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Magma
magma: G := TransitiveGroup(20, 7);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $7$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5:D_4$ | ||
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: | (1,11,2,12)(3,9,4,10)(5,8,6,7)(13,19,14,20)(15,17,16,18), (1,8,13,20,6,12,18,4,10,16)(2,7,14,19,5,11,17,3,9,15) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ $10$: $D_{5}$ $20$: $D_{10}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: $D_{5}$
Degree 10: $D_{10}$
Low degree siblings
20T11, 40T11Siblings 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 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $10$ | $2$ | $( 3,20)( 4,19)( 5,17)( 6,18)( 7,16)( 8,15)( 9,14)(10,13)(11,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ | |
$ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 2, 4)( 5,20, 6,19)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$ | |
$ 10, 10 $ | $2$ | $10$ | $( 1, 3, 6, 7,10,11,13,15,18,19)( 2, 4, 5, 8, 9,12,14,16,17,20)$ | |
$ 10, 10 $ | $2$ | $10$ | $( 1, 4, 6, 8,10,12,13,16,18,20)( 2, 3, 5, 7, 9,11,14,15,17,19)$ | |
$ 10, 10 $ | $2$ | $10$ | $( 1, 5,10,14,18, 2, 6, 9,13,17)( 3, 8,11,16,19, 4, 7,12,15,20)$ | |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 6,10,13,18)( 2, 5, 9,14,17)( 3, 7,11,15,19)( 4, 8,12,16,20)$ | |
$ 10, 10 $ | $2$ | $10$ | $( 1, 7,13,19, 6,11,18, 3,10,15)( 2, 8,14,20, 5,12,17, 4, 9,16)$ | |
$ 10, 10 $ | $2$ | $10$ | $( 1, 8,13,20, 6,12,18, 4,10,16)( 2, 7,14,19, 5,11,17, 3, 9,15)$ | |
$ 10, 10 $ | $2$ | $10$ | $( 1, 9,18, 5,13, 2,10,17, 6,14)( 3,12,19, 8,15, 4,11,20, 7,16)$ | |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,10,18, 6,13)( 2, 9,17, 5,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1,11)( 2,12)( 3,13)( 4,14)( 5,16)( 6,15)( 7,18)( 8,17)( 9,20)(10,19)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $40=2^{3} \cdot 5$ | 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: | 40.8 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A | 5A1 | 5A2 | 10A1 | 10A3 | 10B1 | 10B-1 | 10B3 | 10B-3 | ||
Size | 1 | 1 | 2 | 10 | 10 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 1A | 1A | 2A | 5A2 | 5A1 | 5A2 | 5A1 | 5A2 | 5A1 | 5A1 | 5A2 | |
5 P | 1A | 2A | 2B | 2C | 4A | 1A | 1A | 2A | 2A | 2B | 2B | 2B | 2B | |
Type | ||||||||||||||
40.8.1a | R | |||||||||||||
40.8.1b | R | |||||||||||||
40.8.1c | R | |||||||||||||
40.8.1d | R | |||||||||||||
40.8.2a | R | |||||||||||||
40.8.2b1 | R | |||||||||||||
40.8.2b2 | R | |||||||||||||
40.8.2c1 | R | |||||||||||||
40.8.2c2 | R | |||||||||||||
40.8.2d1 | C | |||||||||||||
40.8.2d2 | C | |||||||||||||
40.8.2d3 | C | |||||||||||||
40.8.2d4 | C |
magma: CharacterTable(G);