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Magma
magma: G := TransitiveGroup(20, 19);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $19$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_{10}:C_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,18,10,13)(2,17,9,14)(3,11,7,20)(4,12,8,19)(5,6), (1,15,10,4,17,12,5,19,14,8)(2,16,9,3,18,11,6,20,13,7) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $D_{4}$ x 2, $C_4\times C_2$ $16$: $C_2^2:C_4$ $20$: $F_5$ $40$: $F_{5}\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: $F_5$
Degree 10: $F_{5}\times C_2$
Low degree siblings
20T19, 20T22 x 2, 40T26, 40T45, 40T55 x 2Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 4, 4, 4, 4, 2, 1, 1 $ | $10$ | $4$ | $( 3, 8,20,15)( 4, 7,19,16)( 5,14,17,10)( 6,13,18, 9)(11,12)$ | |
$ 4, 4, 4, 4, 2, 1, 1 $ | $10$ | $4$ | $( 3,15,20, 8)( 4,16,19, 7)( 5,10,17,14)( 6, 9,18,13)(11,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $5$ | $2$ | $( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,13)(10,14)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $5$ | $2$ | $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 3)( 2, 4)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$ | |
$ 10, 10 $ | $4$ | $10$ | $( 1, 3, 5, 7,10,11,14,16,17,20)( 2, 4, 6, 8, 9,12,13,15,18,19)$ | |
$ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 9, 8)( 2, 4,10, 7)( 5,16, 6,15)(11,13,19,17)(12,14,20,18)$ | |
$ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3,18,15)( 2, 4,17,16)( 5,11,13, 8)( 6,12,14, 7)( 9,19,10,20)$ | |
$ 10, 10 $ | $4$ | $10$ | $( 1, 4, 5, 8,10,12,14,15,17,19)( 2, 3, 6, 7, 9,11,13,16,18,20)$ | |
$ 5, 5, 5, 5 $ | $4$ | $5$ | $( 1, 5,10,14,17)( 2, 6, 9,13,18)( 3, 7,11,16,20)( 4, 8,12,15,19)$ | |
$ 10, 10 $ | $4$ | $10$ | $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,11,15,20, 4, 7,12,16,19)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,19)(10,20)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $80=2^{4} \cdot 5$ | 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: | 80.34 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 2D | 2E | 4A1 | 4A-1 | 4B1 | 4B-1 | 5A | 10A | 10B1 | 10B3 | ||
Size | 1 | 1 | 2 | 5 | 5 | 10 | 10 | 10 | 10 | 10 | 4 | 4 | 4 | 4 | |
2 P | 1A | 1A | 1A | 1A | 1A | 1A | 2D | 2D | 2C | 2C | 5A | 5A | 5A | 5A | |
5 P | 1A | 2A | 2B | 2C | 2D | 2E | 4B-1 | 4B1 | 4A-1 | 4A1 | 1A | 2A | 2B | 2B | |
Type | |||||||||||||||
80.34.1a | R | ||||||||||||||
80.34.1b | R | ||||||||||||||
80.34.1c | R | ||||||||||||||
80.34.1d | R | ||||||||||||||
80.34.1e1 | C | ||||||||||||||
80.34.1e2 | C | ||||||||||||||
80.34.1f1 | C | ||||||||||||||
80.34.1f2 | C | ||||||||||||||
80.34.2a | R | ||||||||||||||
80.34.2b | R | ||||||||||||||
80.34.4a | R | ||||||||||||||
80.34.4b | R | ||||||||||||||
80.34.4c1 | R | ||||||||||||||
80.34.4c2 | R |
magma: CharacterTable(G);