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Magma
magma: G := TransitiveGroup(22, 34);
Group action invariants
Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $34$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_2^{10}.F_{11}$ | ||
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: | (3,16,12,6,7,22,10,13,20,17)(4,15,11,5,8,21,9,14,19,18), (1,15,14,7,12,2,16,13,8,11)(3,21,9,18,20)(4,22,10,17,19)(5,6) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $10$: $C_{10}$ $110$: $F_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 11: $F_{11}$
Low degree siblings
22T35, 44T312, 44T313, 44T316Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
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, 2, 2, 1, 1 $ | $11$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ |
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 3, 4)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $110$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 1, 2)( 3, 4)(11,12)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $2$ | $( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(11,12)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $2$ | $( 9,10)(13,14)(15,16)(17,18)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $110$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $55$ | $2$ | $( 5, 6)( 7, 8)( 9,10)(17,18)(19,20)(21,22)$ |
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $2$ | $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$ |
$ 11, 11 $ | $10240$ | $11$ | $( 1,19,15,11, 8, 3,21,18,13,10, 5)( 2,20,16,12, 7, 4,22,17,14, 9, 6)$ |
$ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3,12, 7,10,20)( 4,11, 8, 9,19)( 5,21,14,18,15)( 6,22,13,17,16)$ |
$ 10, 5, 5, 2 $ | $5632$ | $10$ | $( 1, 2)( 3,11, 7, 9,20)( 4,12, 8,10,19)( 5,22,14,17,15, 6,21,13,18,16)$ |
$ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3,12, 7,10,20, 4,11, 8, 9,19)( 5,21,14,18,16, 6,22,13,17,15)$ |
$ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3, 7,20,12,10)( 4, 8,19,11, 9)( 5,14,15,21,18)( 6,13,16,22,17)$ |
$ 10, 5, 5, 2 $ | $5632$ | $10$ | $( 1, 2)( 3, 8,20,11,10)( 4, 7,19,12, 9)( 5,13,15,22,18, 6,14,16,21,17)$ |
$ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3, 7,20,12,10, 4, 8,19,11, 9)( 5,14,16,22,17, 6,13,15,21,18)$ |
$ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3,20,10, 7,12)( 4,19, 9, 8,11)( 5,15,18,14,21)( 6,16,17,13,22)$ |
$ 10, 5, 5, 2 $ | $5632$ | $10$ | $( 1, 2)( 3,19,10, 8,12)( 4,20, 9, 7,11)( 5,16,18,13,21, 6,15,17,14,22)$ |
$ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3,20,10, 7,12, 4,19, 9, 8,11)( 5,16,17,13,22, 6,15,18,14,21)$ |
$ 5, 5, 5, 5, 1, 1 $ | $2816$ | $5$ | $( 3,10,12,20, 7)( 4, 9,11,19, 8)( 5,18,21,15,14)( 6,17,22,16,13)$ |
$ 10, 5, 5, 2 $ | $5632$ | $10$ | $( 1, 2)( 3, 9,12,19, 7)( 4,10,11,20, 8)( 5,17,21,16,14, 6,18,22,15,13)$ |
$ 10, 10, 1, 1 $ | $2816$ | $10$ | $( 3,10,12,20, 7, 4, 9,11,19, 8)( 5,18,21,16,13, 6,17,22,15,14)$ |
$ 10, 10, 1, 1 $ | $5632$ | $10$ | $( 3,16,12, 6, 7,22,10,13,20,17)( 4,15,11, 5, 8,21, 9,14,19,18)$ |
$ 20, 2 $ | $5632$ | $20$ | $( 1, 2)( 3,15,12, 5, 7,21,10,14,20,18, 4,16,11, 6, 8,22, 9,13,19,17)$ |
$ 10, 10, 1, 1 $ | $5632$ | $10$ | $( 3, 6,10,17,12,22,20,16, 7,13)( 4, 5, 9,18,11,21,19,15, 8,14)$ |
$ 20, 2 $ | $5632$ | $20$ | $( 1, 2)( 3, 5,10,18,12,21,20,15, 7,14, 4, 6, 9,17,11,22,19,16, 8,13)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $352$ | $2$ | $( 3,22)( 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$ |
$ 4, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1760$ | $4$ | $( 1, 2)( 3,21, 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)$ |
$ 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $1760$ | $4$ | $( 3,22, 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16,10,15)(11,14)(12,13)$ |
$ 4, 4, 4, 2, 2, 2, 2, 2 $ | $1760$ | $4$ | $( 1, 2)( 3,21, 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$ |
$ 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ | $1760$ | $4$ | $( 3,22, 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14,12,13)$ |
$ 4, 4, 4, 4, 2, 2, 1, 1 $ | $1760$ | $4$ | $( 3,21, 4,22)( 5,20, 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$ |
$ 4, 4, 4, 2, 2, 2, 2, 2 $ | $1760$ | $4$ | $( 1, 2)( 3,22, 4,21)( 5,19, 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14,12,13)$ |
$ 4, 4, 4, 4, 4, 2 $ | $352$ | $4$ | $( 1, 2)( 3,22, 4,21)( 5,19, 6,20)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$ |
$ 10, 10, 1, 1 $ | $5632$ | $10$ | $( 3,17,20,13,10,22, 7, 6,12,16)( 4,18,19,14, 9,21, 8, 5,11,15)$ |
$ 20, 2 $ | $5632$ | $20$ | $( 1, 2)( 3,18,20,14,10,21, 7, 5,12,15, 4,17,19,13, 9,22, 8, 6,11,16)$ |
$ 10, 10, 1, 1 $ | $5632$ | $10$ | $( 3,13, 7,16,20,22,12,17,10, 6)( 4,14, 8,15,19,21,11,18, 9, 5)$ |
$ 20, 2 $ | $5632$ | $20$ | $( 1, 2)( 3,14, 7,15,20,21,12,18,10, 5, 4,13, 8,16,19,22,11,17, 9, 6)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $112640=2^{11} \cdot 5 \cdot 11$ | 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: | yes | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 112640.a | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);