Show commands:
Magma
magma: G := TransitiveGroup(12, 250);
Group action invariants
| Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $250$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\wr (C_2\times S_4)$ | ||
| CHM label: | $[D(4)^{3}]S(3)=D(4)wrS(3)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (3,6,9,12), (3,9), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12) | 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 $6$: $S_3$ $8$: $C_2^3$ $12$: $D_{6}$ x 3 $24$: $S_4$ x 3, $S_3 \times C_2^2$ $48$: $S_4\times C_2$ x 9 $96$: $V_4^2:S_3$, 12T48 x 3 $192$: 12T100 x 3 $384$: 12T139 $768$: 16T1055 $1536$: 24T3386 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $S_3$
Degree 4: None
Degree 6: $S_4\times C_2$
Low degree siblings
12T250 x 15, 24T5413 x 4, 24T5427 x 4, 24T5587 x 4, 24T5588 x 4, 24T5707 x 4, 24T5752 x 4, 24T6222 x 4, 24T6316 x 4, 24T6797 x 4, 24T7182 x 8, 24T7183 x 8, 24T7184 x 8, 24T7185 x 8, 24T7186 x 8, 24T7187 x 8, 24T7188 x 8, 24T7189 x 8, 24T7190 x 8, 24T7191 x 8, 24T7192 x 8, 24T7193 x 8, 24T7194 x 8, 24T7195 x 8, 24T7196 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
The 65 conjugacy class representatives for $C_2\wr (C_2\times S_4)$
magma: ConjugacyClasses(G);
Group invariants
| Order: | $3072=2^{10} \cdot 3$ | 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: | 3072.dm | magma: IdentifyGroup(G);
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| Character table: | 65 x 65 character table |
magma: CharacterTable(G);