Show commands:
Magma
magma: G := TransitiveGroup(24, 5017);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5017$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_4.\PGOPlus(4,3)$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,2)(3,4)(5,11,6,12)(7,9,8,10)(17,22,18,21)(19,23,20,24), (1,19,2,20)(3,17,4,18)(5,14)(6,13)(7,15)(8,16)(9,22,10,21)(11,24,12,23), (1,21,2,22)(3,23,4,24)(5,15,6,16)(7,13,8,14)(9,20)(10,19)(11,18)(12,17) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_2^2$ x 7 $6$: $S_3$ x 2 $8$: $C_2^3$ $12$: $D_{6}$ x 6 $16$: $Q_8:C_2$ $24$: $S_3 \times C_2^2$ x 2 $36$: $S_3^2$ $48$: 24T19 x 2 $72$: 12T37 $144$: 24T228 $576$: $(A_4\wr C_2):C_2$ $1152$: 12T195 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: $S_3^2$
Degree 8: None
Degree 12: 12T37
Low degree siblings
24T5017, 32T205456 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 52 conjugacy class representatives for $C_4.\PGOPlus(4,3)$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2304=2^{8} \cdot 3^{2}$ | 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: | 2304.gg | magma: IdentifyGroup(G);
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| Character table: | 52 x 52 character table |
magma: CharacterTable(G);