Properties

 Label 16T5 Degree $16$ Order $16$ Cyclic no Abelian yes Solvable yes Primitive no $p$-group yes Group: $C_8\times C_2$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(16, 5);

Group action invariants

 Degree $n$: $16$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_8\times C_2$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $16$ magma: Order(Centralizer(SymmetricGroup(n), G)); 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);

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 4: $C_4$ x 2, $C_2^2$

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 Index Representative 1A $1^{16}$ $1$ $1$ $0$ $()$ 2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$ 2B $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$ 2C $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$ 4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 5, 9,13)( 2, 6,10,14)( 3, 7,12,15)( 4, 8,11,16)$ 4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,13, 9, 5)( 2,14,10, 6)( 3,15,12, 7)( 4,16,11, 8)$ 4B1 $4^{4}$ $1$ $4$ $12$ $( 1,14, 9, 6)( 2,13,10, 5)( 3,16,12, 8)( 4,15,11, 7)$ 4B-1 $4^{4}$ $1$ $4$ $12$ $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,12,16)( 4, 7,11,15)$ 8A1 $8^{2}$ $1$ $8$ $14$ $( 1,11, 5,16, 9, 4,13, 8)( 2,12, 6,15,10, 3,14, 7)$ 8A-1 $8^{2}$ $1$ $8$ $14$ $( 1, 4, 5, 8, 9,11,13,16)( 2, 3, 6, 7,10,12,14,15)$ 8A3 $8^{2}$ $1$ $8$ $14$ $( 1, 7,13, 3, 9,15, 5,12)( 2, 8,14, 4,10,16, 6,11)$ 8A-3 $8^{2}$ $1$ $8$ $14$ $( 1, 3, 5, 7, 9,12,13,15)( 2, 4, 6, 8,10,11,14,16)$ 8B1 $8^{2}$ $1$ $8$ $14$ $( 1,15,13,12, 9, 7, 5, 3)( 2,16,14,11,10, 8, 6, 4)$ 8B-1 $8^{2}$ $1$ $8$ $14$ $( 1,16,13,11, 9, 8, 5, 4)( 2,15,14,12,10, 7, 6, 3)$ 8B3 $8^{2}$ $1$ $8$ $14$ $( 1,12, 5,15, 9, 3,13, 7)( 2,11, 6,16,10, 4,14, 8)$ 8B-3 $8^{2}$ $1$ $8$ $14$ $( 1, 8,13, 4, 9,16, 5,11)( 2, 7,14, 3,10,15, 6,12)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/8$

Group invariants

 Order: $16=2^{4}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 16.5 magma: IdentifyGroup(G); 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 2C 2C 2C 2C 4A1 4A1 4A-1 4A1 4A-1 4A-1 4A1 4A-1 Type 16.5.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 16.5.1b R $1$ $1$ $−1$ $−1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $1$ $1$ 16.5.1c R $1$ $1$ $−1$ $−1$ $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ 16.5.1d R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ $−1$ 16.5.1e1 C $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $−i$ $i$ $i$ $−i$ $−i$ $i$ $i$ $−i$ 16.5.1e2 C $1$ $1$ $−1$ $−1$ $−1$ $−1$ $1$ $1$ $i$ $−i$ $−i$ $i$ $i$ $−i$ $−i$ $i$ 16.5.1f1 C $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $−i$ $i$ $i$ $−i$ $i$ $−i$ $−i$ $i$ 16.5.1f2 C $1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ $−1$ $i$ $−i$ $−i$ $i$ $−i$ $i$ $i$ $−i$ 16.5.1g1 C $1$ $−1$ $−1$ $1$ $−ζ82$ $ζ82$ $−ζ82$ $ζ82$ $ζ83$ $−ζ8$ $ζ8$ $−ζ83$ $ζ8$ $−ζ83$ $ζ83$ $−ζ8$ 16.5.1g2 C $1$ $−1$ $−1$ $1$ $ζ82$ $−ζ82$ $ζ82$ $−ζ82$ $−ζ8$ $ζ83$ $−ζ83$ $ζ8$ $−ζ83$ $ζ8$ $−ζ8$ $ζ83$ 16.5.1g3 C $1$ $−1$ $−1$ $1$ $−ζ82$ $ζ82$ $−ζ82$ $ζ82$ $−ζ83$ $ζ8$ $−ζ8$ $ζ83$ $−ζ8$ $ζ83$ $−ζ83$ $ζ8$ 16.5.1g4 C $1$ $−1$ $−1$ $1$ $ζ82$ $−ζ82$ $ζ82$ $−ζ82$ $ζ8$ $−ζ83$ $ζ83$ $−ζ8$ $ζ83$ $−ζ8$ $ζ8$ $−ζ83$ 16.5.1h1 C $1$ $−1$ $1$ $−1$ $−ζ82$ $ζ82$ $ζ82$ $−ζ82$ $ζ83$ $−ζ8$ $ζ8$ $−ζ83$ $−ζ8$ $ζ83$ $−ζ83$ $ζ8$ 16.5.1h2 C $1$ $−1$ $1$ $−1$ $ζ82$ $−ζ82$ $−ζ82$ $ζ82$ $−ζ8$ $ζ83$ $−ζ83$ $ζ8$ $ζ83$ $−ζ8$ $ζ8$ $−ζ83$ 16.5.1h3 C $1$ $−1$ $1$ $−1$ $−ζ82$ $ζ82$ $ζ82$ $−ζ82$ $−ζ83$ $ζ8$ $−ζ8$ $ζ83$ $ζ8$ $−ζ83$ $ζ83$ $−ζ8$ 16.5.1h4 C $1$ $−1$ $1$ $−1$ $ζ82$ $−ζ82$ $−ζ82$ $ζ82$ $ζ8$ $−ζ83$ $ζ83$ $−ζ8$ $−ζ83$ $ζ8$ $−ζ8$ $ζ83$

magma: CharacterTable(G);