Properties

Label 32T1369
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^4:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 1369);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1369$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $16$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,28)(2,27)(3,26)(4,25)(5,14)(6,13)(7,16)(8,15)(9,20)(10,19)(11,18)(12,17)(21,32)(22,31)(23,30)(24,29), (1,21)(2,22)(3,23)(4,24)(5,17)(6,18)(7,19)(8,20)(9,15)(10,16)(11,13)(12,14)(25,29)(26,30)(27,31)(28,32), (1,16,9,22)(2,15,10,21)(3,14,11,24)(4,13,12,23)(5,26,29,18)(6,25,30,17)(7,28,31,20)(8,27,32,19), (13,19)(14,20)(15,17)(16,18)(21,25)(22,26)(23,27)(24,28), (1,24,5,20)(2,23,6,19)(3,22,7,18)(4,21,8,17)(9,14,29,28)(10,13,30,27)(11,16,31,26)(12,15,32,25)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 31
$4$:  $C_2^2$ x 155
$8$:  $D_{4}$ x 24, $C_2^3$ x 155
$16$:  $D_4\times C_2$ x 84, $C_2^4$ x 31
$32$:  $C_2^2 \wr C_2$ x 16, $C_2^2 \times D_4$ x 42, 32T39
$64$:  16T105 x 12, 32T273 x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 12

Degree 8: $C_2^3$, $D_4\times C_2$ x 18, $C_2^2 \wr C_2$ x 16

Degree 16: $C_2^2 \times D_4$ x 3, 16T105 x 12

Low degree siblings

32T1369 x 127

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 56 conjugacy classes of elements. Data not shown.

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:  $2$
Label:  128.2163
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);