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Properties

Label 16T211
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 211);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $211$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,6)(2,5)(3,8)(4,7)(9,13)(10,14)(11,16)(12,15), (1,9,2,10)(3,11,4,12)(5,13,6,14)(7,16,8,15), (1,4,6,7)(2,3,5,8)(9,14)(10,13)(11,15)(12,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
$32$:  $C_4\wr C_2$ x 4, $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  16T111 x 2, 32T239

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_4\wr C_2$ x 2, $C_2^2 \wr C_2$

Low degree siblings

16T211 x 15, 32T461 x 4, 32T462 x 8, 32T1895 x 4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $128=2^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  128.628
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);