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Properties

Label 16T33
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 33);
 

Group invariants

Abstract group:  $C_2^3:C_4$
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:  $3$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,11,6,8)(2,12,5,7)(3,10,16,13)(4,9,15,14)$, $(1,9,3,7)(2,10,4,8)(5,13,15,11)(6,14,16,12)$
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, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $C_2^3 : C_4 $, $C_2^3: C_4$

Low degree siblings

8T19 x 2, 8T20, 8T21, 16T33, 16T52, 16T53, 32T19

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

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 4A 4B1 4B-1 4C1 4C-1
Size 1 1 2 2 2 4 4 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2D 2D 2C 2A 2C
Type
32.6.1a R 1 1 1 1 1 1 1 1 1 1 1
32.6.1b R 1 1 1 1 1 −1 −1 −1 −1 1 1
32.6.1c R 1 1 1 1 1 −1 −1 1 1 −1 −1
32.6.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1
32.6.1e1 C 1 1 1 −1 −1 −1 1 −i i i −i
32.6.1e2 C 1 1 1 −1 −1 −1 1 i −i −i i
32.6.1f1 C 1 1 1 −1 −1 1 −1 −i i −i i
32.6.1f2 C 1 1 1 −1 −1 1 −1 i −i i −i
32.6.2a R 2 2 −2 −2 2 0 0 0 0 0 0
32.6.2b R 2 2 −2 2 −2 0 0 0 0 0 0
32.6.4a R 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed