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Properties

Label 16T509
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4^2:C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 509);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $509$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4^2:C_2^2$
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:  (1,15)(2,16)(3,13)(4,14)(5,11)(6,12)(7,10)(8,9), (7,16)(8,15), (1,10)(2,9)(3,4)(5,6)(7,8)(11,12)(13,14)(15,16), (1,5)(2,6)(9,14)(10,13)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $D_{4}$ x 12, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 18, $C_2^4$
$32$:  $C_2^2 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T105
$128$:  $C_2 \wr C_2\wr C_2$ x 2, 16T245

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$, $C_2 \wr C_2\wr C_2$ x 2

Low degree siblings

16T509 x 31, 16T602 x 32, 32T2525 x 8, 32T2526 x 8, 32T2527 x 16, 32T2528 x 16, 32T2529 x 16, 32T2530 x 8, 32T2531 x 16, 32T2953 x 8, 32T2954 x 8, 32T2955 x 8, 32T4749 x 16, 32T6221 x 8, 32T6319 x 8

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

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:  $4$
Label:  256.26531
magma: IdentifyGroup(G);
 
Character table:    40 x 40 character table

magma: CharacterTable(G);