Show commands:
Magma
magma: G := TransitiveGroup(22, 38);
Group action invariants
| Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $38$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $M_{22}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| 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);
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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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $1155$ | $2$ | $( 1,11)( 2,22)( 3,21)( 6,18)( 8,17)( 9,20)(10,13)(12,16)$ | |
| $ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $12320$ | $3$ | $( 2, 8,21)( 3,22,17)( 4, 7,15)( 5,14,19)( 6,12,13)(10,18,16)$ | |
| $ 6, 6, 3, 3, 2, 2 $ | $36960$ | $6$ | $( 1,11)( 2, 3, 8,22,21,17)( 4,15, 7)( 5,19,14)( 6,10,12,18,13,16)( 9,20)$ | |
| $ 4, 4, 4, 4, 2, 2, 1, 1 $ | $27720$ | $4$ | $( 1,13,18,16)( 2, 9, 3, 8)( 4, 5)( 6,12,11,10)( 7,19)(17,22,20,21)$ | |
| $ 4, 4, 4, 4, 2, 2, 1, 1 $ | $13860$ | $4$ | $( 1,10,11,13)( 2,17,22, 8)( 3,20,21, 9)( 4,19)( 5, 7)( 6,16,18,12)$ | |
| $ 8, 8, 4, 2 $ | $55440$ | $8$ | $( 1,17,10,22,11, 8,13, 2)( 3, 6,20,16,21,18, 9,12)( 4, 7,19, 5)(14,15)$ | |
| $ 5, 5, 5, 5, 1, 1 $ | $88704$ | $5$ | $( 1,18,21, 2,10)( 3,11, 8,15, 4)( 5,17, 7,12, 9)( 6,16,22,20,19)$ | |
| $ 11, 11 $ | $40320$ | $11$ | $( 1,21, 2,17, 5, 7, 9, 6,14, 8,20)( 3,22,19,10,13,15, 4,12,11,18,16)$ | |
| $ 11, 11 $ | $40320$ | $11$ | $( 1,20, 8,14, 6, 9, 7, 5,17, 2,21)( 3,16,18,11,12, 4,15,13,10,19,22)$ | |
| $ 7, 7, 7, 1 $ | $63360$ | $7$ | $( 1, 8,21, 6,22,14,17)( 2,12,11,19, 7,16,18)( 4,20,10, 5,13, 9,15)$ | |
| $ 7, 7, 7, 1 $ | $63360$ | $7$ | $( 1,17,14,22, 6,21, 8)( 2,18,16, 7,19,11,12)( 4,15, 9,13, 5,10,20)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $443520=2^{7} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$ | 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: | 443520.a | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 7 P | |
| 11 P | |
| Type |
magma: CharacterTable(G);