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Magma
magma: G := TransitiveGroup(22, 11);
Group action invariants
Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $11$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{11}:F_{11}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,18,10,14,11,16,5,15,8,21)(2,20,4,13,3,22,9,12,6,17)(7,19), (1,18,8,12,5,13,11,22,10,15)(2,14,6,20,9,19,3,21,4,17)(7,16) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $10$: $C_{10}$ $55$: $C_{11}:C_5$ $110$: $F_{11}$, 22T5 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 11: None
Low degree siblings
22T11 x 4Siblings 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, 11 $ | $10$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,16,20,13,17,21,14,18,22,15,19)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,20,17,14,22,19,16,13,21,18,15)$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $11$ | $(12,13,14,15,16,17,18,19,20,21,22)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,17,22,16,21,15,20,14,19,13,18)$ | |
$ 11, 11 $ | $5$ | $11$ | $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,21,19,17,15,13,22,20,18,16,14)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,18,13,19,14,20,15,21,16,22,17)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,21,20,19,18,17,16,15,14,13)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,16,20,13,17,21,14,18,22,15,19)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,19,15,22,18,14,21,17,13,20,16)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,15,18,21,13,16,19,22,14,17,20)$ | |
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $10$ | $11$ | $(12,14,16,18,20,22,13,15,17,19,21)$ | |
$ 11, 11 $ | $5$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,19,15,22,18,14,21,17,13,20,16)$ | |
$ 11, 11 $ | $10$ | $11$ | $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,13,14,15,16,17,18,19,20,21,22)$ | |
$ 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, 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 $ | $121$ | $5$ | $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22)$ | |
$ 5, 5, 5, 5, 1, 1 $ | $121$ | $5$ | $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18)$ | |
$ 10, 10, 2 $ | $121$ | $10$ | $( 1,18,10,14,11,16, 5,15, 8,21)( 2,20, 4,13, 3,22, 9,12, 6,17)( 7,19)$ | |
$ 10, 10, 2 $ | $121$ | $10$ | $( 1,20, 5,16,10,22, 8,13,11,21)( 2,19, 9,12, 4,17, 6,15, 3,18)( 7,14)$ | |
$ 22 $ | $55$ | $22$ | $( 1,19, 8,17, 4,15,11,13, 7,22, 3,20,10,18, 6,16, 2,14, 9,12, 5,21)$ | |
$ 22 $ | $55$ | $22$ | $( 1,13,10,12, 8,22, 6,21, 4,20, 2,19,11,18, 9,17, 7,16, 5,15, 3,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $11$ | $2$ | $( 1,18)( 2,13)( 3,19)( 4,14)( 5,20)( 6,15)( 7,21)( 8,16)( 9,22)(10,17)(11,12)$ | |
$ 10, 10, 2 $ | $121$ | $10$ | $( 1,22, 7,20, 6,13, 8,16, 4,21)( 2,18, 5,17,10,19,11,15, 9,12)( 3,14)$ | |
$ 10, 10, 2 $ | $121$ | $10$ | $( 1,14, 8,15, 7,18, 4,16, 6,21)( 2,22,11,17, 5,13, 9,12,10,20)( 3,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $1210=2 \cdot 5 \cdot 11^{2}$ | 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: | 1210.12 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 5A1 | 5A-1 | 5A2 | 5A-2 | 10A1 | 10A-1 | 10A3 | 10A-3 | 11A1 | 11A-1 | 11B | 11C1 | 11C-1 | 11D1 | 11D-1 | 11E1 | 11E-1 | 11F1 | 11F-1 | 11G1 | 11G-1 | 22A1 | 22A-1 | ||
Size | 1 | 11 | 121 | 121 | 121 | 121 | 121 | 121 | 121 | 121 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 55 | 55 | |
2 P | 1A | 1A | 5A2 | 5A-2 | 5A-1 | 5A1 | 5A-1 | 5A1 | 5A2 | 5A-2 | 11A-1 | 11A1 | 11D-1 | 11E1 | 11F1 | 11E-1 | 11G-1 | 11C1 | 11F-1 | 11G1 | 11D1 | 11C-1 | 11B | 11A1 | 11A-1 | |
5 P | 1A | 2A | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 11A1 | 11A-1 | 11D1 | 11E-1 | 11F-1 | 11E1 | 11G1 | 11C-1 | 11F1 | 11G-1 | 11D-1 | 11C1 | 11B | 22A1 | 22A-1 | |
11 P | 1A | 2A | 5A1 | 5A-1 | 5A2 | 5A-2 | 10A-1 | 10A1 | 10A-3 | 10A3 | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 1A | 2A | 2A | |
Type | ||||||||||||||||||||||||||
1210.12.1a | R | |||||||||||||||||||||||||
1210.12.1b | R | |||||||||||||||||||||||||
1210.12.1c1 | C | |||||||||||||||||||||||||
1210.12.1c2 | C | |||||||||||||||||||||||||
1210.12.1c3 | C | |||||||||||||||||||||||||
1210.12.1c4 | C | |||||||||||||||||||||||||
1210.12.1d1 | C | |||||||||||||||||||||||||
1210.12.1d2 | C | |||||||||||||||||||||||||
1210.12.1d3 | C | |||||||||||||||||||||||||
1210.12.1d4 | C | |||||||||||||||||||||||||
1210.12.5a1 | C | |||||||||||||||||||||||||
1210.12.5a2 | C | |||||||||||||||||||||||||
1210.12.5b1 | C | |||||||||||||||||||||||||
1210.12.5b2 | C | |||||||||||||||||||||||||
1210.12.10a | R | |||||||||||||||||||||||||
1210.12.10b1 | C | |||||||||||||||||||||||||
1210.12.10b2 | C | |||||||||||||||||||||||||
1210.12.10c1 | C | |||||||||||||||||||||||||
1210.12.10c2 | C | |||||||||||||||||||||||||
1210.12.10d1 | C | |||||||||||||||||||||||||
1210.12.10d2 | C | |||||||||||||||||||||||||
1210.12.10e1 | C | |||||||||||||||||||||||||
1210.12.10e2 | C | |||||||||||||||||||||||||
1210.12.10f1 | C | |||||||||||||||||||||||||
1210.12.10f2 | C |
magma: CharacterTable(G);