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Magma
magma: G := TransitiveGroup(22, 8);
Group action invariants
Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $8$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{11}^2:C_4$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,8)(2,7)(3,6)(4,5)(9,11)(12,19)(13,18)(14,17)(15,16)(20,22), (1,21,11,15)(2,16,10,20)(3,22,9,14)(4,17,8,19)(5,12,7,13)(6,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 11: None
Low degree siblings
22T8 x 5, 44T28 x 6Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(12,13,14,15,16,17,18,19,20,21,22)$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(12,14,16,18,20,22,13,15,17,19,21)$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(12,16,20,13,17,21,14,18,22,15,19)$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(12,17,22,16,21,15,20,14,19,13,18)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $121$ | $2$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,13,14,15,16,17,18,19,20,21,22)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,14,16,18,20,22,13,15,17,19,21)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,16,20,13,17,21,14,18,22,15,19)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,17,22,16,21,15,20,14,19,13,18)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,19,15,22,18,14,21,17,13,20,16)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,20,17,14,22,19,16,13,21,18,15)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,21,19,17,15,13,22,20,18,16,14)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,21,20,19,18,17,16,15,14,13)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,13,14,15,16,17,18,19,20,21,22)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,14,16,18,20,22,13,15,17,19,21)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,16,20,13,17,21,14,18,22,15,19)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,19,15,22,18,14,21,17,13,20,16)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,20,17,14,22,19,16,13,21,18,15)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,21,19,17,15,13,22,20,18,16,14)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,14,16,18,20,22,13,15,17,19,21)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,16,20,13,17,21,14,18,22,15,19)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,20,17,14,22,19,16,13,21,18,15)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,21,19,17,15,13,22,20,18,16,14)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,14,16,18,20,22,13,15,17,19,21)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,20,17,14,22,19,16,13,21,18,15)$ | |
$ 11, 11 $ | $4$ | $11$ | $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 4, 4, 4, 4, 4, 2 $ | $121$ | $4$ | $( 1,12)( 2,17,11,18)( 3,22,10,13)( 4,16, 9,19)( 5,21, 8,14)( 6,15, 7,20)$ | |
$ 4, 4, 4, 4, 4, 2 $ | $121$ | $4$ | $( 1,12)( 2,18,11,17)( 3,13,10,22)( 4,19, 9,16)( 5,14, 8,21)( 6,20, 7,15)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $484=2^{2} \cdot 11^{2}$ | 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: | 484.8 | magma: IdentifyGroup(G);
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Character table: | 34 x 34 character table |
magma: CharacterTable(G);