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Magma
magma: G := TransitiveGroup(15, 35);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $35$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3^4:D_5$ | ||
CHM label: | $1/2[3^{4}:2]D(5)$ | ||
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: | (1,6,11)(4,14,9), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $10$: $D_{5}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: None
Degree 5: $D_{5}$
Low degree siblings
15T34 x 4, 15T35 x 3, 30T191 x 4, 30T192 x 4, 45T121 x 8, 45T122 x 8, 45T123, 45T124Siblings 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 $ | $1$ | $1$ | $()$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $3$ | $( 1, 6,11)( 4,14, 9)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $3$ | $( 1,11, 6)( 4, 9,14)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $3$ | $( 1, 6,11)( 2,12, 7)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2,12, 7)( 4,14, 9)$ |
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $5$ | $3$ | $( 1,11, 6)( 2, 7,12)$ |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 4,14, 9)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $5$ | $3$ | $( 1, 6,11)( 2,12, 7)( 4, 9,14)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $5$ | $3$ | $( 1, 6,11)( 2, 7,12)( 4,14, 9)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $10$ | $3$ | $( 1,11, 6)( 2, 7,12)( 4, 9,14)( 5,15,10)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $5$ | $3$ | $( 1,11, 6)( 2,12, 7)( 4, 9,14)( 5,10,15)$ |
$ 3, 3, 3, 3, 1, 1, 1 $ | $5$ | $3$ | $( 1,11, 6)( 2, 7,12)( 4,14, 9)( 5,10,15)$ |
$ 3, 3, 3, 3, 3 $ | $10$ | $3$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,10,15)$ |
$ 5, 5, 5 $ | $162$ | $5$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
$ 5, 5, 5 $ | $162$ | $5$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
$ 6, 2, 2, 2, 2, 1 $ | $45$ | $6$ | $( 2,10)( 3, 4, 8,14,13, 9)( 5, 7)( 6,11)(12,15)$ |
$ 6, 2, 2, 2, 2, 1 $ | $45$ | $6$ | $( 1, 6)( 2,10)( 3,14,13, 4, 8, 9)( 5, 7)(12,15)$ |
$ 2, 2, 2, 2, 2, 2, 2, 1 $ | $45$ | $2$ | $( 1,11)( 2,10)( 3, 9)( 4, 8)( 5, 7)(12,15)(13,14)$ |
$ 6, 2, 2, 2, 2, 1 $ | $45$ | $6$ | $( 2,10,12,15, 7, 5)( 3, 9)( 4, 8)( 6,11)(13,14)$ |
$ 6, 6, 2, 1 $ | $45$ | $6$ | $( 1, 6)( 2,10,12,15, 7, 5)( 3, 4, 8,14,13, 9)$ |
$ 6, 6, 2, 1 $ | $45$ | $6$ | $( 1,11)( 2,10,12,15, 7, 5)( 3,14,13, 4, 8, 9)$ |
$ 6, 6, 2, 1 $ | $45$ | $6$ | $( 2,10, 7, 5,12,15)( 3,14,13, 4, 8, 9)( 6,11)$ |
$ 6, 2, 2, 2, 2, 1 $ | $45$ | $6$ | $( 1, 6)( 2,10, 7, 5,12,15)( 3, 9)( 4, 8)(13,14)$ |
$ 6, 6, 2, 1 $ | $45$ | $6$ | $( 1,11)( 2,10, 7, 5,12,15)( 3, 4, 8,14,13, 9)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $810=2 \cdot 3^{4} \cdot 5$ | 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: | 810.101 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);