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Magma
magma: G := TransitiveGroup(20, 18);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $18$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{20}:C_4$ | ||
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: | (1,15,9,3,17,12,6,20,14,8,2,16,10,4,18,11,5,19,13,7), (1,8,10,4)(2,7,9,3)(5,15)(6,16)(11,18,20,13)(12,17,19,14) | 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, $C_2^2$ $8$: $D_{4}$, $C_4\times C_2$, $Q_8$ $16$: $C_4:C_4$ $20$: $F_5$ $40$: $F_{5}\times C_2$ 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
20T18, 40T52, 40T54Siblings 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 $ | $1$ | $1$ | $()$ |
$ 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 $ | $5$ | $2$ | $( 1, 2)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,14)(10,13)(11,12)$ |
$ 4, 4, 4, 4, 4 $ | $10$ | $4$ | $( 1, 3, 2, 4)( 5,20, 6,19)( 7,18, 8,17)( 9,15,10,16)(11,13,12,14)$ |
$ 20 $ | $4$ | $20$ | $( 1, 3, 6, 8,10,11,13,15,17,20, 2, 4, 5, 7, 9,12,14,16,18,19)$ |
$ 4, 4, 4, 4, 2, 2 $ | $10$ | $4$ | $( 1, 3,10, 7)( 2, 4, 9, 8)( 5,16)( 6,15)(11,14,20,17)(12,13,19,18)$ |
$ 4, 4, 4, 4, 2, 2 $ | $10$ | $4$ | $( 1, 3,17,16)( 2, 4,18,15)( 5,11,14, 7)( 6,12,13, 8)( 9,19)(10,20)$ |
$ 20 $ | $4$ | $20$ | $( 1, 4, 6, 7,10,12,13,16,17,19, 2, 3, 5, 8, 9,11,14,15,18,20)$ |
$ 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 $ | $4$ | $10$ | $( 1, 6,10,13,17, 2, 5, 9,14,18)( 3, 8,11,15,20, 4, 7,12,16,19)$ |
$ 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1,11, 2,12)( 3,13, 4,14)( 5,16, 6,15)( 7,18, 8,17)( 9,19,10,20)$ |
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: | 80.31 | magma: IdentifyGroup(G);
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Character table: |
2 4 3 3 4 4 4 3 2 3 3 2 2 2 3 5 1 . . . 1 . . 1 . . 1 1 1 1 1a 4a 4b 2a 2b 2c 4c 20a 4d 4e 20b 5a 10a 4f 2P 1a 2a 2a 1a 1a 1a 2b 10a 2a 2a 10a 5a 5a 2b 3P 1a 4b 4a 2a 2b 2c 4c 20a 4e 4d 20b 5a 10a 4f 5P 1a 4a 4b 2a 2b 2c 4c 4f 4d 4e 4f 1a 2b 4f 7P 1a 4b 4a 2a 2b 2c 4c 20a 4e 4d 20b 5a 10a 4f 11P 1a 4b 4a 2a 2b 2c 4c 20b 4e 4d 20a 5a 10a 4f 13P 1a 4a 4b 2a 2b 2c 4c 20b 4d 4e 20a 5a 10a 4f 17P 1a 4a 4b 2a 2b 2c 4c 20b 4d 4e 20a 5a 10a 4f 19P 1a 4b 4a 2a 2b 2c 4c 20b 4e 4d 20a 5a 10a 4f X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 1 -1 -1 1 1 -1 1 1 -1 X.3 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 X.4 1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 -1 X.5 1 A -A -1 1 -1 -1 1 A -A 1 1 1 1 X.6 1 -A A -1 1 -1 -1 1 -A A 1 1 1 1 X.7 1 A -A -1 1 -1 1 -1 -A A -1 1 1 -1 X.8 1 -A A -1 1 -1 1 -1 A -A -1 1 1 -1 X.9 2 . . -2 -2 2 . . . . . 2 -2 . X.10 2 . . 2 -2 -2 . . . . . 2 -2 . X.11 4 . . . 4 . . -1 . . -1 -1 -1 4 X.12 4 . . . 4 . . 1 . . 1 -1 -1 -4 X.13 4 . . . -4 . . B . . -B -1 1 . X.14 4 . . . -4 . . -B . . B -1 1 . A = -E(4) = -Sqrt(-1) = -i B = -E(20)-E(20)^9+E(20)^13+E(20)^17 = -Sqrt(-5) = -i5 |
magma: CharacterTable(G);