Show commands:
Magma
magma: G := TransitiveGroup(30, 19);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $19$ | 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: | (1,29,17,8,25,14,3,22,20,10,27,16,5,23,12)(2,30,18,7,26,13,4,21,19,9,28,15,6,24,11), (1,2)(3,9)(4,10)(5,7)(6,8)(11,29,12,30)(13,27,14,28)(15,25,16,26)(17,24,18,23)(19,22,20,21) | 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, 30T31, 40T63Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $3$ | $2$ | $(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ | $30$ | $2$ | $( 3,10)( 4, 9)( 5, 8)( 6, 7)(11,29)(12,30)(13,27)(14,28)(15,25)(16,26)(17,24) (18,23)(19,22)(20,21)$ | |
$ 4, 4, 4, 4, 4, 2, 2, 2, 2, 2 $ | $30$ | $4$ | $( 1, 2)( 3, 9)( 4,10)( 5, 7)( 6, 8)(11,29,12,30)(13,27,14,28)(15,25,16,26) (17,24,18,23)(19,22,20,21)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 3, 5, 8,10)( 2, 4, 6, 7, 9)(11,13,15,18,19)(12,14,16,17,20) (21,24,26,28,30)(22,23,25,27,29)$ | |
$ 10, 10, 5, 5 $ | $6$ | $10$ | $( 1, 3, 5, 8,10)( 2, 4, 6, 7, 9)(11,14,15,17,19,12,13,16,18,20) (21,23,26,27,30,22,24,25,28,29)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 5,10, 3, 8)( 2, 6, 9, 4, 7)(11,15,19,13,18)(12,16,20,14,17) (21,26,30,24,28)(22,25,29,23,27)$ | |
$ 10, 10, 5, 5 $ | $6$ | $10$ | $( 1, 5,10, 3, 8)( 2, 6, 9, 4, 7)(11,16,19,14,18,12,15,20,13,17) (21,25,30,23,28,22,26,29,24,27)$ | |
$ 15, 15 $ | $8$ | $15$ | $( 1,11,23, 5,15,27,10,19,22, 3,13,25, 8,18,29)( 2,12,24, 6,16,28, 9,20,21, 4, 14,26, 7,17,30)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $8$ | $3$ | $( 1,13,27)( 2,14,28)( 3,15,29)( 4,16,30)( 5,18,22)( 6,17,21)( 7,20,24) ( 8,19,23)( 9,12,26)(10,11,25)$ | |
$ 15, 15 $ | $8$ | $15$ | $( 1,15,22, 8,11,27, 3,18,23,10,13,29, 5,19,25)( 2,16,21, 7,12,28, 4,17,24, 9, 14,30, 6,20,26)$ | |
$ 15, 15 $ | $8$ | $15$ | $( 1,17,26, 3,20,28, 5,12,30, 8,14,21,10,16,24)( 2,18,25, 4,19,27, 6,11,29, 7, 13,22, 9,15,23)$ | |
$ 15, 15 $ | $8$ | $15$ | $( 1,19,30,10,18,28, 8,15,26, 5,13,24, 3,11,21)( 2,20,29, 9,17,27, 7,16,25, 6, 14,23, 4,12,22)$ |
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: |
1A | 2A | 2B | 3A | 4A | 5A1 | 5A2 | 10A1 | 10A3 | 15A1 | 15A2 | 15A4 | 15A7 | ||
Size | 1 | 3 | 30 | 8 | 30 | 2 | 2 | 6 | 6 | 8 | 8 | 8 | 8 | |
2 P | 1A | 1A | 1A | 3A | 2A | 5A2 | 5A1 | 5A2 | 5A1 | 15A7 | 15A1 | 15A2 | 15A4 | |
3 P | 1A | 2A | 2B | 1A | 4A | 5A2 | 5A1 | 10A3 | 10A1 | 5A1 | 5A2 | 5A1 | 5A2 | |
5 P | 1A | 2A | 2B | 3A | 4A | 1A | 1A | 2A | 2A | 3A | 3A | 3A | 3A | |
Type | ||||||||||||||
120.38.1a | R | |||||||||||||
120.38.1b | R | |||||||||||||
120.38.2a | R | |||||||||||||
120.38.2b1 | R | |||||||||||||
120.38.2b2 | R | |||||||||||||
120.38.2c1 | R | |||||||||||||
120.38.2c2 | R | |||||||||||||
120.38.2c3 | R | |||||||||||||
120.38.2c4 | R | |||||||||||||
120.38.3a | R | |||||||||||||
120.38.3b | R | |||||||||||||
120.38.6a1 | R | |||||||||||||
120.38.6a2 | R |
magma: CharacterTable(G);