Show commands:
Magma
magma: G := TransitiveGroup(38, 43);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $43$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{19}^2:(C_3\times D_9)$ | ||
| 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: | (1,20,15,29)(2,22,14,27)(3,24,13,25)(4,26,12,23)(5,28,11,21)(6,30,10,38)(7,32,9,36)(8,34)(16,31,19,37)(17,33,18,35), (1,11,16,9,15,18,10,6,4,3,12,7,14,8,5,13,17,19)(20,38,22,32,21,35,31,24,26)(23,29,30,27,36,28,33,37,25) | 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 $8$: $D_{4}$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$, $D_{9}$ $24$: $(C_6\times C_2):C_2$, $D_4 \times C_3$ $36$: $C_6\times S_3$, $D_{18}$ $54$: 18T19 $72$: 12T42, 36T24 $108$: 36T69 $216$: 36T189 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
There are 75 conjugacy classes of elements. Data not shown.
magma: ConjugacyClasses(G);
Group invariants
| Order: | $77976=2^{3} \cdot 3^{3} \cdot 19^{2}$ | 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: | 77976.o | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);