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Magma
magma: G := TransitiveGroup(12, 188);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $188$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\wr A_4$ | ||
CHM label: | $[2^{6}]A_{4}=2wrA_{4}(6)$ | ||
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: | (1,12), (2,8)(3,9)(4,10)(5,11), (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) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ x 5 $24$: $A_4\times C_2$ x 5 $48$: $C_2^4:C_3$ $96$: $((C_2 \times D_4): C_2):C_3$ x 2, 12T56 $192$: 16T424 $384$: 16T731 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: $A_4$
Low degree siblings
12T187 x 8, 12T188 x 7, 24T2559 x 8, 24T2560 x 4, 24T2561 x 8, 24T2562 x 8, 24T2563 x 4, 24T2564 x 8, 24T2565 x 8, 24T2566 x 8, 24T2567 x 8, 24T2568 x 4, 24T2569 x 4, 24T2570 x 8, 24T2571 x 8, 24T2572 x 8, 24T2573 x 8, 24T2574 x 4, 24T2575 x 4, 24T2576 x 4, 24T2577 x 4, 32T34825 x 8Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $12$ | $2$ | $( 1,12)( 4, 5)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 4, 5)(10,11)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 1,12)( 4, 5)(10,11)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1,12)( 4, 5)( 8, 9)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $4$ | $2$ | $( 1,12)( 8, 9)(10,11)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $6$ | $2$ | $( 4, 5)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $12$ | $2$ | $( 1,12)( 4, 5)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 3)( 4, 5)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 2, 1, 1 $ | $6$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $12$ | $2$ | $( 2, 8)( 3, 9)( 4,10)( 5,11)$ |
$ 2, 2, 2, 2, 2, 1, 1 $ | $24$ | $2$ | $( 1,12)( 2, 8)( 3, 9)( 4,10)( 5,11)$ |
$ 4, 2, 2, 1, 1, 1, 1 $ | $12$ | $4$ | $( 2, 8)( 3, 9)( 4,10, 5,11)$ |
$ 4, 2, 2, 2, 1, 1 $ | $24$ | $4$ | $( 1,12)( 2, 8)( 3, 9)( 4,10, 5,11)$ |
$ 4, 2, 2, 1, 1, 1, 1 $ | $12$ | $4$ | $( 2, 9, 3, 8)( 4,10)( 5,11)$ |
$ 4, 2, 2, 2, 1, 1 $ | $24$ | $4$ | $( 1,12)( 2, 9, 3, 8)( 4,10)( 5,11)$ |
$ 4, 4, 1, 1, 1, 1 $ | $12$ | $4$ | $( 2, 9, 3, 8)( 4,10, 5,11)$ |
$ 4, 4, 2, 1, 1 $ | $24$ | $4$ | $( 1,12)( 2, 9, 3, 8)( 4,10, 5,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $12$ | $2$ | $( 1,12)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6, 7)$ |
$ 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1,12)( 2, 8)( 3, 9)( 4,10, 5,11)( 6, 7)$ |
$ 4, 2, 2, 2, 2 $ | $12$ | $4$ | $( 1,12)( 2, 9, 3, 8)( 4,10)( 5,11)( 6, 7)$ |
$ 4, 4, 2, 2 $ | $12$ | $4$ | $( 1,12)( 2, 9, 3, 8)( 4,10, 5,11)( 6, 7)$ |
$ 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 6, 3, 3 $ | $64$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 6,10)( 3, 7,11)$ |
$ 6, 3, 3 $ | $64$ | $6$ | $( 1, 5, 9)( 2, 6,11, 3, 7,10)( 4, 8,12)$ |
$ 6, 6 $ | $64$ | $6$ | $( 1, 5, 9,12, 4, 8)( 2, 6,11, 3, 7,10)$ |
$ 3, 3, 3, 3 $ | $64$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
$ 6, 3, 3 $ | $64$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2,10, 6)( 3,11, 7)$ |
$ 6, 3, 3 $ | $64$ | $6$ | $( 1, 9, 5)( 2,11, 7, 3,10, 6)( 4,12, 8)$ |
$ 6, 6 $ | $64$ | $6$ | $( 1, 9, 5,12, 8, 4)( 2,11, 7, 3,10, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $768=2^{8} \cdot 3$ | 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: | 768.1084555 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);