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Properties

Label 16T1251
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.C_2\wr C_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1251);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1251$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4.C_2\wr C_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,7,11,4,15,5,9,2,13,8,12,3,16,6,10), (1,7)(2,8)(3,6,4,5)(9,13,11,15)(10,14,12,16)
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
$256$:  16T682
$512$:  16T944

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $((C_8 : C_2):C_2):C_2$

Low degree siblings

16T1251 x 7, 32T36617 x 4, 32T36618 x 8, 32T36619 x 4, 32T36620 x 4, 32T36621 x 16, 32T36622 x 4, 32T36623 x 4, 32T36624 x 4, 32T36625 x 8, 32T36626 x 4, 32T57005 x 4, 32T57008 x 4, 32T70806 x 2, 32T70840 x 2, 32T71345, 32T71348

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 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, 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 $ $4$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)( 9,12)(10,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 7, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $32$ $4$ $( 3, 4)( 5, 7)( 6, 8)(13,16,14,15)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 2)( 5, 8)( 6, 7)( 9,10)(11,12)(13,15,14,16)$
$ 8, 8 $ $32$ $8$ $( 1, 7, 4, 5, 2, 8, 3, 6)( 9,13,12,16,10,14,11,15)$
$ 8, 8 $ $32$ $8$ $( 1, 7, 4, 5, 2, 8, 3, 6)( 9,14,11,16,10,13,12,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,16,10,15)(11,13,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,15)(10,16)(11,14)(12,13)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,15)(10,16)(11,14)(12,13)$
$ 16 $ $64$ $16$ $( 1,14, 7,11, 4,15, 5, 9, 2,13, 8,12, 3,16, 6,10)$
$ 16 $ $64$ $16$ $( 1,13, 8,12, 3,15, 5, 9, 2,14, 7,11, 4,16, 6,10)$
$ 16 $ $64$ $16$ $( 1,11, 5,13, 3,10, 7,15, 2,12, 6,14, 4, 9, 8,16)$
$ 16 $ $64$ $16$ $( 1,11, 5,14, 4, 9, 8,15, 2,12, 6,13, 3,10, 7,16)$
$ 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $16$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 9,10)(11,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 1, 2)( 5, 6)( 7, 8)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,14)$
$ 4, 4, 2, 2, 2, 1, 1 $ $32$ $4$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(15,16)$
$ 4, 4, 4, 2, 2 $ $64$ $4$ $( 1, 7, 4, 6)( 2, 8, 3, 5)( 9,13)(10,14)(11,15,12,16)$
$ 4, 4, 4, 2, 2 $ $64$ $4$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,15)(10,16)(11,14,12,13)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,14, 6,10)( 2,13, 5, 9)( 3,15, 8,12)( 4,16, 7,11)$
$ 8, 8 $ $64$ $8$ $( 1,13, 5, 9, 2,14, 6,10)( 3,16, 7,11, 4,15, 8,12)$
$ 8, 8 $ $64$ $8$ $( 1,11, 5,13, 2,12, 6,14)( 3, 9, 8,16, 4,10, 7,15)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,11, 5,14)( 2,12, 6,13)( 3, 9, 8,15)( 4,10, 7,16)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1024=2^{10}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $7$
Label:  1024.dhh
magma: IdentifyGroup(G);
 
Character table:    34 x 34 character table

magma: CharacterTable(G);