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Magma
magma: G := TransitiveGroup(12, 123);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $123$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\times S_5$ | ||
CHM label: | $L(6):2[x]2$ | ||
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: | (1,3,5,7,9)(2,4,6,8,12), (1,11)(2,4)(3,5)(6,8)(7,9)(10,12), (1,12)(2,3)(4,5)(6,7)(8,9)(10,11) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $120$: $S_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 4: None
Degree 6: $\PGL(2,5)$
Low degree siblings
10T22 x 2, 12T123, 20T62 x 2, 20T65 x 2, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 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 $ | $1$ | $1$ | $()$ |
$ 4, 4, 1, 1, 1, 1 $ | $30$ | $4$ | $( 4, 6, 8,10)( 5, 7, 9,11)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1 $ | $15$ | $2$ | $( 4, 8)( 5, 9)( 6,10)( 7,11)$ |
$ 5, 5, 1, 1 $ | $24$ | $5$ | $( 2, 4,10, 6, 8)( 3, 5,11, 7, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,12)( 4, 5)( 6,11)( 7,10)( 8, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 2)( 3,12)( 4, 7)( 5, 6)( 8,11)( 9,10)$ |
$ 6, 6 $ | $20$ | $6$ | $( 1, 2, 5,12, 3, 4)( 6,11, 8, 7,10, 9)$ |
$ 10, 2 $ | $24$ | $10$ | $( 1, 2, 5, 6, 9,12, 3, 4, 7, 8)(10,11)$ |
$ 6, 6 $ | $20$ | $6$ | $( 1, 2, 5, 8,11, 6)( 3, 4, 9,10, 7,12)$ |
$ 4, 4, 2, 2 $ | $30$ | $4$ | $( 1, 2, 5,10)( 3, 4,11,12)( 6, 7)( 8, 9)$ |
$ 6, 6 $ | $20$ | $6$ | $( 1, 3, 5, 9,11, 7)( 2, 4, 8,10, 6,12)$ |
$ 3, 3, 3, 3 $ | $20$ | $3$ | $( 1, 3, 5)( 2, 4,12)( 6,10, 8)( 7,11, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2,12)( 4, 6)( 5, 7)( 8,10)( 9,11)$ |
$ 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: | $240=2^{4} \cdot 3 \cdot 5$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 240.189 | magma: IdentifyGroup(G);
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Character table: |
2 4 3 4 1 4 3 2 1 2 3 2 2 3 4 3 1 . . . . 1 1 . 1 . 1 1 1 1 5 1 . . 1 . . . 1 . . . . . 1 1a 4a 2a 5a 2b 2c 6a 10a 6b 4b 6c 3a 2d 2e 2P 1a 2a 1a 5a 1a 1a 3a 5a 3a 2a 3a 3a 1a 1a 3P 1a 4a 2a 5a 2b 2c 2e 10a 2c 4b 2d 1a 2d 2e 5P 1a 4a 2a 1a 2b 2c 6a 2e 6b 4b 6c 3a 2d 2e 7P 1a 4a 2a 5a 2b 2c 6a 10a 6b 4b 6c 3a 2d 2e X.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 X.3 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 X.5 4 . . -1 . -2 1 -1 1 . 1 1 -2 4 X.6 4 . . -1 . 2 1 -1 -1 . -1 1 2 4 X.7 4 . . -1 . -2 -1 1 1 . -1 1 2 -4 X.8 4 . . -1 . 2 -1 1 -1 . 1 1 -2 -4 X.9 5 -1 1 . 1 1 -1 . 1 -1 1 -1 1 5 X.10 5 1 1 . 1 -1 -1 . -1 1 -1 -1 -1 5 X.11 5 1 1 . -1 1 1 . 1 -1 -1 -1 -1 -5 X.12 5 -1 1 . -1 -1 1 . -1 1 1 -1 1 -5 X.13 6 . -2 1 -2 . . 1 . . . . . 6 X.14 6 . -2 1 2 . . -1 . . . . . -6 |
magma: CharacterTable(G);