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Magma
magma: G := TransitiveGroup(21, 48);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $48$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_7^3:\He_3$ | ||
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: | (1,6,5)(2,3,7)(8,12,13)(9,14,10), (1,15,9,3,20,10,5,18,11,7,16,12,2,21,13,4,19,14,6,17,8) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $27$: $C_3^2:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
21T48 x 3Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $(1,6,4,2,7,5,3)$ |
$ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $(1,2,3,4,5,6,7)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)(15,17,19,21,16,18,20)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)(15,17,19,21,16,18,20)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)(15,21,20,19,18,17,16)$ |
$ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $27$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $27$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
$ 7, 7, 7 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $27$ | $7$ | $( 1, 3, 5, 7, 2, 4, 6)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $27$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,21,20,19,18,17,16)$ |
$ 7, 7, 7 $ | $9$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ |
$ 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, 7 $ | $9$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,21,20,19,18,17,16)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,10,12)(11,14,13)$ |
$ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 2, 5, 3)( 4, 6, 7)( 9,10,12)(11,14,13)(15,17,19,21,16,18,20)$ |
$ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 2, 5, 3)( 4, 6, 7)( 9,10,12)(11,14,13)(15,21,20,19,18,17,16)$ |
$ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,12,10)(11,13,14)$ |
$ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 2, 3, 5)( 4, 7, 6)( 9,12,10)(11,13,14)(15,17,19,21,16,18,20)$ |
$ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 2, 3, 5)( 4, 7, 6)( 9,12,10)(11,13,14)(15,21,20,19,18,17,16)$ |
$ 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 $ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ |
$ 21 $ | $441$ | $21$ | $( 1,15, 9, 3,20,10, 5,18,11, 7,16,12, 2,21,13, 4,19,14, 6,17, 8)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $147$ | $3$ | $( 1,15, 9)( 2,21,13)( 3,20,10)( 4,19,14)( 5,18,11)( 6,17, 8)( 7,16,12)$ |
$ 21 $ | $441$ | $21$ | $( 1,15, 9, 4,19,14, 7,16,12, 3,20,10, 6,17, 8, 2,21,13, 5,18,11)$ |
$ 21 $ | $441$ | $21$ | $( 1,15,10, 3,20,12, 5,18,14, 7,16, 9, 2,21,11, 4,19,13, 6,17, 8)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $147$ | $3$ | $( 1,15,10)( 2,21,11)( 3,20,12)( 4,19,13)( 5,18,14)( 6,17, 8)( 7,16, 9)$ |
$ 21 $ | $441$ | $21$ | $( 1,15,10, 4,19,13, 7,16, 9, 3,20,12, 6,17, 8, 2,21,11, 5,18,14)$ |
$ 21 $ | $441$ | $21$ | $( 1,15,12, 3,20, 9, 5,18,13, 7,16,10, 2,21,14, 4,19,11, 6,17, 8)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $147$ | $3$ | $( 1,15,12)( 2,21,14)( 3,20, 9)( 4,19,11)( 5,18,13)( 6,17, 8)( 7,16,10)$ |
$ 21 $ | $441$ | $21$ | $( 1,15,12, 4,19,11, 7,16,10, 3,20, 9, 6,17, 8, 2,21,14, 5,18,13)$ |
$ 21 $ | $441$ | $21$ | $( 1, 9,20, 5,11,16, 2,13,19, 6, 8,15, 3,10,18, 7,12,21, 4,14,17)$ |
$ 21 $ | $441$ | $21$ | $( 1, 9,20, 6, 8,15, 4,14,17, 2,13,19, 7,12,21, 5,11,16, 3,10,18)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $147$ | $3$ | $( 1, 9,20)( 2,13,19)( 3,10,18)( 4,14,17)( 5,11,16)( 6, 8,15)( 7,12,21)$ |
$ 21 $ | $441$ | $21$ | $( 1,10,18, 4,13,19, 7, 9,20, 3,12,21, 6, 8,15, 2,11,16, 5,14,17)$ |
$ 21 $ | $441$ | $21$ | $( 1,10,18, 2,11,16, 3,12,21, 4,13,19, 5,14,17, 6, 8,15, 7, 9,20)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $147$ | $3$ | $( 1,10,18)( 2,11,16)( 3,12,21)( 4,13,19)( 5,14,17)( 6, 8,15)( 7, 9,20)$ |
$ 21 $ | $441$ | $21$ | $( 1,12,21, 7,10,18, 6, 8,15, 5,13,19, 4,11,16, 3, 9,20, 2,14,17)$ |
$ 21 $ | $441$ | $21$ | $( 1,12,21, 5,13,19, 2,14,17, 6, 8,15, 3, 9,20, 7,10,18, 4,11,16)$ |
$ 3, 3, 3, 3, 3, 3, 3 $ | $147$ | $3$ | $( 1,12,21)( 2,14,17)( 3, 9,20)( 4,11,16)( 5,13,19)( 6, 8,15)( 7,10,18)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $9261=3^{3} \cdot 7^{3}$ | 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: | 9261.e | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);