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Magma
magma: G := TransitiveGroup(21, 49);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $49$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_7^3:C_9:C_3$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,11,17)(2,12,15,3,13,20,5,8,16)(4,14,18,7,10,19,6,9,21), (1,7,3)(2,4,5)(8,11,14,10,13,9,12)(15,16,18)(17,20,19) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ x 4 $9$: $C_3^2$ $27$: $C_9:C_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: None
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | 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 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,11, 9,14,12,10)(15,20,18,16,21,19,17)$ | |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 8,11,14,10,13, 9,12)(15,16,17,18,19,20,21)$ | |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)(15,20,18,16,21,19,17)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,14,13,12,11,10, 9)(15,19,16,20,17,21,18)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,13,11, 9,14,12,10)(15,18,21,17,20,16,19)$ | |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,10,12,14, 9,11,13)$ | |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $27$ | $7$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,14,13,12,11,10, 9)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,12, 9,13,10,14,11)(15,20,18,16,21,19,17)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8, 9,10,11,12,13,14)(15,17,19,21,16,18,20)$ | |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $(15,21,20,19,18,17,16)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 6, 4, 2, 7, 5, 3)( 8,14,13,12,11,10, 9)(15,20,18,16,21,19,17)$ | |
| $ 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $9$ | $7$ | $( 8,11,14,10,13, 9,12)$ | |
| $ 7, 7, 7 $ | $27$ | $7$ | $( 1, 4, 7, 3, 6, 2, 5)( 8, 9,10,11,12,13,14)(15,20,18,16,21,19,17)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 3, 5)( 4, 7, 6)( 9,10,12)(11,14,13)(16,17,19)(18,21,20)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $343$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ | |
| $ 9, 9, 3 $ | $1029$ | $9$ | $( 1,11,17)( 2,12,15, 3,13,20, 5, 8,16)( 4,14,18, 7,10,19, 6, 9,21)$ | |
| $ 9, 9, 3 $ | $1029$ | $9$ | $( 1,17,11)( 2,15,13, 5,16,12, 3,20, 8)( 4,18,10, 6,21,14, 7,19, 9)$ | |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $3$ | $( 9,10,12)(11,14,13)(16,19,17)(18,20,21)$ | |
| $ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12,13)( 9,14,10)(15,19,21)(17,20,18)$ | |
| $ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13, 9)(11,12,14)(15,20,19)(16,17,21)$ | |
| $ 9, 9, 3 $ | $1029$ | $9$ | $( 1,11,18, 5, 8,16, 6, 9,19)( 2,12,21, 7,10,15, 3,13,17)( 4,14,20)$ | |
| $ 9, 9, 3 $ | $1029$ | $9$ | $( 1,17,12, 4,18, 8, 2,15,13)( 3,20,14, 5,16, 9, 6,21,10)( 7,19,11)$ | |
| $ 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $147$ | $3$ | $( 9,12,10)(11,13,14)(16,17,19)(18,21,20)$ | |
| $ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 1, 2, 3, 4, 5, 6, 7)( 8,12,14)(10,13,11)(15,19,20)(16,21,17)$ | |
| $ 7, 3, 3, 3, 3, 1, 1 $ | $441$ | $21$ | $( 1, 4, 7, 3, 6, 2, 5)( 8,13,12)( 9,10,14)(15,20,16)(18,19,21)$ | |
| $ 9, 9, 3 $ | $1029$ | $9$ | $( 1,11,20, 7,10,21, 5, 8,16)( 2,12,19)( 3,13,18, 4,14,17, 6, 9,15)$ | |
| $ 9, 9, 3 $ | $1029$ | $9$ | $( 1,17, 9, 3,20,10, 4,18,14)( 2,15,13, 7,19,12, 6,21, 8)( 5,16,11)$ |
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.f | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 3 P | |
| 7 P | |
| Type |
magma: CharacterTable(G);