
Regular extensions
Data not computed