# Properties

 Label 37T11 Degree $37$ Order $1.376\times 10^{43}$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $S_{37}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(37, 11);

## Group action invariants

 Degree $n$: $37$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $11$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_{37}$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,2), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

The 21637 conjugacy class representatives for $S_{37}$ are not computed

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     not computed

## Group invariants

 Order: $13763753091226345046315979581580902400000000=2^{34} \cdot 3^{17} \cdot 5^{8} \cdot 7^{5} \cdot 11^{3} \cdot 13^{2} \cdot 17^{2} \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 13763753091226345046315979581580902400000000.a magma: IdentifyGroup(G); Character table: not computed

magma: CharacterTable(G);