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Magma
magma: G := TransitiveGroup(21, 21);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7^2:C_3^2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,4)(3,6,5)(8,12,13)(9,14,10)(15,18,17)(16,20,21), (1,8,21,7,10,17,6,12,20,5,14,16,4,9,19,3,11,15,2,13,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $21$: $C_7:C_3$ x 2 $63$: 21T7 x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T21Siblings 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, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8, 9,10,11,12,13,14)(15,16,17,18,19,20,21)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,11,14,10,13, 9,12)(15,18,21,17,20,16,19)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $49$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $49$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,10,12,14, 9,11,13)(15,16,17,18,19,20,21)$ |
$ 7, 7, 7 $ | $3$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,11,14,10,13, 9,12)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,18,21,17,20,16,19)$ |
$ 7, 7, 7 $ | $3$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,13,11, 9,14,12,10)(15,19,16,20,17,21,18)$ |
$ 7, 7, 7 $ | $3$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ |
$ 7, 7, 7 $ | $3$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,10,12,14, 9,11,13)(15,21,20,19,18,17,16)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 8,15)( 2,11,17)( 3,14,19)( 4,10,21)( 5,13,16)( 6, 9,18)( 7,12,20)$ |
$ 21 $ | $21$ | $21$ | $( 1, 8,16, 4,10,15, 7,12,21, 3,14,20, 6, 9,19, 2,11,18, 5,13,17)$ |
$ 21 $ | $21$ | $21$ | $( 1, 8,18, 3,14,15, 5,13,19, 7,12,16, 2,11,20, 4,10,17, 6, 9,21)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1, 8,15)( 2,13,19)( 3,11,16)( 4, 9,20)( 5,14,17)( 6,12,21)( 7,10,18)$ |
$ 21 $ | $21$ | $21$ | $( 1, 8,16, 2,13,20, 3,11,17, 4, 9,21, 5,14,18, 6,12,15, 7,10,19)$ |
$ 21 $ | $21$ | $21$ | $( 1, 8,18, 4, 9,16, 7,10,21, 3,11,19, 6,12,17, 2,13,15, 5,14,20)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1, 8,15)( 2,14,16)( 3,13,17)( 4,12,18)( 5,11,19)( 6,10,20)( 7, 9,21)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $49$ | $3$ | $( 1,15, 8)( 2,16,14)( 3,17,13)( 4,18,12)( 5,19,11)( 6,20,10)( 7,21, 9)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1,15, 8)( 2,17,11)( 3,19,14)( 4,21,10)( 5,16,13)( 6,18, 9)( 7,20,12)$ |
$ 21 $ | $21$ | $21$ | $( 1,15, 9, 7,20,13, 6,18,10, 5,16,14, 4,21,11, 3,19, 8, 2,17,12)$ |
$ 21 $ | $21$ | $21$ | $( 1,15,11, 5,16, 9, 2,17,14, 6,18,12, 3,19,10, 7,20, 8, 4,21,13)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $7$ | $3$ | $( 1,15, 8)( 2,19,13)( 3,16,11)( 4,20, 9)( 5,17,14)( 6,21,12)( 7,18,10)$ |
$ 21 $ | $21$ | $21$ | $( 1,15, 9, 5,17, 8, 2,19,14, 6,21,13, 3,16,12, 7,18,11, 4,20,10)$ |
$ 21 $ | $21$ | $21$ | $( 1,15,11, 6,21, 8, 4,20,12, 2,19, 9, 7,18,13, 5,17,10, 3,16,14)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $441=3^{2} \cdot 7^{2}$ | 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: | 441.9 | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);