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Magma
magma: G := TransitiveGroup(12, 48);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $48$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^2\times S_4$ | ||
CHM label: | $2S_{4}(6)[x]2=[1/4.2^{6}]S(3)$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $4$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,7)(6,12), (1,5,9)(2,6,10)(3,7,11)(4,8,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11), (2,10)(3,11)(4,8)(5,9) | 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 $6$: $S_3$ $8$: $C_2^3$ $12$: $D_{6}$ x 3 $24$: $S_4$, $S_3 \times C_2^2$ $48$: $S_4\times C_2$ x 3 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: None
Degree 6: $D_{6}$, $S_4\times C_2$ x 2
Low degree siblings
12T48 x 11, 16T182 x 4, 24T125, 24T126 x 6, 24T150 x 3, 24T151 x 4, 24T152 x 4, 32T388Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 4,10)( 5,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $6$ | $2$ | $( 2, 4)( 3, 5)( 8,10)( 9,11)$ |
$ 4, 4, 1, 1, 1, 1 $ | $6$ | $4$ | $( 2, 4, 8,10)( 3, 5, 9,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2, 8)( 3, 9)( 4,10)( 5,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3,12)( 4, 5)( 6, 9)( 7, 8)(10,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)$ |
$ 6, 6 $ | $8$ | $6$ | $( 1, 2, 5,12, 3, 4)( 6, 9,10, 7, 8,11)$ |
$ 6, 6 $ | $8$ | $6$ | $( 1, 2, 5, 6, 9,10)( 3, 4, 7, 8,11,12)$ |
$ 4, 4, 2, 2 $ | $6$ | $4$ | $( 1, 2, 7, 8)( 3, 6, 9,12)( 4, 5)(10,11)$ |
$ 4, 4, 2, 2 $ | $6$ | $4$ | $( 1, 2, 7, 8)( 3, 6, 9,12)( 4,11)( 5,10)$ |
$ 6, 6 $ | $8$ | $6$ | $( 1, 3, 5, 7, 9,11)( 2, 4, 6, 8,10,12)$ |
$ 3, 3, 3, 3 $ | $8$ | $3$ | $( 1, 3, 5)( 2, 4,12)( 6, 8,10)( 7, 9,11)$ |
$ 4, 4, 2, 2 $ | $6$ | $4$ | $( 1, 3, 7, 9)( 2, 6, 8,12)( 4,10)( 5,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $6$ | $2$ | $( 1, 3)( 2,12)( 4,10)( 5,11)( 6, 8)( 7, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 6)( 2, 3)( 4, 5)( 7,12)( 8, 9)(10,11)$ |
$ 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 6)( 2, 3)( 4,11)( 5,10)( 7,12)( 8, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 6)( 2, 9)( 3, 8)( 4,11)( 5,10)( 7,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 7)( 2, 8)( 3, 9)( 4,10)( 5,11)( 6,12)$ |
$ 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,12)( 2, 3)( 4, 5)( 6, 7)( 8, 9)(10,11)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $96=2^{5} \cdot 3$ | 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: | 96.226 | magma: IdentifyGroup(G);
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Character table: |
2 5 5 4 4 5 4 4 2 2 4 4 2 2 4 4 5 5 5 5 5 3 1 . . . . . . 1 1 . . 1 1 . . . . 1 1 1 1a 2a 2b 4a 2c 2d 2e 6a 6b 4b 4c 6c 3a 4d 2f 2g 2h 2i 2j 2k 2P 1a 1a 1a 2c 1a 1a 1a 3a 3a 2c 2c 3a 3a 2c 1a 1a 1a 1a 1a 1a 3P 1a 2a 2b 4a 2c 2d 2e 2k 2i 4b 4c 2j 1a 4d 2f 2g 2h 2i 2j 2k 5P 1a 2a 2b 4a 2c 2d 2e 6a 6b 4b 4c 6c 3a 4d 2f 2g 2h 2i 2j 2k X.1 1 1 1 1 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 -1 1 -1 -1 1 X.3 1 -1 -1 1 1 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 1 -1 1 -1 -1 X.5 1 -1 1 -1 1 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 1 1 1 1 1 X.7 1 1 -1 -1 1 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 -1 -1 -1 1 -1 X.9 2 -2 . . 2 . . -1 1 . . 1 -1 . . -2 2 -2 -2 2 X.10 2 -2 . . 2 . . 1 -1 . . 1 -1 . . 2 -2 2 -2 -2 X.11 2 2 . . 2 . . -1 -1 . . -1 -1 . . 2 2 2 2 2 X.12 2 2 . . 2 . . 1 1 . . -1 -1 . . -2 -2 -2 2 -2 X.13 3 -1 -1 1 -1 -1 -1 . . 1 1 . . 1 -1 -1 -1 3 3 3 X.14 3 -1 -1 1 -1 1 1 . . -1 -1 . . 1 -1 1 1 -3 3 -3 X.15 3 -1 1 -1 -1 -1 -1 . . 1 1 . . -1 1 1 1 -3 3 -3 X.16 3 -1 1 -1 -1 1 1 . . -1 -1 . . -1 1 -1 -1 3 3 3 X.17 3 1 -1 -1 -1 -1 1 . . -1 1 . . 1 1 1 -1 -3 -3 3 X.18 3 1 -1 -1 -1 1 -1 . . 1 -1 . . 1 1 -1 1 3 -3 -3 X.19 3 1 1 1 -1 -1 1 . . -1 1 . . -1 -1 -1 1 3 -3 -3 X.20 3 1 1 1 -1 1 -1 . . 1 -1 . . -1 -1 1 -1 -3 -3 3 |
magma: CharacterTable(G);