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Magma
magma: G := TransitiveGroup(12, 6);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $6$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $A_4\times C_2$ | ||
CHM label: | $A_{4}(12)x2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,5)(2,4,3)(6,8,7)(10,12,11), (1,7)(2,11)(3,12)(4,10)(5,8)(6,9), (1,11,6)(2,9,7)(3,10,5)(4,8,12) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $3$: $C_3$ $6$: $C_6$ $12$: $A_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: $C_3$
Degree 4: None
Degree 6: $A_4$, $A_4\times C_2$
Low degree siblings
6T6, 8T13, 12T7, 24T9Siblings 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, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 5)( 3, 6)( 8,11)( 9,12)$ |
$ 6, 6 $ | $4$ | $6$ | $( 1, 2, 6, 7,11, 9)( 3, 4, 5,12,10, 8)$ |
$ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 2,12)( 3, 7,11)( 4, 5, 6)( 8, 9,10)$ |
$ 6, 6 $ | $4$ | $6$ | $( 1, 3, 2, 7,12,11)( 4, 9, 5,10, 6, 8)$ |
$ 3, 3, 3, 3 $ | $4$ | $3$ | $( 1, 3, 8)( 2,10, 6)( 4, 9,11)( 5, 7,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2,11)( 3,12)( 4,10)( 5, 8)( 6, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $24=2^{3} \cdot 3$ | 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: | 24.13 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 1 1 1 1 3 3 3 1 . 1 1 1 1 . 1 1a 2a 6a 3a 6b 3b 2b 2c 2P 1a 1a 3b 3b 3a 3a 1a 1a 3P 1a 2a 2c 1a 2c 1a 2b 2c 5P 1a 2a 6b 3b 6a 3a 2b 2c X.1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 1 -1 X.3 1 -1 A -A /A -/A 1 -1 X.4 1 -1 /A -/A A -A 1 -1 X.5 1 1 -/A -/A -A -A 1 1 X.6 1 1 -A -A -/A -/A 1 1 X.7 3 1 . . . . -1 -3 X.8 3 -1 . . . . -1 3 A = -E(3) = (1-Sqrt(-3))/2 = -b3 |
magma: CharacterTable(G);