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Magma
magma: G := TransitiveGroup(12, 28);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $28$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_3\times D_4$ | ||
CHM label: | $D(4)[x]S(3)$ | ||
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: | (1,7)(3,9)(5,11), (1,5)(2,10)(4,8)(7,11), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,4,7,10)(2,5,8,11)(3,6,9,12) | 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 $6$: $S_3$ $8$: $D_{4}$ x 2, $C_2^3$ $12$: $D_{6}$ x 3 $16$: $D_4\times C_2$ $24$: $S_3 \times C_2^2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $D_{4}$
Degree 6: $D_{6}$
Low degree siblings
12T28 x 3, 24T52 x 2, 24T53 x 2, 24T54 x 2Siblings 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 $ | $3$ | $2$ | $( 2, 6)( 3,11)( 5, 9)( 8,12)$ |
$ 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $2$ | $2$ | $( 2, 8)( 4,10)( 6,12)$ |
$ 2, 2, 2, 2, 2, 1, 1 $ | $6$ | $2$ | $( 2,12)( 3,11)( 4,10)( 5, 9)( 6, 8)$ |
$ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ |
$ 12 $ | $4$ | $12$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12)$ |
$ 4, 4, 4 $ | $6$ | $4$ | $( 1, 2, 7, 8)( 3,12, 9, 6)( 4, 5,10,11)$ |
$ 6, 6 $ | $4$ | $6$ | $( 1, 2, 9,10, 5, 6)( 3, 4,11,12, 7, 8)$ |
$ 6, 6 $ | $2$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 3)( 2, 8)( 4, 6)( 5,11)( 7, 9)(10,12)$ |
$ 6, 3, 3 $ | $4$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2,10, 6)( 4,12, 8)$ |
$ 4, 4, 4 $ | $2$ | $4$ | $( 1, 4, 7,10)( 2, 5, 8,11)( 3, 6, 9,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 4)( 2,11)( 3, 6)( 5, 8)( 7,10)( 9,12)$ |
$ 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $48=2^{4} \cdot 3$ | 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: | 48.38 | magma: IdentifyGroup(G);
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Character table: |
2 4 4 3 3 3 2 3 2 3 4 2 3 3 3 4 3 1 . 1 . . 1 . 1 1 . 1 1 1 1 1 1a 2a 2b 2c 2d 12a 4a 6a 6b 2e 6c 4b 2f 3a 2g 2P 1a 1a 1a 1a 1a 6b 2g 3a 3a 1a 3a 2g 1a 3a 1a 3P 1a 2a 2b 2c 2d 4b 4a 2f 2g 2e 2b 4b 2f 1a 2g 5P 1a 2a 2b 2c 2d 12a 4a 6a 6b 2e 6c 4b 2f 3a 2g 7P 1a 2a 2b 2c 2d 12a 4a 6a 6b 2e 6c 4b 2f 3a 2g 11P 1a 2a 2b 2c 2d 12a 4a 6a 6b 2e 6c 4b 2f 3a 2g X.1 1 1 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 1 1 1 X.3 1 -1 -1 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 1 1 1 X.5 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 1 X.6 1 1 -1 -1 -1 1 1 -1 1 1 -1 1 -1 1 1 X.7 1 1 -1 -1 1 -1 -1 1 1 1 -1 -1 1 1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 -1 -1 1 1 X.9 2 . -2 . . -1 . 1 -1 . 1 2 -2 -1 2 X.10 2 . -2 . . 1 . -1 -1 . 1 -2 2 -1 2 X.11 2 . 2 . . -1 . -1 -1 . -1 2 2 -1 2 X.12 2 . 2 . . 1 . 1 -1 . -1 -2 -2 -1 2 X.13 2 -2 . . . . . . -2 2 . . . 2 -2 X.14 2 2 . . . . . . -2 -2 . . . 2 -2 X.15 4 . . . . . . . 2 . . . . -2 -4 |
magma: CharacterTable(G);