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Properties

Label 16T4
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 4);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2$
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:  (1,7,5,11)(2,8,6,12)(3,9,15,13)(4,10,16,14), (1,13,2,14)(3,8,4,7)(5,9,6,10)(11,15,12,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 6, $C_2^2$
$8$:  $C_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 6, $C_2^2$

Degree 8: $C_4\times C_2$ x 3

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

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:  16.2
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1 4E1 4E-1 4F1 4F-1
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 2B 2A 2A 2C 2C 2C 2A 2B 2A 2C 2B 2B
Type
16.2.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.2.1b R 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 1 1
16.2.1c R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 −1 −1
16.2.1d R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1
16.2.1e1 C 1 −1 −1 1 −i i i −i −1 −1 i −i i −i 1 1
16.2.1e2 C 1 −1 −1 1 i −i −i i −1 −1 −i i −i i 1 1
16.2.1f1 C 1 −1 −1 1 −i i −i i 1 1 i −i −i i −1 −1
16.2.1f2 C 1 −1 −1 1 i −i i −i 1 1 −i i i −i −1 −1
16.2.1g1 C 1 −1 1 −1 −i i −1 −1 i −i −i i 1 1 i −i
16.2.1g2 C 1 −1 1 −1 i −i −1 −1 −i i i −i 1 1 −i i
16.2.1h1 C 1 −1 1 −1 −i i 1 1 −i i −i i −1 −1 −i i
16.2.1h2 C 1 −1 1 −1 i −i 1 1 i −i i −i −1 −1 i −i
16.2.1i1 C 1 1 −1 −1 −1 −1 −i i −i i 1 1 i −i i −i
16.2.1i2 C 1 1 −1 −1 −1 −1 i −i i −i 1 1 −i i −i i
16.2.1j1 C 1 1 −1 −1 1 1 −i i i −i −1 −1 i −i −i i
16.2.1j2 C 1 1 −1 −1 1 1 i −i −i i −1 −1 −i i i −i

magma: CharacterTable(G);