Show commands:
Magma
magma: G := TransitiveGroup(24, 5013);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $5013$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_4\times \PGOPlus(4,3)$ | ||
| 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: | (1,22,5,16,12,17)(2,21,6,15,11,18)(3,24,8,13,9,20)(4,23,7,14,10,19), (1,21,5,18,12,16)(2,22,6,17,11,15)(3,23,8,19,9,13)(4,24,7,20,10,14), (1,24)(2,23)(3,21)(4,22)(5,13)(6,14)(7,16)(8,15)(9,17,10,18)(11,20,12,19) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 7 $4$: $C_4$ x 4, $C_2^2$ x 7 $6$: $S_3$ x 2 $8$: $C_4\times C_2$ x 6, $C_2^3$ $12$: $D_{6}$ x 6 $16$: $C_4\times C_2^2$ $24$: $S_3 \times C_2^2$ x 2, $S_3 \times C_4$ x 4 $36$: $S_3^2$ $48$: 24T27 x 2 $72$: 12T37 $144$: 24T224 $576$: $(A_4\wr C_2):C_2$ $1152$: 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
24T5013, 32T205457 x 2, 36T3162 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 64 conjugacy class representatives for $C_4\times \PGOPlus(4,3)$
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.gd | magma: IdentifyGroup(G);
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| Character table: | 64 x 64 character table |
magma: CharacterTable(G);