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Magma
magma: G := TransitiveGroup(20, 33);
Group action invariants
Degree $n$: | $20$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_5:S_4$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,9,17,5,13)(2,11,20,6,15,4,10,19,8,14,3,12,18,7,16), (1,7,4,6)(2,5,3,8)(9,19,12,18)(10,17,11,20)(13,15,16,14) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $10$: $D_{5}$ $24$: $S_4$ $30$: $D_{15}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: None
Degree 4: $S_4$
Degree 5: $D_{5}$
Degree 10: None
Low degree siblings
30T19, 30T31, 40T63Siblings 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 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $30$ | $2$ | $( 3, 4)( 5,17)( 6,18)( 7,20)( 8,19)( 9,13)(10,14)(11,16)(12,15)$ |
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ | $8$ | $3$ | $( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $3$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$ |
$ 4, 4, 4, 4, 4 $ | $30$ | $4$ | $( 1, 2, 3, 4)( 5,18, 7,20)( 6,19, 8,17)( 9,14,11,16)(10,15,12,13)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1, 5, 9,13,17)( 2, 8,11,14,20, 3, 6,12,15,18, 4, 7,10,16,19)$ |
$ 10, 10 $ | $6$ | $10$ | $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$ |
$ 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$ |
$ 15, 5 $ | $8$ | $15$ | $( 1, 9,17, 5,13)( 2,12,19, 6,16, 3,10,20, 7,14, 4,11,18, 8,15)$ |
$ 10, 10 $ | $6$ | $10$ | $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $120=2^{3} \cdot 3 \cdot 5$ | 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: | 120.38 | magma: IdentifyGroup(G);
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Character table: |
2 3 2 . 3 2 2 . . 2 2 . . 2 3 1 . 1 . . 1 1 1 . 1 1 1 . 5 1 . 1 1 . 1 1 1 1 1 1 1 1 1a 2a 3a 2b 4a 5a 15a 15b 10a 5b 15c 15d 10b 2P 1a 1a 3a 1a 2b 5b 15d 15c 5b 5a 15a 15b 5a 3P 1a 2a 1a 2b 4a 5b 5b 5b 10b 5a 5a 5a 10a 5P 1a 2a 3a 2b 4a 1a 3a 3a 2b 1a 3a 3a 2b 7P 1a 2a 3a 2b 4a 5b 15c 15d 10b 5a 15b 15a 10a 11P 1a 2a 3a 2b 4a 5a 15b 15a 10a 5b 15d 15c 10b 13P 1a 2a 3a 2b 4a 5b 15d 15c 10b 5a 15a 15b 10a X.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 X.3 2 . -1 2 . 2 -1 -1 2 2 -1 -1 2 X.4 2 . 2 2 . A A A A *A *A *A *A X.5 2 . 2 2 . *A *A *A *A A A A A X.6 2 . -1 2 . *A C D *A A F E A X.7 2 . -1 2 . *A D C *A A E F A X.8 2 . -1 2 . A E F A *A C D *A X.9 2 . -1 2 . A F E A *A D C *A X.10 3 -1 . -1 1 3 . . -1 3 . . -1 X.11 3 1 . -1 -1 3 . . -1 3 . . -1 X.12 6 . . -2 . B . . -A *B . . -*A X.13 6 . . -2 . *B . . -*A B . . -A A = E(5)+E(5)^4 = (-1+Sqrt(5))/2 = b5 B = 3*E(5)+3*E(5)^4 = (-3+3*Sqrt(5))/2 = 3b5 C = E(15)^4+E(15)^11 D = E(15)+E(15)^14 E = E(15)^7+E(15)^8 F = E(15)^2+E(15)^13 |
magma: CharacterTable(G);