# Properties

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

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);