# Properties

 Label 31T12 Degree $31$ Order $8.223\times 10^{33}$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $S_{31}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(31, 12);

## Group action invariants

 Degree $n$: $31$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $12$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_{31}$ Parity: $-1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (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), (1,2) 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

There are 6,842 conjugacy classes of elements. Data not shown.

magma: ConjugacyClasses(G);

## Group invariants

 Order: $8222838654177922817725562880000000=2^{26} \cdot 3^{14} \cdot 5^{7} \cdot 7^{4} \cdot 11^{2} \cdot 13^{2} \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 8222838654177922817725562880000000.a magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);