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Properties

Label 16T1775
Degree $16$
Order $16384$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\wr D_4.D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1775);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1775$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\wr D_4.D_4$
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:  (1,14,2,13)(3,4)(5,9)(6,10)(7,15,8,16), (15,16), (1,3,6,7,9,11,13,15,2,4,5,8,10,12,14,16), (1,10,2,9)(7,16,8,15)
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 20, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 30, $C_2^4$
$32$:  $C_2^2 \wr C_2$ x 8, $Q_8:C_2^2$ x 2, $C_2^2 \times D_4$ x 5
$64$:  $(C_4^2 : C_2):C_2$ x 4, $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T87, 16T105 x 2, 16T109 x 4
$128$:  $C_2 \wr C_2\wr C_2$ x 4, 16T245 x 2, 16T265 x 2, 32T1237
$256$:  16T477 x 2, 16T509 x 2, 16T511, 16T531, 16T538
$512$:  32T12264 x 2, 32T12969
$1024$:  16T1177
$2048$:  32T128074
$4096$:  16T1556
$8192$:  32T519843

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(C_4^2 : C_2):C_2$

Low degree siblings

16T1775 x 15, 16T1779 x 16, 32T723864 x 8, 32T723865 x 8, 32T723866 x 8, 32T723867 x 8, 32T723868 x 8, 32T723869 x 8, 32T723870 x 8, 32T723871 x 8, 32T723872 x 8, 32T723873 x 8, 32T723874 x 8, 32T723875 x 16, 32T723876 x 8, 32T723877 x 8, 32T723878 x 8, 32T723879 x 8, 32T723880 x 8, 32T723881 x 8, 32T723882 x 8, 32T723883 x 8, 32T723884 x 8, 32T723885 x 8, 32T723886 x 8, 32T723887 x 8, 32T723888 x 8, 32T723889 x 8, 32T723890 x 8, 32T723891 x 8, 32T723892 x 8, 32T723893 x 8, 32T723894 x 8, 32T723986 x 8, 32T723987 x 8, 32T723988 x 8, 32T723989 x 8, 32T723990 x 8, 32T723991 x 8, 32T723992 x 8, 32T723993 x 8, 32T723994 x 8, 32T723995 x 8, 32T723996 x 8, 32T723997 x 8, 32T723998 x 8, 32T723999 x 8, 32T724000 x 8, 32T724001 x 8, 32T724002 x 8, 32T724003 x 8, 32T724004 x 8, 32T724005 x 8, 32T724006 x 8, 32T724007 x 8, 32T724008 x 8, 32T724009 x 8, 32T724010 x 8, 32T724011 x 8, 32T724012 x 8, 32T724013 x 8, 32T724014 x 8, 32T724015 x 8, 32T727404 x 8, 32T728417 x 8, 32T743356 x 4, 32T743433 x 4, 32T743528 x 4, 32T743568 x 4, 32T743678 x 4, 32T743789 x 4, 32T862501 x 4, 32T862596 x 4, 32T873083 x 4, 32T873832 x 4, 32T968027 x 4, 32T968141 x 4, 32T1037995 x 4, 32T1038046 x 4, 32T1061662 x 4, 32T1061665 x 4, 32T1099771 x 4, 32T1099774 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

The 136 conjugacy class representatives for $C_2\wr D_4.D_4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $16384=2^{14}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $8$
Label:  16384.iu
magma: IdentifyGroup(G);
 
Character table:    136 x 136 character table

magma: CharacterTable(G);