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Properties

Label 16T27
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 27);
 

Group invariants

Abstract group:  $C_4^2:C_2$
magma: IdentifyGroup(G);
 
Order:  $32=2^{5}$
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 $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,5,9,14)(2,6,10,13)(3,16,12,7)(4,15,11,8)$, $(1,2)(3,12)(4,11)(7,15)(8,16)(9,10)$, $(1,4,2,3)(5,8,6,7)(9,11,10,12)(13,16,14,15)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$
$16$:  $Q_8:C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$ x 3

Low degree siblings

32T13

Siblings are shown 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 $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
2D $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3,12)( 4,11)( 7,15)( 8,16)( 9,10)$
4A1 $4^{4}$ $2$ $4$ $12$ $( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,15)( 4, 8,12,16)$
4A-1 $4^{4}$ $2$ $4$ $12$ $( 1,14,10, 6)( 2,13, 9, 5)( 3,15,11, 7)( 4,16,12, 8)$
4B1 $4^{4}$ $2$ $4$ $12$ $( 1, 8, 2, 7)( 3,14, 4,13)( 5,11, 6,12)( 9,15,10,16)$
4B-1 $4^{4}$ $2$ $4$ $12$ $( 1,16, 2,15)( 3, 6, 4, 5)( 7,10, 8, 9)(11,14,12,13)$
4C1 $4^{4}$ $2$ $4$ $12$ $( 1,12, 9, 3)( 2,11,10, 4)( 5,15,14, 8)( 6,16,13, 7)$
4C-1 $4^{4}$ $2$ $4$ $12$ $( 1, 3, 9,12)( 2, 4,10,11)( 5, 8,14,15)( 6, 7,13,16)$
4D $4^{4}$ $4$ $4$ $12$ $( 1,13, 9, 6)( 2,14,10, 5)( 3, 8,12,15)( 4, 7,11,16)$
4E $4^{4}$ $4$ $4$ $12$ $( 1,15,10, 7)( 2,16, 9, 8)( 3,13,11, 5)( 4,14,12, 6)$
4F $4^{4}$ $4$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,15, 6,16)( 7,14, 8,13)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D 4E 4F
Size 1 1 1 1 4 2 2 2 2 2 2 4 4 4
2 P 1A 1A 1A 1A 1A 2C 2C 2A 2A 2B 2B 2B 2C 2A
Type
32.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.33.1b R 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1 1 1
32.33.1c R 1 1 1 1 −1 −1 −1 1 1 −1 −1 1 −1 1
32.33.1d R 1 1 1 1 −1 1 1 −1 −1 −1 −1 1 1 −1
32.33.1e R 1 1 1 1 −1 1 1 1 1 1 1 −1 −1 −1
32.33.1f R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1
32.33.1g R 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 1 −1
32.33.1h R 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1
32.33.2a1 C 2 −2 −2 2 0 0 0 −2i 2i 0 0 0 0 0
32.33.2a2 C 2 −2 −2 2 0 0 0 2i −2i 0 0 0 0 0
32.33.2b1 C 2 −2 2 −2 0 −2i 2i 0 0 0 0 0 0 0
32.33.2b2 C 2 −2 2 −2 0 2i −2i 0 0 0 0 0 0 0
32.33.2c1 C 2 2 −2 −2 0 0 0 0 0 −2i 2i 0 0 0
32.33.2c2 C 2 2 −2 −2 0 0 0 0 0 2i −2i 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed