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Magma
magma: G := TransitiveGroup(21, 44);
Group action invariants
Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_3\times A_7$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $3$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,11,17,21,7,5,13,2,12,18,19,8,6,14,3,10,16,20,9,4,15), (1,16,15,10,2,17,13,11,3,18,14,12)(4,8,6,7,5,9)(19,20,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $3$: $C_3$ $2520$: $A_7$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 7: $A_7$
Low degree siblings
45T442 x 2Siblings are shown 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 $ | $1$ | $1$ | $()$ | |
$ 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$ | |
$ 3, 3, 3, 3, 3, 3, 3 $ | $1$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ | |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $105$ | $2$ | $( 1, 6)( 2, 4)( 3, 5)( 7,11)( 8,12)( 9,10)$ | |
$ 6, 6, 3, 3, 3 $ | $105$ | $6$ | $( 1, 5, 2, 6, 3, 4)( 7,10, 8,11, 9,12)(13,15,14)(16,18,17)(19,21,20)$ | |
$ 6, 6, 3, 3, 3 $ | $105$ | $6$ | $( 1, 4, 3, 6, 2, 5)( 7,12, 9,11, 8,10)(13,14,15)(16,17,18)(19,20,21)$ | |
$ 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $70$ | $3$ | $(1,6,9)(2,4,7)(3,5,8)$ | |
$ 3, 3, 3, 3, 3, 3, 3 $ | $70$ | $3$ | $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,12,11)(13,15,14)(16,18,17)(19,21,20)$ | |
$ 3, 3, 3, 3, 3, 3, 3 $ | $70$ | $3$ | $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,11,12)(13,14,15)(16,17,18)(19,20,21)$ | |
$ 3, 3, 3, 2, 2, 2, 2, 2, 2 $ | $210$ | $6$ | $( 1, 6, 9)( 2, 4, 7)( 3, 5, 8)(10,13)(11,14)(12,15)(16,19)(17,20)(18,21)$ | |
$ 6, 6, 3, 3, 3 $ | $210$ | $6$ | $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,15,11,13,12,14)(16,21,17,19,18,20)$ | |
$ 6, 6, 3, 3, 3 $ | $210$ | $6$ | $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,14,12,13,11,15)(16,20,18,19,17,21)$ | |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $280$ | $3$ | $( 1, 6, 9)( 2, 4, 7)( 3, 5, 8)(10,13,18)(11,14,16)(12,15,17)$ | |
$ 3, 3, 3, 3, 3, 3, 3 $ | $280$ | $3$ | $( 1, 5, 7)( 2, 6, 8)( 3, 4, 9)(10,15,16)(11,13,17)(12,14,18)(19,21,20)$ | |
$ 3, 3, 3, 3, 3, 3, 3 $ | $280$ | $3$ | $( 1, 4, 8)( 2, 5, 9)( 3, 6, 7)(10,14,17)(11,15,18)(12,13,16)(19,20,21)$ | |
$ 4, 4, 4, 2, 2, 2, 1, 1, 1 $ | $630$ | $4$ | $( 1, 6, 9,10)( 2, 4, 7,11)( 3, 5, 8,12)(13,18)(14,16)(15,17)$ | |
$ 12, 6, 3 $ | $630$ | $12$ | $( 1, 5, 7,10, 3, 4, 9,12, 2, 6, 8,11)(13,17,14,18,15,16)(19,21,20)$ | |
$ 12, 6, 3 $ | $630$ | $12$ | $( 1, 4, 8,10, 2, 5, 9,11, 3, 6, 7,12)(13,16,15,18,14,17)(19,20,21)$ | |
$ 5, 5, 5, 1, 1, 1, 1, 1, 1 $ | $504$ | $5$ | $( 1, 6, 9,10,13)( 2, 4, 7,11,14)( 3, 5, 8,12,15)$ | |
$ 15, 3, 3 $ | $504$ | $15$ | $( 1, 5, 7,10,15, 2, 6, 8,11,13, 3, 4, 9,12,14)(16,18,17)(19,21,20)$ | |
$ 15, 3, 3 $ | $504$ | $15$ | $( 1, 4, 8,10,14, 3, 6, 7,12,13, 2, 5, 9,11,15)(16,17,18)(19,20,21)$ | |
$ 7, 7, 7 $ | $360$ | $7$ | $( 1, 6, 9,10,13,18,21)( 2, 4, 7,11,14,16,19)( 3, 5, 8,12,15,17,20)$ | |
$ 21 $ | $360$ | $21$ | $( 1, 5, 7,10,15,16,21, 3, 4, 9,12,14,18,20, 2, 6, 8,11,13,17,19)$ | |
$ 21 $ | $360$ | $21$ | $( 1, 4, 8,10,14,17,21, 2, 5, 9,11,15,18,19, 3, 6, 7,12,13,16,20)$ | |
$ 7, 7, 7 $ | $360$ | $7$ | $( 1, 6, 9,10,13,21,18)( 2, 4, 7,11,14,19,16)( 3, 5, 8,12,15,20,17)$ | |
$ 21 $ | $360$ | $21$ | $( 1, 5, 7,10,15,19,18, 3, 4, 9,12,14,21,17, 2, 6, 8,11,13,20,16)$ | |
$ 21 $ | $360$ | $21$ | $( 1, 4, 8,10,14,20,18, 2, 5, 9,11,15,21,16, 3, 6, 7,12,13,19,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $7560=2^{3} \cdot 3^{3} \cdot 5 \cdot 7$ | 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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 7560.e | magma: IdentifyGroup(G);
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Character table: |
Size | |
2 P | |
3 P | |
5 P | |
7 P | |
Type |
magma: CharacterTable(G);