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Magma
magma: G := TransitiveGroup(24, 21211);
Group invariants
| Abstract group: | $C_3^8:(C_2^3:\OD_{16})$ | magma: IdentifyGroup(G);
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| Order: | $839808=2^{7} \cdot 3^{8}$ | 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 | magma: NilpotencyClass(G);
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Group action invariants
| Degree $n$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $21211$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | $(1,8,6,10,2,7,4,12,3,9,5,11)(13,22,17,21,14,23,18,19,15,24,16,20)$, $(4,6,5)(13,18,14,17,15,16)(19,22,21,23,20,24)$, $(1,16,12,19,6,13,8,22)(2,18,11,20,5,14,9,24)(3,17,10,21,4,15,7,23)$ | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ 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 4, $C_4\times C_2$ x 6, $C_2^3$ $16$: $C_8:C_2$ x 4, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$ $32$: $(C_8:C_2):C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_8:C_2)$ x 2, $C_2 \times (C_2^2:C_4)$ $64$: 16T72, 16T76, 16T95 $128$: 16T252 $2592$: 12T244, 12T245 $5184$: 24T7671, 24T7683 $10368$: 24T10026, 24T10032 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: $C_4$
Degree 6: None
Degree 8: $C_8:C_2$
Degree 12: None
Low degree siblings
24T21211 x 3, 36T34605 x 4Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Conjugacy classes not computedmagma: ConjugacyClasses(G);
Character table
Character table not computed
magma: CharacterTable(G);
Regular extensions
Data not computed