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Properties

Label 16T867
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4.(C_4\times D_4)$

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Show commands: Magma

magma: G := TransitiveGroup(16, 867);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $867$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4.(C_4\times D_4)$
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:  (1,13,3,15)(2,14,4,16)(5,12,7,9)(6,11,8,10), (1,12,14,6)(2,11,13,5)(3,9,15,7)(4,10,16,8), (5,12,7,10)(6,11,8,9)(13,15)(14,16)
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, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239
$128$:  16T208, 16T218, 16T230
$256$:  32T3729

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T824 x 4, 16T867 x 3, 16T915 x 2, 16T926 x 2, 32T9932 x 4, 32T9933 x 2, 32T9934 x 2, 32T10284 x 4, 32T10285 x 2, 32T10286 x 2, 32T10548 x 2, 32T10549 x 2, 32T10550, 32T10551, 32T10612, 32T10613 x 2, 32T10614, 32T19547, 32T19551, 32T19563, 32T19568, 32T20024, 32T20025, 32T20090, 32T20093, 32T25962 x 2, 32T26009 x 2, 32T26480 x 2, 32T28811, 32T33434 x 2, 32T33768, 32T33770

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $512=2^{9}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $4$
Label:  512.46939
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);