Show commands:
Magma
magma: G := TransitiveGroup(38, 37);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $37$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{19}^2:(C_3\times D_{18})$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,20,9,21,17,22,6,23,14,24,3,25,11,26,19,27,8,28,16,29,5,30,13,31,2,32,10,33,18,34,7,35,15,36,4,37,12,38), (1,22,16,25,10,20)(2,26,8,31,17,29)(3,30,19,37,5,38)(4,34,11,24,12,28)(6,23,14,36,7,27)(9,35)(13,32,15,21,18,33) | 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 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$, $D_{9}$ $36$: $C_6\times S_3$, $D_{18}$ $54$: 18T19 $108$: 36T69 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
The 51 conjugacy class representatives for $C_{19}^2:(C_3\times D_{18})$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $38988=2^{2} \cdot 3^{3} \cdot 19^{2}$ | 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 | ||
| Label: | 38988.n | magma: IdentifyGroup(G);
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| Character table: | 51 x 51 character table |
magma: CharacterTable(G);