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Properties

Label 16T8
Degree $16$
Order $16$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 8);
 

Group invariants

Abstract group:  $C_4:C_4$
magma: IdentifyGroup(G);
 
Order:  $16=2^{4}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $16$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,15,6,3)(2,16,5,4)(7,14,11,9)(8,13,12,10)$, $(1,8,6,12)(2,7,5,11)(3,9,15,14)(4,10,16,13)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2

Degree 8: $C_4\times C_2$, $D_4$, $Q_8$

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 TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,13)(10,14)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,11)( 8,12)( 9,14)(10,13)$
4A $4^{4}$ $2$ $4$ $12$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$
4B $4^{4}$ $2$ $4$ $12$ $( 1,13, 2,14)( 3, 7, 4, 8)( 5, 9, 6,10)(11,16,12,15)$
4C1 $4^{4}$ $2$ $4$ $12$ $( 1,15, 6, 3)( 2,16, 5, 4)( 7,14,11, 9)( 8,13,12,10)$
4C-1 $4^{4}$ $2$ $4$ $12$ $( 1, 3, 6,15)( 2, 4, 5,16)( 7, 9,11,14)( 8,10,12,13)$
4D1 $4^{4}$ $2$ $4$ $12$ $( 1, 7, 6,11)( 2, 8, 5,12)( 3,10,15,13)( 4, 9,16,14)$
4D-1 $4^{4}$ $2$ $4$ $12$ $( 1,11, 6, 7)( 2,12, 5, 8)( 3,13,15,10)( 4,14,16, 9)$

Malle's constant $a(G)$:     $1/8$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 4B 4C1 4C-1 4D1 4D-1
Size 1 1 1 1 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2B 2B 2C 2C 2C 2C
Type
16.4.1a R 1 1 1 1 1 1 1 1 1 1
16.4.1b R 1 1 1 1 −1 −1 1 1 −1 −1
16.4.1c R 1 1 1 1 −1 1 −1 −1 −1 1
16.4.1d R 1 1 1 1 1 −1 −1 −1 1 −1
16.4.1e1 C 1 −1 1 −1 −1 −i −i i 1 i
16.4.1e2 C 1 −1 1 −1 −1 i i −i 1 −i
16.4.1f1 C 1 −1 1 −1 1 −i i −i −1 i
16.4.1f2 C 1 −1 1 −1 1 i −i i −1 −i
16.4.2a R 2 −2 −2 2 0 0 0 0 0 0
16.4.2b S 2 2 −2 −2 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed