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Magma
magma: G := TransitiveGroup(30, 3);
Group action invariants
Degree $n$: | $30$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $D_{15}$ | ||
Parity: | $-1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $30$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,27)(2,28)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,17)(13,16)(14,15)(29,30), (1,10,18,26,4,12,20,28,6,14,22,29,8,16,24)(2,9,17,25,3,11,19,27,5,13,21,30,7,15,23) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $10$: $D_{5}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 3: $S_3$
Degree 5: $D_{5}$
Degree 6: $S_3$
Degree 10: $D_5$
Degree 15: $D_{15}$
Low degree siblings
15T2Siblings 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, 2, 2, 2 $ | $15$ | $2$ | $( 1, 2)( 3,29)( 4,30)( 5,28)( 6,27)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21) (13,20)(14,19)(15,18)(16,17)$ | |
$ 15, 15 $ | $2$ | $15$ | $( 1, 4, 6, 8,10,12,14,16,18,20,22,24,26,28,29)( 2, 3, 5, 7, 9,11,13,15,17,19, 21,23,25,27,30)$ | |
$ 15, 15 $ | $2$ | $15$ | $( 1, 6,10,14,18,22,26,29, 4, 8,12,16,20,24,28)( 2, 5, 9,13,17,21,25,30, 3, 7, 11,15,19,23,27)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1, 8,14,20,26)( 2, 7,13,19,25)( 3, 9,15,21,27)( 4,10,16,22,28) ( 5,11,17,23,30)( 6,12,18,24,29)$ | |
$ 15, 15 $ | $2$ | $15$ | $( 1,10,18,26, 4,12,20,28, 6,14,22,29, 8,16,24)( 2, 9,17,25, 3,11,19,27, 5,13, 21,30, 7,15,23)$ | |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1,12,22)( 2,11,21)( 3,13,23)( 4,14,24)( 5,15,25)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,19,30)(10,20,29)$ | |
$ 5, 5, 5, 5, 5, 5 $ | $2$ | $5$ | $( 1,14,26, 8,20)( 2,13,25, 7,19)( 3,15,27, 9,21)( 4,16,28,10,22) ( 5,17,30,11,23)( 6,18,29,12,24)$ | |
$ 15, 15 $ | $2$ | $15$ | $( 1,16,29,14,28,12,26,10,24, 8,22, 6,20, 4,18)( 2,15,30,13,27,11,25, 9,23, 7, 21, 5,19, 3,17)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $30=2 \cdot 3 \cdot 5$ | 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: | 30.3 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 3A | 5A1 | 5A2 | 15A1 | 15A2 | 15A4 | 15A7 | ||
Size | 1 | 15 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
2 P | 1A | 1A | 3A | 5A2 | 5A1 | 15A2 | 15A4 | 15A7 | 15A1 | |
3 P | 1A | 2A | 1A | 5A2 | 5A1 | 5A1 | 5A2 | 5A1 | 5A2 | |
5 P | 1A | 2A | 3A | 1A | 1A | 3A | 3A | 3A | 3A | |
Type | ||||||||||
30.3.1a | R | |||||||||
30.3.1b | R | |||||||||
30.3.2a | R | |||||||||
30.3.2b1 | R | |||||||||
30.3.2b2 | R | |||||||||
30.3.2c1 | R | |||||||||
30.3.2c2 | R | |||||||||
30.3.2c3 | R | |||||||||
30.3.2c4 | R |
magma: CharacterTable(G);