Show commands:
Magma
magma: G := TransitiveGroup(22, 16);
Group action invariants
| Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $C_{11}^2:C_5:C_4$ | ||
| 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: | (1,16,3,20)(2,18)(4,22,11,14)(5,13,10,12)(6,15,9,21)(7,17,8,19), (1,16,3,19)(2,12)(4,15,11,20)(5,22,10,13)(6,18,9,17)(7,14,8,21) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $10$: $D_{5}$ $20$: 20T2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 11: None
Low degree siblings
44T60Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 11, 11 $ | $20$ | $11$ | $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,14,16,18,20,22,13,15,17,19,21)$ | |
| $ 11, 11 $ | $20$ | $11$ | $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,16,20,13,17,21,14,18,22,15,19)$ | |
| $ 11, 11 $ | $20$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,20,17,14,22,19,16,13,21,18,15)$ | |
| $ 11, 11 $ | $20$ | $11$ | $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,17,22,16,21,15,20,14,19,13,18)$ | |
| $ 11, 11 $ | $20$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,22,21,20,19,18,17,16,15,14,13)$ | |
| $ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $11$ | $(12,16,20,13,17,21,14,18,22,15,19)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $121$ | $2$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$ | |
| $ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,21,16,15,17)(14,19,20,18,22)$ | |
| $ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,14,16,20,17,22,21,19,15,18)$ | |
| $ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,16,17,21,15)(14,20,22,19,18)$ | |
| $ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,19,17,14,15,22,16,18,21,20)$ | |
| $ 4, 4, 4, 4, 4, 2 $ | $605$ | $4$ | $( 1,16, 3,20)( 2,18)( 4,22,11,14)( 5,13,10,12)( 6,15, 9,21)( 7,17, 8,19)$ | |
| $ 4, 4, 4, 4, 4, 2 $ | $605$ | $4$ | $( 1,19, 6,20)( 2,17, 5,22)( 3,15, 4,13)( 7,18,11,21)( 8,16,10,12)( 9,14)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2420=2^{2} \cdot 5 \cdot 11^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 2420.bn | magma: IdentifyGroup(G);
| |
| Character table: |
| Size | |
| 2 P | |
| 5 P | |
| 11 P | |
| Type |
magma: CharacterTable(G);