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Magma
magma: G := TransitiveGroup(21, 35);
Group action invariants
Degree $n$: | $21$ | 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: | $C_7^3:C_9$ | ||
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: | (1,12,16,5,10,21,7,9,20)(2,8,19)(3,11,15,6,13,17,4,14,18), (1,5,2,6,3,7,4)(8,11,14,10,13,9,12)(15,16,17,18,19,20,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $9$: $C_9$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T35 x 18Siblings 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, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,10,12,14, 9,11,13)(15,20,18,16,21,19,17)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,19,16,20,17,21,18)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,10,12,14, 9,11,13)(15,17,19,21,16,18,20)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $(15,20,18,16,21,19,17)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,11,14,10,13, 9,12)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $(1,7,6,5,4,3,2)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,13,11, 9,14,12,10)(15,16,17,18,19,20,21)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,16,17,18,19,20,21)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,10,12,14, 9,11,13)(15,19,16,20,17,21,18)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)(15,16,17,18,19,20,21)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,13,11, 9,14,12,10)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8, 9,10,11,12,13,14)(15,18,21,17,20,16,19)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
$ 9, 9, 3 $ | $343$ | $9$ | $( 1,12,16, 5,10,21, 7, 9,20)( 2, 8,19)( 3,11,15, 6,13,17, 4,14,18)$ |
$ 9, 9, 3 $ | $343$ | $9$ | $( 1,10,18, 2, 8,17, 6,14,20)( 3,13,16)( 4,11,15, 7,12,19, 5, 9,21)$ |
$ 9, 9, 3 $ | $343$ | $9$ | $( 1, 9,18, 5,12,17, 7,10,20)( 2, 8,16)( 3,14,21, 6,11,15, 4,13,19)$ |
$ 9, 9, 3 $ | $343$ | $9$ | $( 1,16,10, 7,20,12, 5,21, 9)( 2,19, 8)( 3,15,13, 4,18,11, 6,17,14)$ |
$ 9, 9, 3 $ | $343$ | $9$ | $( 1,19, 8, 5,18,13, 6,16, 9)( 2,17,11, 7,21,12, 3,15,14)( 4,20,10)$ |
$ 9, 9, 3 $ | $343$ | $9$ | $( 1,17,13, 7,18,14, 5,20, 9)( 2,16,12)( 3,15,11, 4,21,10, 6,19, 8)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $3087=3^{2} \cdot 7^{3}$ | 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: | 3087.f | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);