Show commands:
Magma
magma: G := TransitiveGroup(24, 4996);
Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $4996$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $(C_3^2\times D_4^2):C_4$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,6)(2,5)(7,8)(9,11)(10,12)(13,18)(14,17)(15,16)(19,23)(20,24)(21,22), (1,17,8,19)(2,18,7,20)(3,13,11,24)(4,14,12,23)(5,16,9,21)(6,15,10,22), (1,20,6,21)(2,19,5,22)(3,23)(4,24)(7,18,8,17)(9,16,11,13)(10,15,12,14) | 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 $8$: $D_{4}$ x 12, $C_4\times C_2$ x 6, $C_2^3$ $16$: $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$ $32$: $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3 $36$: $C_3^2:C_4$ $64$: $(((C_4 \times C_2): C_2):C_2):C_2$, 16T79, 16T146 $72$: 12T40 x 3 $128$: $C_2 \wr C_2\wr C_2$ x 2, 32T1151 $144$: 12T82 x 6, 24T241 $256$: 16T542 $288$: 24T674 x 3 $576$: 24T1452 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $D_{4}$
Degree 6: $C_3^2:C_4$
Degree 8: $C_2 \wr C_2\wr C_2$
Degree 12: 12T82
Low degree siblings
24T4996 x 63Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 90 conjugacy class representatives for $(C_3^2\times D_4^2):C_4$
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.fr | magma: IdentifyGroup(G);
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| Character table: | 90 x 90 character table |
magma: CharacterTable(G);