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Properties

Label 16T3
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 3);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4$
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:  (1,6)(2,5)(3,16)(4,15)(7,12)(8,11)(9,14)(10,13), (1,10)(2,9)(3,11)(4,12)(5,14)(6,13)(7,15)(8,16), (1,15)(2,16)(3,5)(4,6)(7,10)(8,9)(11,14)(12,13), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15

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

LabelCycle TypeSizeOrderRepresentative
$ 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,11)( 8,12)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,12)( 8,11)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,11)( 6,12)( 9,16)(10,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 7)( 3,13)( 4,14)( 5,12)( 6,11)( 9,15)(10,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5, 9)( 6,10)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,14)( 2,13)( 3, 7)( 4, 8)( 5,10)( 6, 9)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,15)( 2,16)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)$

magma: ConjugacyClasses(G);
 

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:  16.14
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A
Type
16.14.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.14.1b R 1 −1 −1 −1 −1 1 −1 −1 1 1 1 1 −1 1 −1 1
16.14.1c R 1 −1 −1 −1 1 −1 −1 1 −1 1 −1 −1 1 1 1 1
16.14.1d R 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 1
16.14.1e R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1
16.14.1f R 1 −1 1 −1 −1 −1 1 1 1 −1 −1 1 1 1 −1 −1
16.14.1g R 1 −1 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1 1 −1
16.14.1h R 1 −1 1 1 −1 1 −1 1 −1 1 −1 1 −1 −1 1 −1
16.14.1i R 1 −1 1 1 1 −1 −1 −1 1 1 1 −1 1 −1 −1 −1
16.14.1j R 1 1 −1 −1 −1 −1 1 1 1 1 1 −1 −1 −1 1 −1
16.14.1k R 1 1 −1 −1 1 1 1 −1 −1 1 −1 1 1 −1 −1 −1
16.14.1l R 1 1 −1 1 −1 1 −1 1 −1 −1 1 −1 1 1 −1 −1
16.14.1m R 1 1 −1 1 1 −1 −1 −1 1 −1 −1 1 −1 1 1 −1
16.14.1n R 1 1 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1 1
16.14.1o R 1 1 1 −1 1 −1 −1 1 −1 −1 1 1 −1 −1 −1 1
16.14.1p R 1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 −1 1

magma: CharacterTable(G);