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Properties

Label 32T3449
Degree $32$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3.C_2^5$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(32, 3449);
 

Group invariants

Abstract group:  $C_2^3.C_2^5$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $2$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3449$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,5)(2,6)(3,7)(4,8)(9,16)(10,15)(11,13)(12,14)(17,30)(18,29)(19,32)(20,31)(21,26)(22,25)(23,28)(24,27)$, $(1,14)(2,13)(3,16)(4,15)(5,12)(6,11)(7,9)(8,10)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$, $(1,7,15,11)(2,8,16,12)(3,5,13,10)(4,6,14,9)(17,28,24,29)(18,27,23,30)(19,26,22,31)(20,25,21,32)$, $(1,5,16,9)(2,6,15,10)(3,7,14,12)(4,8,13,11)(17,27,23,29)(18,28,24,30)(19,25,21,31)(20,26,22,32)$, $(1,19,13,24)(2,20,14,23)(3,17,15,22)(4,18,16,21)(5,25,11,30)(6,26,12,29)(7,27,10,32)(8,28,9,31)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 31
$4$:  $C_2^2$ x 155
$8$:  $D_{4}$ x 8, $C_2^3$ x 155
$16$:  $D_4\times C_2$ x 28, $C_2^4$ x 31
$32$:  $Q_8:C_2^2$ x 12, $C_2^2 \times D_4$ x 14, 32T39
$64$:  16T69 x 6, 16T83 x 4, 32T273
$128$:  16T198 x 3, 16T206 x 3, 32T1020

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$, $Q_8:C_2^2$ x 6

Degree 16: 16T83, 16T198 x 3, 16T206 x 3

Low degree siblings

32T3449 x 127

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

Copy content magma:ConjugacyClasses(G);
 

Character table

52 x 52 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed