# Properties

 Label 37T10 Degree $37$ Order $6.882\times 10^{42}$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $A_{37}$

Show commands: Magma

magma: G := TransitiveGroup(37, 10);

## Group action invariants

 Degree $n$: $37$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $10$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $A_{37}$ 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: (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), (1,2,3) magma: Generators(G);

## Low degree resolvents

none

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 10871 conjugacy class representatives for $A_{37}$ are not computed

magma: ConjugacyClasses(G);

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

## Group invariants

 Order: $6881876545613172523157989790790451200000000=2^{33} \cdot 3^{17} \cdot 5^{8} \cdot 7^{5} \cdot 11^{3} \cdot 13^{2} \cdot 17^{2} \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 6881876545613172523157989790790451200000000.a magma: IdentifyGroup(G); Character table: not computed

magma: CharacterTable(G);