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Magma
magma: G := TransitiveGroup(12, 76);
Group action invariants
Degree $n$: | $12$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $76$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2\times A_5$ | ||
CHM label: | $[2]L(6)_{6}$ | ||
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: | (2,4,6,8,10)(3,5,7,9,11), (4,10)(5,11)(6,8)(7,9), (1,2)(3,12)(4,11)(5,10), (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) $2$: $C_2$ $60$: $A_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 3: None
Degree 4: None
Degree 6: $\PSL(2,5)$
Low degree siblings
10T11, 12T75, 20T31, 20T36, 24T203, 30T29, 30T30, 40T61Siblings 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 $ | $15$ | $2$ | $( 4,10)( 5,11)( 6, 8)( 7, 9)$ |
$ 5, 5, 1, 1 $ | $12$ | $5$ | $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$ |
$ 5, 5, 1, 1 $ | $12$ | $5$ | $( 2, 6,10, 4, 8)( 3, 7,11, 5, 9)$ |
$ 2, 2, 2, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,12)( 4, 5)( 6, 8)( 7, 9)(10,11)$ |
$ 6, 6 $ | $20$ | $6$ | $( 1, 2, 5,12, 3, 4)( 6,11, 9, 7,10, 8)$ |
$ 10, 2 $ | $12$ | $10$ | $( 1, 2, 6, 5, 9,12, 3, 7, 4, 8)(10,11)$ |
$ 3, 3, 3, 3 $ | $20$ | $3$ | $( 1, 2, 6)( 3, 7,12)( 4,11, 9)( 5,10, 8)$ |
$ 10, 2 $ | $12$ | $10$ | $( 1, 3, 5, 8,10,12, 2, 4, 9,11)( 6, 7)$ |
$ 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: | $120=2^{3} \cdot 3 \cdot 5$ | 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: | 120.35 | magma: IdentifyGroup(G);
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Character table: |
2 3 3 1 1 3 1 1 1 1 3 3 1 . . . . 1 . 1 . 1 5 1 . 1 1 . . 1 . 1 1 1a 2a 5a 5b 2b 6a 10a 3a 10b 2c 2P 1a 1a 5b 5a 1a 3a 5b 3a 5a 1a 3P 1a 2a 5b 5a 2b 2c 10b 1a 10a 2c 5P 1a 2a 1a 1a 2b 6a 2c 3a 2c 2c 7P 1a 2a 5b 5a 2b 6a 10b 3a 10a 2c X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 1 -1 -1 1 -1 -1 X.3 3 -1 A *A -1 . A . *A 3 X.4 3 -1 *A A -1 . *A . A 3 X.5 3 1 A *A -1 . -A . -*A -3 X.6 3 1 *A A -1 . -*A . -A -3 X.7 4 . -1 -1 . 1 -1 1 -1 4 X.8 4 . -1 -1 . -1 1 1 1 -4 X.9 5 1 . . 1 -1 . -1 . 5 X.10 5 -1 . . 1 1 . -1 . -5 A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 |
magma: CharacterTable(G);