Show commands:
Magma
magma: G := TransitiveGroup(38, 44);
Group action invariants
| Degree $n$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{19}^2:(S_3\times C_9)$ | ||
| 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: | (1,29,19,24,14,37,8,26,16,28,18,38,9,31,2,34,5,30)(3,20,10,36,7,21,11,22,12,27,17,33,4,25,15,23,13,32)(6,35), (1,27,15,34,12,23,14,24,19,36,3,28)(2,37,8,21,4,38,13,33,7,30,11,32)(5,29,6,20,18,26,10,22,9,31,16,25)(17,35) | 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 $8$: $D_{4}$ $9$: $C_9$ $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$, $C_{18}$ x 3 $24$: $(C_6\times C_2):C_2$, $D_4 \times C_3$ $36$: $C_6\times S_3$, 36T2 $54$: $C_9\times S_3$ $72$: 12T42, 36T15 $108$: 36T63 $216$: 36T181 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 90 conjugacy class representatives for $D_{19}^2:(S_3\times C_9)$
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.p | magma: IdentifyGroup(G);
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| Character table: | 90 x 90 character table |
magma: CharacterTable(G);