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Properties

Label 36T31119
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

magma: G := TransitiveGroup(36, 31119);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31119$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3^6.(C_2^5.S_4)$
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:  (1,17,2,18)(3,13,4,14)(5,16,6,15)(7,21,8,22)(9,24,10,23)(11,20,12,19)(25,34,27,36)(26,33,28,35)(29,31,30,32), (1,34,13,2,33,14)(3,32,15,6,35,18)(4,31,16,5,36,17)(7,30,24,8,29,23)(9,28,20,11,26,22)(10,27,19,12,25,21)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$24$:  $S_4$ x 3
$48$:  12T27 x 3
$96$:  $V_4^2:S_3$, 12T62 x 2
$192$:  12T98 x 2, 12T102
$384$:  24T728
$768$:  24T1590
$139968$:  18T824

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: 12T98

Degree 18: 18T824

Low degree siblings

36T31118 x 4, 36T31119 x 3, 36T31120 x 4, 36T31121 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 148 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $559872=2^{8} \cdot 3^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  559872.mk
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);