# Properties

 Label 12T74 Degree $12$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(12, 74);

## Group action invariants

 Degree $n$: $12$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $74$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ CHM label: $S_{5}(12)$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (2,4,6,8,10)(3,5,7,9,11), (1,10)(2,7)(3,12)(4,5)(6,11)(8,9) magma: Generators(G);

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 4: None

Degree 6: $\PGL(2,5)$

## Low degree siblings

5T5, 6T14, 10T12, 10T13, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 30T25, 30T27, 40T62

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1, 1, 1$ $15$ $2$ $( 3, 7)( 4,10)( 6, 8)( 9,11)$ $5, 5, 1, 1$ $24$ $5$ $( 2, 4, 6, 8,10)( 3, 5, 7, 9,11)$ $2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 2)( 3, 6)( 4,11)( 5,12)( 7, 8)( 9,10)$ $6, 6$ $20$ $6$ $( 1, 2, 3, 4,11, 6)( 5,10, 7, 8, 9,12)$ $4, 4, 2, 2$ $30$ $4$ $( 1, 2, 3, 8)( 4, 7)( 5,10,11,12)( 6, 9)$ $3, 3, 3, 3$ $20$ $3$ $( 1, 3, 9)( 2, 4, 8)( 5, 7,11)( 6,12,10)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $120=2^{3} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Label: 120.34 magma: IdentifyGroup(G);
 Character table:  2 3 3 . 2 1 2 1 3 1 . . 1 1 . 1 5 1 . 1 . . . . 1a 2a 5a 2b 6a 4a 3a 2P 1a 1a 5a 1a 3a 2a 3a 3P 1a 2a 5a 2b 2b 4a 1a 5P 1a 2a 1a 2b 6a 4a 3a X.1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 -1 1 X.3 4 . -1 -2 1 . 1 X.4 4 . -1 2 -1 . 1 X.5 5 1 . 1 1 -1 -1 X.6 5 1 . -1 -1 1 -1 X.7 6 -2 1 . . . . 

magma: CharacterTable(G);