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Magma
magma: G := TransitiveGroup(22, 38);
Group invariants
| Abstract group: | $M_{22}$ | magma: IdentifyGroup(G);
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| 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 | magma: NilpotencyClass(G);
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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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| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | yes | magma: IsPrimitive(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 | Index | Representative |
| 1A | $1^{22}$ | $1$ | $1$ | $0$ | $()$ |
| 2A | $2^{8},1^{6}$ | $1155$ | $2$ | $8$ | $( 2,10)( 3,13)( 4, 5)( 8,20)(11,18)(12,22)(14,15)(16,21)$ |
| 3A | $3^{6},1^{4}$ | $12320$ | $3$ | $12$ | $( 1, 9,19)( 3,22,16)( 4, 5,10)( 8,14,13)(11,12,15)(18,20,21)$ |
| 4A | $4^{4},2^{2},1^{2}$ | $13860$ | $4$ | $14$ | $( 1, 6)( 2,22,10,12)( 3,21,13,16)( 4,11, 5,18)( 8,15,20,14)( 9,17)$ |
| 4B | $4^{4},2^{2},1^{2}$ | $27720$ | $4$ | $14$ | $( 1,17)( 2,18,10,11)( 3,14,13,15)( 4,22, 5,12)( 6, 9)( 8,21,20,16)$ |
| 5A | $5^{4},1^{2}$ | $88704$ | $5$ | $16$ | $( 1,19,18,22, 5)( 2,16, 4, 6, 9)( 3,20,17,12,13)( 7,14,11,10, 8)$ |
| 6A | $6^{2},3^{2},2^{2}$ | $36960$ | $6$ | $16$ | $( 1,11, 9,12,19,15)( 2, 6)( 3,16,22)( 4,10, 5)( 7,17)( 8,20,14,21,13,18)$ |
| 7A1 | $7^{3},1$ | $63360$ | $7$ | $18$ | $( 1, 3, 6,10, 2,13, 7)( 4,16,15,12,17, 8, 5)( 9,14,11,18,21,20,22)$ |
| 7A-1 | $7^{3},1$ | $63360$ | $7$ | $18$ | $( 1,10, 7, 6,13, 3, 2)( 4,12, 5,15, 8,16,17)( 9,18,22,11,20,14,21)$ |
| 8A | $8^{2},4,2$ | $55440$ | $8$ | $18$ | $( 1, 9, 6,17)( 2,20,22,14,10, 8,12,15)( 3, 4,21,11,13, 5,16,18)( 7,19)$ |
| 11A1 | $11^{2}$ | $40320$ | $11$ | $20$ | $( 1, 6,20,19, 2, 3, 8,14,13, 5,16)( 4,22,15,11,12,10, 7,18, 9,17,21)$ |
| 11A-1 | $11^{2}$ | $40320$ | $11$ | $20$ | $( 1,20, 2, 8,13,16, 6,19, 3,14, 5)( 4,15,12, 7, 9,21,22,11,10,18,17)$ |
Malle's constant $a(G)$: $1/8$
magma: ConjugacyClasses(G);
Character table
| 1A | 2A | 3A | 4A | 4B | 5A | 6A | 7A1 | 7A-1 | 8A | 11A1 | 11A-1 | ||
| Size | 1 | 1155 | 12320 | 13860 | 27720 | 88704 | 36960 | 63360 | 63360 | 55440 | 40320 | 40320 | |
| 2 P | 1A | 1A | 3A | 2A | 2A | 5A | 3A | 7A1 | 7A-1 | 4A | 11A-1 | 11A1 | |
| 3 P | 1A | 2A | 1A | 4A | 4B | 5A | 2A | 7A-1 | 7A1 | 8A | 11A1 | 11A-1 | |
| 5 P | 1A | 2A | 3A | 4A | 4B | 1A | 6A | 7A-1 | 7A1 | 8A | 11A1 | 11A-1 | |
| 7 P | 1A | 2A | 3A | 4A | 4B | 5A | 6A | 1A | 1A | 8A | 11A-1 | 11A1 | |
| 11 P | 1A | 2A | 3A | 4A | 4B | 5A | 6A | 7A1 | 7A-1 | 8A | 1A | 1A | |
| Type |
magma: CharacterTable(G);
Regular extensions
| $f_{ 1 } =$ |
$\left(5 x^{4}+34 x^{3}-119 x^{2}+212 x-164\right)^{4} \left(19 x^{3}-12 x^{2}+28 x+32\right)^{2}-2^{22} \left(x^{2}-x+3\right)^{11}/\left(11 t^{2}+1\right)$
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