Show commands:
Magma
magma: G := TransitiveGroup(22, 38);
Group action invariants
| Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $M_{22}$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,17,9,5,10,22,8)(3,20,15,12,19,11,14)(4,21,16,13,7,18,6), (1,5,10)(2,17,12)(3,8,4)(6,16,19)(9,18,21)(14,20,22) | magma: Generators(G);
|
Low degree resolvents
noneResolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 11: 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 |
| $1^{22}$ | $1$ | $1$ | $()$ | |
| $7^{3},1$ | $63360$ | $7$ | $( 1,16,22,18, 3,12,14)( 2, 8,11, 6,21,17, 7)( 4,20, 5,15,13, 9,10)$ | |
| $7^{3},1$ | $63360$ | $7$ | $( 1,14,12, 3,18,22,16)( 2, 7,17,21, 6,11, 8)( 4,10, 9,13,15, 5,20)$ | |
| $11^{2}$ | $40320$ | $11$ | $( 1,22,15, 9, 3,18,16,10, 5, 8,19)( 2,13,14,21, 7,11,12, 6,17,20, 4)$ | |
| $11^{2}$ | $40320$ | $11$ | $( 1,19, 8, 5,10,16,18, 3, 9,15,22)( 2, 4,20,17, 6,12,11, 7,21,14,13)$ | |
| $2^{8},1^{6}$ | $1155$ | $2$ | $( 2, 6)( 3, 8)( 5, 7)( 9,16)(11,13)(12,22)(14,19)(18,20)$ | |
| $4^{4},2^{2},1^{2}$ | $13860$ | $4$ | $( 2,20, 6,18)( 3,13, 8,11)( 4,15)( 5,14, 7,19)( 9,22,16,12)(17,21)$ | |
| $8^{2},4,2$ | $55440$ | $8$ | $( 1,10)( 2,11,20, 3, 6,13,18, 8)( 4,17,15,21)( 5, 9,14,22, 7,16,19,12)$ | |
| $3^{6},1^{4}$ | $12320$ | $3$ | $( 1,21, 4)( 2,20,22)( 3,14,11)( 6,18,12)( 8,19,13)(10,17,15)$ | |
| $6^{2},3^{2},2^{2}$ | $36960$ | $6$ | $( 1, 4,21)( 2, 8,20,19,22,13)( 3,18,14,12,11, 6)( 5, 9)( 7,16)(10,15,17)$ | |
| $4^{4},2^{2},1^{2}$ | $27720$ | $4$ | $( 1,10)( 2, 3,16,11)( 5,12,14,18)( 6, 8, 9,13)( 7,22,19,20)(17,21)$ | |
| $5^{4},1^{2}$ | $88704$ | $5$ | $( 1, 8,11,21, 9)( 3, 5, 4,15,20)( 6,13,16,19,12)( 7,18,10,14,22)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $443520=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | no | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 443520.a | magma: IdentifyGroup(G);
| |
| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| 11 P | |
| Type |
magma: CharacterTable(G);