Show commands:
Magma
magma: G := TransitiveGroup(21, 29);
Group action invariants
| Degree $n$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_7^2:(C_6\times S_3)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,4,3)(2,6,7)(8,15,11,16,9,20)(10,18,12,21,13,19)(14,17), (1,15,5,17,6,21)(2,19,7,18,3,16)(4,20)(8,12,14)(10,13,11), (1,4,6,5,2,7)(8,19,10,20,14,15)(9,16,12,21,11,17)(13,18) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $3$: $C_3$ $4$: $C_2^2$ $6$: $S_3$, $C_6$ x 3 $12$: $D_{6}$, $C_6\times C_2$ $18$: $S_3\times C_3$ $36$: $C_6\times S_3$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 7: None
Low degree siblings
14T37, 21T29, 28T170, 42T223 x 2, 42T224 x 2, 42T225 x 2, 42T252, 42T253, 42T254, 42T255Siblings 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 $ | $18$ | $7$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,11,14,10,13, 9,12)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 1, 1, 1, 1, 1, 1, 1 $ | $18$ | $7$ | $( 8,12, 9,13,10,14,11)(15,19,16,20,17,21,18)$ |
| $ 7, 7, 7 $ | $12$ | $7$ | $( 1, 5, 2, 6, 3, 7, 4)( 8,14,13,12,11,10, 9)(15,17,19,21,16,18,20)$ |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1 $ | $49$ | $3$ | $( 2, 5, 3)( 4, 6, 7)( 9,12,10)(11,13,14)(16,19,17)(18,20,21)$ |
| $ 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, 3 $ | $98$ | $3$ | $( 1,14,21)( 2,13,15)( 3,12,16)( 4,11,17)( 5,10,18)( 6, 9,19)( 7, 8,20)$ |
| $ 21 $ | $84$ | $21$ | $( 1,11,16, 2,14,18, 3,10,20, 4,13,15, 5, 9,17, 6,12,19, 7, 8,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1,13,20)( 2, 9,15)( 3,12,17)( 4, 8,19)( 5,11,21)( 6,14,16)( 7,10,18)$ |
| $ 21 $ | $84$ | $21$ | $( 1,13,15, 3, 9,16, 5,12,17, 7, 8,18, 2,11,19, 4,14,20, 6,10,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3 $ | $14$ | $3$ | $( 1,10,17)( 2, 8,21)( 3,13,18)( 4,11,15)( 5, 9,19)( 6,14,16)( 7,12,20)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $147$ | $6$ | $( 2, 3, 5)( 4, 7, 6)( 8,20,10,16,11,21)( 9,18,14,15,13,17)(12,19)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ | $21$ | $2$ | $( 8,21)( 9,20)(10,19)(11,18)(12,17)(13,16)(14,15)$ |
| $ 14, 7 $ | $126$ | $14$ | $( 1, 7, 6, 5, 4, 3, 2)( 8,18,14,19,13,20,12,21,11,15,10,16, 9,17)$ |
| $ 6, 6, 3, 3, 2, 1 $ | $147$ | $6$ | $( 2, 5, 3)( 4, 6, 7)( 8,18,13,19, 9,21)(10,17)(11,20,12,16,14,15)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ | $49$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,12)(16,21)(17,20)(18,19)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $49$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 9,11,10,14,12,13)(16,18,17,21,19,20)$ |
| $ 6, 6, 6, 1, 1, 1 $ | $49$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 9,13,12,14,10,11)(16,20,19,21,17,18)$ |
| $ 6, 6, 6, 3 $ | $98$ | $6$ | $( 1,14,19, 5,11,16)( 2, 8,20, 4,10,15)( 3, 9,21)( 6,12,17, 7,13,18)$ |
| $ 6, 6, 6, 3 $ | $98$ | $6$ | $( 1,11,19, 3,12,16)( 2, 8,21)( 4, 9,18, 7,14,17)( 5,13,20, 6,10,15)$ |
| $ 6, 6, 6, 3 $ | $98$ | $6$ | $( 1,13,21, 5,14,16)( 2, 8,18, 4,12,19)( 3,10,15)( 6, 9,20, 7,11,17)$ |
| $ 6, 6, 6, 2, 1 $ | $147$ | $6$ | $( 2, 6, 5, 7, 3, 4)( 8,20, 9,15,13,16)(10,17)(11,19,14,18,12,21)$ |
| $ 14, 2, 2, 2, 1 $ | $126$ | $14$ | $( 2, 7)( 3, 6)( 4, 5)( 8,21,13,19,11,17, 9,15,14,20,12,18,10,16)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $21$ | $2$ | $( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,16)(10,17)(11,18)(12,19)(13,20)(14,21)$ |
| $ 6, 6, 6, 2, 1 $ | $147$ | $6$ | $( 2, 4, 3, 7, 5, 6)( 8,18, 9,15,11,16)(10,19,13,17,12,20)(14,21)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $1764=2^{2} \cdot 3^{2} \cdot 7^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 1764.134 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);