Show commands:
Magma
magma: G := TransitiveGroup(14, 39);
Group action invariants
| Degree $n$: | $14$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $39$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $\PGL(2,13)$ | ||
| CHM label: | $L(14):2=PGL(2,13)$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | yes | 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: | (1,12)(2,6)(3,4)(7,11)(9,10)(13,14), (1,2,4,8,3,6,12,11,9,5,10,7), (1,2,3,4,5,6,7,8,9,10,11,12,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 7: None
Low degree siblings
28T201, 42T284Siblings 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 $ | $1$ | $1$ | $()$ | |
| $ 3, 3, 3, 3, 1, 1 $ | $182$ | $3$ | $( 1, 4, 7)( 2,14, 9)( 3, 5,13)( 6,12, 8)$ | |
| $ 2, 2, 2, 2, 2, 2, 1, 1 $ | $91$ | $2$ | $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)$ | |
| $ 4, 4, 4, 1, 1 $ | $182$ | $4$ | $( 1, 8,12, 5)( 2, 3,11,10)( 4, 6, 9, 7)$ | |
| $ 6, 6, 1, 1 $ | $182$ | $6$ | $( 1,13, 7, 5, 4, 3)( 2, 8, 9,12,14, 6)$ | |
| $ 12, 1, 1 $ | $182$ | $12$ | $( 1, 7, 8, 9, 2, 6, 3,12,13, 4,14,10)$ | |
| $ 12, 1, 1 $ | $182$ | $12$ | $( 1,12, 8, 4, 2,10, 3, 7,13, 9,14, 6)$ | |
| $ 13, 1 $ | $168$ | $13$ | $( 1, 9, 8, 2, 3, 6,13, 7,10,11, 5, 4,12)$ | |
| $ 2, 2, 2, 2, 2, 2, 2 $ | $78$ | $2$ | $( 1,11)( 2,12)( 3, 8)( 4, 6)( 5,10)( 7, 9)(13,14)$ | |
| $ 7, 7 $ | $156$ | $7$ | $( 1, 6,12, 4, 8, 9,10)( 2,13, 3,14,11, 7, 5)$ | |
| $ 7, 7 $ | $156$ | $7$ | $( 1, 8, 6, 9,12,10, 4)( 2,11,13, 7, 3, 5,14)$ | |
| $ 7, 7 $ | $156$ | $7$ | $( 1,12, 8,10, 6, 4, 9)( 2, 3,11, 5,13,14, 7)$ | |
| $ 14 $ | $156$ | $14$ | $( 1, 2, 6,13,12, 3, 4,14, 8,11, 9, 7,10, 5)$ | |
| $ 14 $ | $156$ | $14$ | $( 1,13, 4,11,10, 2,12,14, 9, 5, 6, 3, 8, 7)$ | |
| $ 14 $ | $156$ | $14$ | $( 1,11,12, 5, 8,13,10,14, 6, 7, 4, 2, 9, 3)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $2184=2^{3} \cdot 3 \cdot 7 \cdot 13$ | 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 | ||
| Label: | 2184.b | magma: IdentifyGroup(G);
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| Character table: |
| Size | |
| 2 P | |
| 3 P | |
| 7 P | |
| 13 P | |
| Type |
magma: CharacterTable(G);