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Magma
magma: G := TransitiveGroup(12, 261);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $261$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\wr C_2^2$ | ||
CHM label: | $[S(3)^{4}]E(4)=S(3)wrE(4)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (4,8), (4,8,12), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12) | 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 $8$: $D_{4}$ x 6, $C_2^3$ $16$: $D_4\times C_2$ x 3 $32$: $C_2^2 \wr C_2$ $64$: $(((C_4 \times C_2): C_2):C_2):C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: None
Degree 4: $C_2^2$
Degree 6: None
Low degree siblings
12T267 x 3, 18T481 x 3, 24T7723, 24T7724 x 3, 24T7725 x 3, 24T7726 x 3, 24T7727 x 3, 24T7728, 24T7729, 24T7762 x 3, 24T7763 x 3, 24T7764 x 3, 24T7765 x 3, 24T7766 x 3, 24T7767 x 3, 24T7768 x 3, 24T7782, 36T6224 x 3, 36T6225 x 3, 36T6226 x 3, 36T6227 x 3, 36T6228 x 3, 36T6229 x 3, 36T6230 x 3, 36T6290 x 3Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 3,11, 7)( 4, 8,12)$ | |
$ 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 4, 8,12)$ | |
$ 3, 3, 3, 3 $ | $16$ | $3$ | $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8,12)$ | |
$ 3, 3, 3, 1, 1, 1 $ | $32$ | $3$ | $( 1, 9, 5)( 2, 6,10)( 4, 8,12)$ | |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 6,10)( 4, 8,12)$ | |
$ 3, 3, 1, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 6,10)( 3, 7,11)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$ | |
$ 6, 2, 2, 2 $ | $144$ | $6$ | $( 1,10)( 2, 5)( 3, 4, 7, 8,11,12)( 6, 9)$ | |
$ 6, 6 $ | $144$ | $6$ | $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 7,11)( 8,12)$ | |
$ 3, 3, 2, 2, 1, 1 $ | $72$ | $6$ | $( 1, 9, 5)( 2, 6,10)( 7,11)( 8,12)$ | |
$ 3, 2, 2, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 2, 6,10)( 7,11)( 8,12)$ | |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $81$ | $2$ | $( 5, 9)( 6,10)( 7,11)( 8,12)$ | |
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 8,12)$ | |
$ 3, 2, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $6$ | $( 3,11, 7)( 4, 8)$ | |
$ 3, 3, 2, 1, 1, 1, 1 $ | $48$ | $6$ | $( 1, 9, 5)( 2, 6,10)( 8,12)$ | |
$ 3, 3, 3, 2, 1 $ | $96$ | $6$ | $( 1, 9, 5)( 2, 6,10)( 3,11, 7)( 4, 8)$ | |
$ 3, 2, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $6$ | $( 2, 6,10)( 8,12)$ | |
$ 3, 3, 2, 1, 1, 1, 1 $ | $48$ | $6$ | $( 2, 6,10)( 3,11, 7)( 4, 8)$ | |
$ 3, 2, 1, 1, 1, 1, 1, 1, 1 $ | $24$ | $6$ | $( 1, 5, 9)( 8,12)$ | |
$ 3, 3, 2, 1, 1, 1, 1 $ | $48$ | $6$ | $( 1, 5, 9)( 3,11, 7)( 4, 8)$ | |
$ 4, 2, 2, 2, 2 $ | $216$ | $4$ | $( 1,10)( 2, 5)( 3,12,11, 8)( 4, 7)( 6, 9)$ | |
$ 6, 4, 2 $ | $432$ | $12$ | $( 1, 2, 5, 6, 9,10)( 3,12,11, 8)( 4, 7)$ | |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $108$ | $2$ | $( 5, 9)( 6,10)( 8,12)$ | |
$ 3, 2, 2, 2, 1, 1, 1 $ | $216$ | $6$ | $( 3,11, 7)( 4, 8)( 5, 9)( 6,10)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 6,10)( 8,12)$ | |
$ 3, 2, 2, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 3,11, 7)( 4, 8)( 6,10)$ | |
$ 3, 3, 2, 2, 1, 1 $ | $72$ | $6$ | $( 1, 9, 5)( 2, 6)( 3,11, 7)( 4, 8)$ | |
$ 4, 4, 2, 2 $ | $324$ | $4$ | $( 1,10, 9, 6)( 2, 5)( 3,12,11, 8)( 4, 7)$ | |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $2$ | $( 6,10)( 7,11)$ | |
$ 3, 2, 2, 1, 1, 1, 1, 1 $ | $72$ | $6$ | $( 3,11)( 4, 8,12)( 6,10)$ | |
$ 3, 3, 2, 2, 1, 1 $ | $72$ | $6$ | $( 1, 9, 5)( 2, 6)( 3,11)( 4, 8,12)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ | |
$ 6, 6 $ | $144$ | $6$ | $( 1, 3, 9,11, 5, 7)( 2,12, 6, 4,10, 8)$ | |
$ 6, 2, 2, 2 $ | $144$ | $6$ | $( 1, 7)( 2,12, 6, 4,10, 8)( 3, 9)( 5,11)$ | |
$ 2, 2, 2, 2, 2, 2 $ | $36$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ | |
$ 6, 6 $ | $144$ | $6$ | $( 1, 8, 5,12, 9, 4)( 2, 7,10, 3, 6,11)$ | |
$ 6, 2, 2, 2 $ | $144$ | $6$ | $( 1, 8, 5,12, 9, 4)( 2,11)( 3, 6)( 7,10)$ | |
$ 4, 4, 2, 2 $ | $324$ | $4$ | $( 1,11, 5, 7)( 2,12, 6, 8)( 3, 9)( 4,10)$ | |
$ 4, 4, 2, 2 $ | $324$ | $4$ | $( 1, 8, 5, 4)( 2, 3, 6,11)( 7,10)( 9,12)$ | |
$ 4, 2, 2, 2, 2 $ | $216$ | $4$ | $( 1, 7)( 2,12, 6, 8)( 3, 9)( 4,10)( 5,11)$ | |
$ 6, 4, 2 $ | $432$ | $12$ | $( 1, 3, 9,11, 5, 7)( 2, 4,10, 8)( 6,12)$ | |
$ 4, 2, 2, 2, 2 $ | $216$ | $4$ | $( 1, 4)( 2, 3, 6,11)( 5, 8)( 7,10)( 9,12)$ | |
$ 6, 4, 2 $ | $432$ | $12$ | $( 1, 8, 5,12, 9, 4)( 2,11)( 3, 6, 7,10)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $5184=2^{6} \cdot 3^{4}$ | 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: | 5184.bv | magma: IdentifyGroup(G);
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Character table: | 45 x 45 character table |
magma: CharacterTable(G);