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Properties

Label 36T31055
Degree $36$
Order $559872$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3^6.C_2^5:S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(36, 31055);
 

Group invariants

Abstract group:  $C_3^6.C_2^5:S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $559872=2^{8} \cdot 3^{7}$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $36$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31055$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,27,21,3,29,20,6,26,23)(2,28,22,4,30,19,5,25,24)(7,17,35,8,18,36)(9,14,34,12,16,31)(10,13,33,11,15,32)$, $(1,22,32,16,3,24,34,13,6,19,36,17)(2,21,31,15,4,23,33,14,5,20,35,18)(7,27,11,30,10,26,8,28,12,29,9,25)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$24$:  $S_4$ x 3
$48$:  $S_4\times C_2$ x 3
$96$:  $V_4^2:S_3$
$192$:  $V_4^2:(S_3\times C_2)$ x 2, 12T100
$768$:  16T1068
$139968$:  18T822

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 4: None

Degree 6: $S_4$

Degree 9: None

Degree 12: 12T110

Degree 18: 18T822

Low degree siblings

36T31054, 36T31058, 36T31059, 36T31064, 36T31065, 36T31068, 36T31069, 36T31070, 36T31071, 36T31074, 36T31075, 36T31080, 36T31081, 36T31084, 36T31085, 36T31198 x 2, 36T31199 x 2, 36T31200 x 2, 36T31201 x 2

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

Character table not computed

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed