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Magma
magma: G := TransitiveGroup(12, 2);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $2$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_6\times C_2$ | ||
CHM label: | $E(4)[x]C(3)=6x2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $12$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,10)(2,5)(3,12)(4,7)(6,9)(8,11), (1,5,9)(2,6,10)(3,7,11)(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 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $C_6$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 3: $C_3$
Degree 4: $C_2^2$
Degree 6: $C_6$ x 3
Low degree siblings
There are no siblings 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 $ | $1$ | $1$ | $()$ |
$ 6, 6 $ | $1$ | $6$ | $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ |
$ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1, 6, 5,10, 9, 2)( 3, 8, 7,12,11, 4)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1, 8, 9, 4, 5,12)( 2, 3,10,11, 6, 7)$ |
$ 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 9, 5)( 2,10, 6)( 3,11, 7)( 4,12, 8)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 5)( 3,12)( 4, 7)( 6, 9)( 8,11)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1,11, 9, 7, 5, 3)( 2,12,10, 8, 6, 4)$ |
$ 6, 6 $ | $1$ | $6$ | $( 1,12, 5, 4, 9, 8)( 2, 7, 6,11,10, 3)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $12=2^{2} \cdot 3$ | magma: Order(G);
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Cyclic: | no | magma: IsCyclic(G);
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Abelian: | yes | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $1$ | ||
Label: | 12.5 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 2 2 2 2 2 2 2 2 3 1 1 1 1 1 1 1 1 1 1 1 1 1a 6a 6b 2a 3a 6c 2b 6d 3b 2c 6e 6f 2P 1a 3b 3a 1a 3b 3a 1a 3b 3a 1a 3b 3a 3P 1a 2c 2b 2a 1a 2c 2b 2a 1a 2c 2b 2a 5P 1a 6c 6e 2a 3b 6a 2b 6f 3a 2c 6b 6d X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 X.3 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 X.4 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 X.5 1 A /A 1 -A /A -1 -A -/A -1 A -/A X.6 1 /A A 1 -/A A -1 -/A -A -1 /A -A X.7 1 A -/A -1 -A /A 1 A -/A -1 -A /A X.8 1 /A -A -1 -/A A 1 /A -A -1 -/A A X.9 1 -/A A -1 -/A -A -1 /A -A 1 /A A X.10 1 -A /A -1 -A -/A -1 A -/A 1 A /A X.11 1 -/A -A 1 -/A -A 1 -/A -A 1 -/A -A X.12 1 -A -/A 1 -A -/A 1 -A -/A 1 -A -/A A = -E(3) = (1-Sqrt(-3))/2 = -b3 |
magma: CharacterTable(G);