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Magma
magma: G := TransitiveGroup(15, 7);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_5\times S_3$ | ||
CHM label: | $D(5)[x]S(3)$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,11)(2,7)(4,14)(5,10)(8,13), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $6$: $S_3$ $10$: $D_{5}$ $12$: $D_{6}$ $20$: $D_{10}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 5: $D_{5}$
Low degree siblings
30T8, 30T10, 30T13Siblings 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, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $5$ | $2$ | $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$ |
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$ |
$ 2, 2, 2, 2, 2, 2, 2, 1 $ | $15$ | $2$ | $( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)$ |
$ 15 $ | $4$ | $15$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$ |
$ 6, 6, 3 $ | $10$ | $6$ | $( 1, 2, 6, 7,11,12)( 3,10, 8,15,13, 5)( 4,14, 9)$ |
$ 10, 5 $ | $6$ | $10$ | $( 1, 2,13,14,10,11, 7, 8, 4, 5)( 3, 9,15, 6,12)$ |
$ 15 $ | $4$ | $15$ | $( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$ |
$ 10, 5 $ | $6$ | $10$ | $( 1, 3,10,12, 4, 6,13,15, 7, 9)( 2,14,11, 8, 5)$ |
$ 5, 5, 5 $ | $2$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
$ 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
$ 5, 5, 5 $ | $2$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $60=2^{2} \cdot 3 \cdot 5$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 60.8 | magma: IdentifyGroup(G);
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Character table: |
2 2 2 2 2 . 1 1 . 1 1 1 1 3 1 1 . . 1 1 . 1 . 1 1 1 5 1 . 1 . 1 . 1 1 1 1 1 1 1a 2a 2b 2c 15a 6a 10a 15b 10b 5a 3a 5b 2P 1a 1a 1a 1a 15b 3a 5a 15a 5b 5b 3a 5a 3P 1a 2a 2b 2c 5a 2a 10b 5b 10a 5b 1a 5a 5P 1a 2a 2b 2c 3a 6a 2b 3a 2b 1a 3a 1a 7P 1a 2a 2b 2c 15b 6a 10b 15a 10a 5b 3a 5a 11P 1a 2a 2b 2c 15a 6a 10a 15b 10b 5a 3a 5b 13P 1a 2a 2b 2c 15b 6a 10b 15a 10a 5b 3a 5a 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 2 -2 . . -1 1 . -1 . 2 -1 2 X.6 2 2 . . -1 -1 . -1 . 2 -1 2 X.7 2 . -2 . A . -A *A -*A *A 2 A X.8 2 . -2 . *A . -*A A -A A 2 *A X.9 2 . 2 . A . A *A *A *A 2 A X.10 2 . 2 . *A . *A A A A 2 *A X.11 4 . . . -A . . -*A . B -2 *B X.12 4 . . . -*A . . -A . *B -2 B A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = 2*E(5)+2*E(5)^4 = -1+Sqrt(5) = 2b5 |
magma: CharacterTable(G);