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Magma
magma: G := TransitiveGroup(30, 31);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $31$ | 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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$\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,23,2,24)(3,22,4,21)(5,29,6,30)(7,28,8,27)(9,26,10,25)(13,19)(14,20)(15,18)(16,17), (1,11,24,5,16,27,9,19,21,3,13,25,7,17,30)(2,12,23,6,15,28,10,20,22,4,14,26,8,18,29) | 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 3: $S_3$
Degree 5: $D_{5}$
Degree 6: $S_4$
Degree 10: None
Degree 15: $D_{15}$
Low degree siblings
20T33, 30T19, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$ |
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 1, 1 $ | $30$ | $4$ | $( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,29,12,30)(13,28,14,27)(15,25,16,26) (17,23,18,24)(19,22,20,21)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $30$ | $2$ | $( 1, 2)( 3,10)( 4, 9)( 5, 8)( 6, 7)(11,29)(12,30)(13,28)(14,27)(15,25)(16,26) (17,23)(18,24)(19,22)(20,21)$ |
$ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,16,17,19)(12,14,15,18,20) (21,24,25,27,30)(22,23,26,28,29)$ |
$ 10, 10, 5, 5 $ | $6$ | $10$ | $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,14,16,18,19,12,13,15,17,20) (21,23,25,28,30,22,24,26,27,29)$ |
$ 10, 10, 5, 5 $ | $6$ | $10$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,14,17,12,16,20,13,18) (21,26,30,23,27,22,25,29,24,28)$ |
$ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,16,19,13,17)(12,15,20,14,18) (21,25,30,24,27)(22,26,29,23,28)$ |
$ 15, 15 $ | $8$ | $15$ | $( 1,11,23, 5,16,28, 9,19,22, 3,13,26, 7,17,29)( 2,12,24, 6,15,27,10,20,21, 4, 14,25, 8,18,30)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,13,27)( 2,14,28)( 3,16,30)( 4,15,29)( 5,17,21)( 6,18,22)( 7,19,24) ( 8,20,23)( 9,11,25)(10,12,26)$ |
$ 15, 15 $ | $8$ | $15$ | $( 1,15,22, 7,12,28, 3,18,23, 9,14,29, 5,20,26)( 2,16,21, 8,11,27, 4,17,24,10, 13,30, 6,19,25)$ |
$ 15, 15 $ | $8$ | $15$ | $( 1,17,26, 3,19,28, 5,11,29, 7,13,22, 9,16,23)( 2,18,25, 4,20,27, 6,12,30, 8, 14,21,10,15,24)$ |
$ 15, 15 $ | $8$ | $15$ | $( 1,19,30, 9,17,27, 7,16,25, 5,13,24, 3,11,21)( 2,20,29,10,18,28, 8,15,26, 6, 14,23, 4,12,22)$ |
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 3 2 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 4a 2b 5a 10a 10b 5b 15a 3a 15b 15c 15d 2P 1a 1a 2a 1a 5b 5b 5a 5a 15c 3a 15d 15b 15a 3P 1a 2a 4a 2b 5b 10b 10a 5a 5b 1a 5b 5a 5a 5P 1a 2a 4a 2b 1a 2a 2a 1a 3a 3a 3a 3a 3a 7P 1a 2a 4a 2b 5b 10b 10a 5a 15d 3a 15c 15a 15b 11P 1a 2a 4a 2b 5a 10a 10b 5b 15b 3a 15a 15d 15c 13P 1a 2a 4a 2b 5b 10b 10a 5a 15c 3a 15d 15b 15a 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 2 . . 2 2 2 2 -1 -1 -1 -1 -1 X.4 2 2 . . A A *A *A A 2 A *A *A X.5 2 2 . . *A *A A A *A 2 *A A A X.6 2 2 . . *A *A A A C -1 D E F X.7 2 2 . . *A *A A A D -1 C F E X.8 2 2 . . A A *A *A E -1 F D C X.9 2 2 . . A A *A *A F -1 E C D X.10 3 -1 -1 1 3 -1 -1 3 . . . . . X.11 3 -1 1 -1 3 -1 -1 3 . . . . . X.12 6 -2 . . B -A -*A *B . . . . . X.13 6 -2 . . *B -*A -A B . . . . . 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);