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Magma
magma: G := TransitiveGroup(22, 24);
Group action invariants
Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $24$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{11}^2:C_5:C_{20}$ | ||
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: | (1,6,10,11,3)(2,9,8,5,7)(12,18,15,22,13)(14,17,21,19,20), (1,22)(2,14,9,13,10,16,7,18,5,12,11,19,4,20,3,17,6,15,8,21) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $5$: $C_5$ $10$: $D_{5}$, $C_{10}$ $20$: 20T1, 20T2 $50$: $D_5\times C_5$ $100$: 20T25 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 11: None
Low degree siblings
44T138Siblings 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 $ | $1$ | $1$ | $()$ |
$ 11, 11 $ | $100$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,14,15,16,17,18,19,20,21,22)$ |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $20$ | $11$ | $(12,14,16,18,20,22,13,15,17,19,21)$ |
$ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,17,15,16,21)(14,22,18,20,19)$ |
$ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,15,21,17,16)(14,18,19,22,20)$ |
$ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,21,16,15,17)(14,19,20,18,22)$ |
$ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,16,17,21,15)(14,20,22,19,18)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $5$ | $(13,21,16,15,17)(14,19,20,18,22)$ |
$ 11, 5, 5, 1 $ | $220$ | $55$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,22,15,18)(14,20,19,21,17)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $5$ | $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)$ |
$ 11, 5, 5, 1 $ | $220$ | $55$ | $( 1,10, 4, 8, 9)( 3, 5,11, 7, 6)(12,13,14,15,16,17,18,19,20,21,22)$ |
$ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,17,15,16,21)(14,22,18,20,19)$ |
$ 5, 5, 5, 5, 1, 1 $ | $121$ | $5$ | $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $5$ | $(13,16,17,21,15)(14,20,22,19,18)$ |
$ 11, 5, 5, 1 $ | $220$ | $55$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,17,22,20)(14,21,16,18,15)$ |
$ 5, 5, 5, 5, 1, 1 $ | $242$ | $5$ | $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,15,21,17,16)(14,18,19,22,20)$ |
$ 5, 5, 5, 5, 1, 1 $ | $121$ | $5$ | $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,17,15,16,21)(14,22,18,20,19)$ |
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $22$ | $5$ | $(13,17,15,16,21)(14,22,18,20,19)$ |
$ 11, 5, 5, 1 $ | $220$ | $55$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,18,21,14)(15,17,16,22,19)$ |
$ 5, 5, 5, 5, 1, 1 $ | $121$ | $5$ | $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,15,21,17,16)(14,18,19,22,20)$ |
$ 5, 5, 5, 5, 1, 1 $ | $121$ | $5$ | $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22)$ |
$ 10, 10, 1, 1 $ | $121$ | $10$ | $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,20,21,18,16,22,15,14,17,19)$ |
$ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,19,17,14,15,22,16,18,21,20)$ |
$ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,14,16,20,17,22,21,19,15,18)$ |
$ 10, 2, 2, 2, 2, 2, 1, 1 $ | $242$ | $10$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,18,15,19,21,22,17,20,16,14)$ |
$ 10, 2, 2, 2, 2, 2, 1, 1 $ | $242$ | $10$ | $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(13,22)(14,21)(15,20)(16,19)(17,18)$ |
$ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,18,15,19,21,22,17,20,16,14)$ |
$ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,20,21,18,16,22,15,14,17,19)$ |
$ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,19,17,14,15,22,16,18,21,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $121$ | $2$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$ |
$ 10, 2, 2, 2, 2, 2, 1, 1 $ | $242$ | $10$ | $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,22)(14,21)(15,20)(16,19)(17,18)$ |
$ 10, 2, 2, 2, 2, 2, 1, 1 $ | $242$ | $10$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,14,16,20,17,22,21,19,15,18)$ |
$ 10, 10, 1, 1 $ | $121$ | $10$ | $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(13,19,17,14,15,22,16,18,21,20)$ |
$ 10, 10, 1, 1 $ | $242$ | $10$ | $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,19,17,14,15,22,16,18,21,20)$ |
$ 10, 10, 1, 1 $ | $121$ | $10$ | $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,14,16,20,17,22,21,19,15,18)$ |
$ 10, 10, 1, 1 $ | $121$ | $10$ | $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,18,15,19,21,22,17,20,16,14)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,22)( 2,14, 9,13,10,16, 7,18, 5,12,11,19, 4,20, 3,17, 6,15, 8,21)$ |
$ 4, 4, 4, 4, 4, 2 $ | $605$ | $4$ | $( 1,18, 2,22)( 3,15,11,14)( 4,19,10,21)( 5,12, 9,17)( 6,16, 8,13)( 7,20)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,20, 8,17, 2,18, 4,14, 7,19, 6,21,10,13, 5,12, 3,16,11,22)( 9,15)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,14,11,20, 9,21, 5,12, 8,16, 3,13, 4,18, 6,17,10,15, 7,22)( 2,19)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,19, 5,12, 7,14, 8,15, 3,21, 6,13, 2,20,11,18,10,17, 4,22)( 9,16)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,15, 2,17, 8,18,11,13, 7,16, 5,12, 4,21, 9,20, 6,14,10,22)( 3,19)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,16, 2,15, 4,13, 8,20, 5,12,10,18, 9,19, 7,21, 3,14, 6,22)(11,17)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,21, 7,13,11,15,10,20, 2,16, 4,17, 9,14, 5,12, 6,18, 3,22)( 8,19)$ |
$ 20, 2 $ | $605$ | $20$ | $( 1,17, 7,15, 5,12, 2,13, 3,20,10,14, 4,16, 6,19, 9,18, 8,22)(11,21)$ |
$ 4, 4, 4, 4, 4, 2 $ | $605$ | $4$ | $( 1,13, 9,22)( 2,21, 8,14)( 3,18, 7,17)( 4,15, 6,20)( 5,12)(10,19,11,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $12100=2^{2} \cdot 5^{2} \cdot 11^{2}$ | 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: | 12100.bk | magma: IdentifyGroup(G);
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Character table: not available. |
magma: CharacterTable(G);