Show commands:
Magma
magma: G := TransitiveGroup(30, 25);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $25$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $S_5$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $6$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,14,17,12)(2,13,18,11)(3,24,8,10)(4,23,7,9)(5,25)(6,26)(15,30,19,21)(16,29,20,22)(27,28), (1,9,13,3,5)(2,10,14,4,6)(7,22,25,15,18)(8,21,26,16,17)(11,28,23,20,30)(12,27,24,19,29) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: None
Degree 5: $S_5$
Degree 6: None
Degree 10: $S_5$
Degree 15: $S_5$
Low degree siblings
5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T27, 40T62Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $15$ | $2$ | $( 3,13)( 4,14)( 5, 9)( 6,10)( 7,22)( 8,21)(11,30)(12,29)(17,26)(18,25)(19,27) (20,28)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $10$ | $2$ | $( 1, 2)( 3,12)( 4,11)( 5, 7)( 6, 8)( 9,10)(13,25)(14,26)(15,23)(16,24)(17,19) (18,20)(21,22)(27,30)(28,29)$ | |
$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $ | $30$ | $4$ | $( 1, 2)( 3,25,20,29)( 4,26,19,30)( 5,10, 8,22)( 6, 9, 7,21)(11,27,17,14) (12,28,18,13)(15,23)(16,24)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $24$ | $5$ | $( 1, 3, 9, 5,13)( 2, 4,10, 6,14)( 7,15,22,18,25)( 8,16,21,17,26) (11,20,28,30,23)(12,19,27,29,24)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $20$ | $3$ | $( 1, 3,28)( 2, 4,27)( 5,16,21)( 6,15,22)( 7,14,29)( 8,13,30)( 9,17,20) (10,18,19)(11,26,23)(12,25,24)$ | |
$ 6, 6, 6, 6, 6 $ | $20$ | $6$ | $( 1, 4,23,22,16,12)( 2, 3,24,21,15,11)( 5,18,30,27,20, 7)( 6,17,29,28,19, 8) ( 9,14,26,10,13,25)$ |
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: | no | magma: IsSolvable(G);
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Nilpotency class: | not nilpotent | ||
Label: | 120.34 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 3A | 4A | 5A | 6A | ||
Size | 1 | 10 | 15 | 20 | 30 | 24 | 20 | |
2 P | 1A | 1A | 1A | 3A | 2B | 5A | 3A | |
3 P | 1A | 2A | 2B | 1A | 4A | 5A | 2A | |
5 P | 1A | 2A | 2B | 3A | 4A | 1A | 6A | |
Type | ||||||||
120.34.1a | R | |||||||
120.34.1b | R | |||||||
120.34.4a | R | |||||||
120.34.4b | R | |||||||
120.34.5a | R | |||||||
120.34.5b | R | |||||||
120.34.6a | R |
magma: CharacterTable(G);