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Magma
magma: G := TransitiveGroup(22, 26);
Group action invariants
| Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $26$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_2\times M_{11}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,21,14,15,10,8,12,3)(2,22,13,16,9,7,11,4)(5,20,6,19)(17,18), (1,3,5,11,18,2,4,6,12,17)(7,16,20,22,14,8,15,19,21,13)(9,10) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $7920$: $M_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 11: $M_{11}$
Low degree siblings
22T27, 24T12204, 44T140Siblings 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 |
| $1^{22}$ | $1$ | $1$ | $()$ | |
| $2^{11}$ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ | |
| $10^{2},2$ | $1584$ | $10$ | $( 1,15, 9, 4,14, 2,16,10, 3,13)( 5,22,19,18, 8, 6,21,20,17, 7)(11,12)$ | |
| $5^{4},1^{2}$ | $1584$ | $5$ | $( 1,16, 9, 3,14)( 2,15,10, 4,13)( 5,21,19,17, 8)( 6,22,20,18, 7)$ | |
| $22$ | $720$ | $22$ | $( 1, 3, 6,14,17, 8,19,22,15,12, 9, 2, 4, 5,13,18, 7,20,21,16,11,10)$ | |
| $11^{2}$ | $720$ | $11$ | $( 1, 4, 6,13,17, 7,19,21,15,11, 9)( 2, 3, 5,14,18, 8,20,22,16,12,10)$ | |
| $22$ | $720$ | $22$ | $( 1,10,11,16,21,20, 7,18,13, 5, 4, 2, 9,12,15,22,19, 8,17,14, 6, 3)$ | |
| $11^{2}$ | $720$ | $11$ | $( 1, 9,11,15,21,19, 7,17,13, 6, 4)( 2,10,12,16,22,20, 8,18,14, 5, 3)$ | |
| $2^{11}$ | $165$ | $2$ | $( 1,18)( 2,17)( 3,10)( 4, 9)( 5,12)( 6,11)( 7, 8)(13,21)(14,22)(15,16)(19,20)$ | |
| $2^{8},1^{6}$ | $165$ | $2$ | $( 1,17)( 2,18)( 3, 9)( 4,10)( 5,11)( 6,12)(13,22)(14,21)$ | |
| $4^{4},2^{2},1^{2}$ | $990$ | $4$ | $( 1, 3,17, 9)( 2, 4,18,10)( 5,13,11,22)( 6,14,12,21)(15,16)(19,20)$ | |
| $4^{4},2,1^{4}$ | $990$ | $4$ | $( 1, 4,17,10)( 2, 3,18, 9)( 5,14,11,21)( 6,13,12,22)( 7, 8)$ | |
| $8^{2},4,1^{2}$ | $990$ | $8$ | $( 1,12, 3,21,17, 6, 9,14)( 2,11, 4,22,18, 5,10,13)(15,19,16,20)$ | |
| $8^{2},4,2$ | $990$ | $8$ | $( 1,11, 3,22,17, 5, 9,13)( 2,12, 4,21,18, 6,10,14)( 7, 8)(15,20,16,19)$ | |
| $8^{2},4,2$ | $990$ | $8$ | $( 1,13, 9, 5,17,22, 3,11)( 2,14,10, 6,18,21, 4,12)( 7, 8)(15,19,16,20)$ | |
| $8^{2},4,1^{2}$ | $990$ | $8$ | $( 1,14, 9, 6,17,21, 3,12)( 2,13,10, 5,18,22, 4,11)(15,20,16,19)$ | |
| $6^{3},2^{2}$ | $440$ | $6$ | $( 1, 8,22, 2, 7,21)( 3, 4)( 5,18,10, 6,17, 9)(11,19,16,12,20,15)(13,14)$ | |
| $3^{6},1^{4}$ | $440$ | $3$ | $( 1, 7,22)( 2, 8,21)( 5,17,10)( 6,18, 9)(11,20,16)(12,19,15)$ | |
| $6^{2},3^{2},2^{2}$ | $1320$ | $6$ | $( 1,18, 7, 9,22, 6)( 2,17, 8,10,21, 5)( 3,14)( 4,13)(11,16,20)(12,15,19)$ | |
| $6^{3},2^{2}$ | $1320$ | $6$ | $( 1,17, 7,10,22, 5)( 2,18, 8, 9,21, 6)( 3,13)( 4,14)(11,15,20,12,16,19)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $15840=2^{5} \cdot 3^{2} \cdot 5 \cdot 11$ | 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: | 15840.q | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 5 P | |
| 11 P | |
| Type |
magma: CharacterTable(G);