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Properties

Label 16T595
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, 595);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $595$
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:  (1,2)(3,4), (9,10)(11,12)(13,14)(15,16), (1,10)(2,9)(3,12)(4,11)(5,15)(6,16)(7,13)(8,14), (1,5)(2,6)(3,7)(4,8)(9,16)(10,15)(11,14)(12,13), (1,12,2,11)(3,9,4,10)
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
$128$:  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: $D_4\times C_2$

Low degree siblings

16T484 x 4, 16T595 x 7, 32T2414 x 4, 32T2415 x 2, 32T2416 x 2, 32T2417 x 4, 32T2418 x 4, 32T2914 x 4, 32T2915 x 4, 32T2916 x 4, 32T2917 x 4, 32T2918 x 4, 32T2919 x 4, 32T4761 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 $ $4$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 5,13, 6,14)( 7,16, 8,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 5,13)( 6,14)( 7,16)( 8,15)( 9,10)(11,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 5,13, 6,14)( 7,16, 8,15)( 9,10)(11,12)$
$ 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,16)( 8,15)( 9,10)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 2)( 3, 4)( 5,13, 6,14)( 7,16, 8,15)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 3, 2, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,11,10,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,13, 8,14)( 9,11,10,12)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3, 2, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,13, 8,14)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 4, 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 4, 2, 3)( 5,15, 6,16)( 7,13, 8,14)( 9,12,10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5,11,13)( 2, 6,12,14)( 3, 7,10,16)( 4, 8, 9,15)$
$ 8, 8 $ $16$ $8$ $( 1, 5,11,13, 2, 6,12,14)( 3, 7,10,16, 4, 8, 9,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,13,10,14)(11,16,12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13)(10,14)(11,16)(12,15)$
$ 8, 8 $ $16$ $8$ $( 1, 7,11,16, 2, 8,12,15)( 3, 6,10,14, 4, 5, 9,13)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 7,11,16)( 2, 8,12,15)( 3, 6,10,14)( 4, 5, 9,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15)( 6,16)( 7,13)( 8,14)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15, 6,16)( 7,13, 8,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,15, 6,16)( 7,13, 8,14)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,13, 6,14)( 7,16, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)$

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:  256.26534
magma: IdentifyGroup(G);
 
Character table:    40 x 40 character table

magma: CharacterTable(G);