Show commands:
Magma
magma: G := TransitiveGroup(15, 1);
Group action invariants
Degree $n$: | $15$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $1$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_{15}$ | ||
CHM label: | $C(15)=5[x]3$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $15$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) | magma: Generators(G);
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Low degree resolvents
$\card{(G/N)}$ Galois groups for stem field(s) $3$: $C_3$ $5$: $C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $C_3$
Degree 5: $C_5$
Low degree siblings
There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Label | Cycle Type | Size | Order | Index | Representative |
1A | $1^{15}$ | $1$ | $1$ | $0$ | $()$ |
3A1 | $3^{5}$ | $1$ | $3$ | $10$ | $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ |
3A-1 | $3^{5}$ | $1$ | $3$ | $10$ | $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ |
5A1 | $5^{3}$ | $1$ | $5$ | $12$ | $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ |
5A-1 | $5^{3}$ | $1$ | $5$ | $12$ | $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ |
5A2 | $5^{3}$ | $1$ | $5$ | $12$ | $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ |
5A-2 | $5^{3}$ | $1$ | $5$ | $12$ | $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ |
15A1 | $15$ | $1$ | $15$ | $14$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$ |
15A-1 | $15$ | $1$ | $15$ | $14$ | $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ |
15A2 | $15$ | $1$ | $15$ | $14$ | $( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$ |
15A-2 | $15$ | $1$ | $15$ | $14$ | $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$ |
15A4 | $15$ | $1$ | $15$ | $14$ | $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$ |
15A-4 | $15$ | $1$ | $15$ | $14$ | $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$ |
15A7 | $15$ | $1$ | $15$ | $14$ | $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$ |
15A-7 | $15$ | $1$ | $15$ | $14$ | $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$ |
Malle's constant $a(G)$: $1/10$
magma: ConjugacyClasses(G);
Group invariants
Order: | $15=3 \cdot 5$ | magma: Order(G);
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Cyclic: | yes | magma: IsCyclic(G);
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Abelian: | yes | magma: IsAbelian(G);
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Solvable: | yes | magma: IsSolvable(G);
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Nilpotency class: | $1$ | ||
Label: | 15.1 | magma: IdentifyGroup(G);
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Character table: |
1A | 3A1 | 3A-1 | 5A1 | 5A-1 | 5A2 | 5A-2 | 15A1 | 15A-1 | 15A2 | 15A-2 | 15A4 | 15A-4 | 15A7 | 15A-7 | ||
Size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
3 P | 1A | 3A-1 | 3A1 | 5A-1 | 5A-2 | 5A1 | 5A2 | 15A2 | 15A-2 | 15A4 | 15A-4 | 15A-7 | 15A-1 | 15A1 | 15A7 | |
5 P | 1A | 1A | 1A | 5A1 | 5A2 | 5A-1 | 5A-2 | 5A1 | 5A-1 | 5A2 | 5A-2 | 5A-1 | 5A2 | 5A-2 | 5A1 | |
Type | ||||||||||||||||
15.1.1a | R | |||||||||||||||
15.1.1b1 | C | |||||||||||||||
15.1.1b2 | C | |||||||||||||||
15.1.1c1 | C | |||||||||||||||
15.1.1c2 | C | |||||||||||||||
15.1.1c3 | C | |||||||||||||||
15.1.1c4 | C | |||||||||||||||
15.1.1d1 | C | |||||||||||||||
15.1.1d2 | C | |||||||||||||||
15.1.1d3 | C | |||||||||||||||
15.1.1d4 | C | |||||||||||||||
15.1.1d5 | C | |||||||||||||||
15.1.1d6 | C | |||||||||||||||
15.1.1d7 | C | |||||||||||||||
15.1.1d8 | C |
magma: CharacterTable(G);