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Magma
magma: G := TransitiveGroup(16, 5);
Group action invariants
Degree $n$: | $16$ | magma: t, n := TransitiveGroupIdentification(G); n;
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Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
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Group: | $C_8\times C_2$ | ||
Parity: | $1$ | magma: IsEven(G);
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Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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$\card{\Aut(F/K)}$: | $16$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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Generators: | (1,4,5,8,9,11,13,16)(2,3,6,7,10,12,14,15), (1,15,13,12,9,7,5,3)(2,16,14,11,10,8,6,4) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_4$ x 2, $C_2^2$ $8$: $C_8$ x 2, $C_4\times C_2$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 8: $C_8$ x 2, $C_4\times C_2$
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 | Representative |
$ 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 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 3, 5, 7, 9,12,13,15)( 2, 4, 6, 8,10,11,14,16)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 4, 5, 8, 9,11,13,16)( 2, 3, 6, 7,10,12,14,15)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,12,15)( 4, 8,11,16)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,12,16)( 4, 7,11,15)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 7,13, 3, 9,15, 5,12)( 2, 8,14, 4,10,16, 6,11)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1, 8,13, 4, 9,16, 5,11)( 2, 7,14, 3,10,15, 6,12)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$ | |
$ 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1,11, 5,16, 9, 4,13, 8)( 2,12, 6,15,10, 3,14, 7)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1,12, 5,15, 9, 3,13, 7)( 2,11, 6,16,10, 4,14, 8)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,13, 9, 5)( 2,14,10, 6)( 3,15,12, 7)( 4,16,11, 8)$ | |
$ 4, 4, 4, 4 $ | $1$ | $4$ | $( 1,14, 9, 6)( 2,13,10, 5)( 3,16,12, 8)( 4,15,11, 7)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1,15,13,12, 9, 7, 5, 3)( 2,16,14,11,10, 8, 6, 4)$ | |
$ 8, 8 $ | $1$ | $8$ | $( 1,16,13,11, 9, 8, 5, 4)( 2,15,14,12,10, 7, 6, 3)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $16=2^{4}$ | magma: Order(G);
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Cyclic: | no | 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: | 16.5 | magma: IdentifyGroup(G);
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Character table: |
1A | 2A | 2B | 2C | 4A1 | 4A-1 | 4B1 | 4B-1 | 8A1 | 8A-1 | 8A3 | 8A-3 | 8B1 | 8B-1 | 8B3 | 8B-3 | ||
Size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |
2 P | 1A | 1A | 1A | 1A | 2A | 2A | 2A | 2A | 4A1 | 4A1 | 4A-1 | 4A-1 | 4A1 | 4A1 | 4A-1 | 4A-1 | |
Type | |||||||||||||||||
16.5.1a | R | ||||||||||||||||
16.5.1b | R | ||||||||||||||||
16.5.1c | R | ||||||||||||||||
16.5.1d | R | ||||||||||||||||
16.5.1e1 | C | ||||||||||||||||
16.5.1e2 | C | ||||||||||||||||
16.5.1f1 | C | ||||||||||||||||
16.5.1f2 | C | ||||||||||||||||
16.5.1g1 | C | ||||||||||||||||
16.5.1g2 | C | ||||||||||||||||
16.5.1g3 | C | ||||||||||||||||
16.5.1g4 | C | ||||||||||||||||
16.5.1h1 | C | ||||||||||||||||
16.5.1h2 | C | ||||||||||||||||
16.5.1h3 | C | ||||||||||||||||
16.5.1h4 | C |
magma: CharacterTable(G);