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Properties

Label 36T85036
Degree $36$
Order $306110016$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no

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Show commands: Magma

magma: G := TransitiveGroup(36, 85036);
 

Group action invariants

Degree $n$:  $36$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $85036$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,14,10,27,35)(2,23,15,12,25,36,3,24,13,11,26,34)(4,9,28,19,16,32)(5,8,29,20,18,31,6,7,30,21,17,33), (1,13,26,3,14,25,2,15,27)(4,6,5)(7,21)(8,19)(9,20)(16,17,18)(22,34,24,36,23,35)(31,33,32), (1,28,2,29,3,30)(4,25,17,15,5,27,16,13,6,26,18,14)(7,24,31,35)(8,22,32,36)(9,23,33,34)(10,19,12,21,11,20), (1,7,13,20)(2,9,14,21)(3,8,15,19)(4,34,18,10,5,35,16,12,6,36,17,11)(22,30)(23,28,24,29)(25,31)(26,33,27,32)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_4$ x 8, $C_2^2$ x 35
$8$:  $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:  $C_4\times C_2^2$ x 14, $C_2^4$
$32$:  $C_2^3 : D_4 $ x 2, 32T34
$36$:  $C_3^2:C_4$
$64$:  16T68
$72$:  12T40 x 7
$144$:  24T241 x 7
$576$:  24T1394
$2592$:  12T242
$5184$:  24T7644
$46656$:  24T14743
$3779136$:  36T50018

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: None

Degree 4: $C_2^2$

Degree 6: None

Degree 9: None

Degree 12: 12T242

Degree 18: None

Low degree siblings

36T85036 x 8

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

There are 2,062 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $306110016=2^{6} \cdot 3^{14}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  not available
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);