# Properties

 Label 24T202 Degree $24$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

Show commands: Magma

magma: G := TransitiveGroup(24, 202);

## Group action invariants

 Degree $n$: $24$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $202$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_5$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $4$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3)(2,4)(5,9)(6,10)(7,13)(8,14)(11,19)(12,20)(15,17)(16,18)(21,23)(22,24), (3,17,11,7,5)(4,18,12,8,6)(9,14,22,20,15)(10,13,21,19,16) 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: $\PGL(2,5)$

Degree 8: None

Degree 12: $S_5$

## Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 1, 1, 1, 1$ $24$ $5$ $( 3, 5, 7,11,17)( 4, 6, 8,12,18)( 9,15,20,22,14)(10,16,19,21,13)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3, 4)( 5,18)( 6,17)( 7,12)( 8,11)( 9,16)(10,15)(13,20)(14,19)(21,22) (23,24)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,13)( 8,14)(11,19)(12,20)(15,17)(16,18)(21,23) (22,24)$ $4, 4, 4, 4, 4, 4$ $30$ $4$ $( 1, 3, 9,17)( 2, 4,10,18)( 5,13, 6,14)( 7,19,23,21)( 8,20,24,22)(11,15,12,16)$ $6, 6, 6, 6$ $20$ $6$ $( 1, 3,13,18,10,11)( 2, 4,14,17, 9,12)( 5,19, 7,15,24,22)( 6,20, 8,16,23,21)$ $3, 3, 3, 3, 3, 3, 3, 3$ $20$ $3$ $( 1, 9,20)( 2,10,19)( 3,12, 5)( 4,11, 6)( 7,17,23)( 8,18,24)(13,21,15) (14,22,16)$

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 2 1 1 3 1 . . 1 . 1 1 5 1 1 . . . . . 1a 5a 2a 2b 4a 6a 3a 2P 1a 5a 1a 1a 2a 3a 3a 3P 1a 5a 2a 2b 4a 2b 1a 5P 1a 1a 2a 2b 4a 6a 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 1 -2 . . . . 

magma: CharacterTable(G);