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Properties

Label 16T675
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.D_8$

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Show commands: Magma

magma: G := TransitiveGroup(16, 675);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $675$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4.D_8$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,2,7)(3,5,4,6)(9,11,10,12)(13,14)(15,16), (1,16,7,11,3,14,6,9,2,15,8,12,4,13,5,10)
magma: Generators(G);
 

Low degree resolvents

|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, $C_4\times C_2$
$16$:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
$32$:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26
$64$:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T163
$128$:  16T330

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $D_{8}$

Low degree siblings

16T675 x 3, 32T3277 x 2, 32T3278 x 4, 32T5767 x 4, 32T7554 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $256=2^{8}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $5$
Label:  256.384
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);