# Properties

 Label 30T25 Degree $30$ Order $120$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $S_5$

Show commands: Magma

magma: G := TransitiveGroup(30, 25);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $25$ 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); $\card{\Aut(F/K)}$: $6$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,14,17,12)(2,13,18,11)(3,24,8,10)(4,23,7,9)(5,25)(6,26)(15,30,19,21)(16,29,20,22)(27,28), (1,9,13,3,5)(2,10,14,4,6)(7,22,25,15,18)(8,21,26,16,17)(11,28,23,20,30)(12,27,24,19,29) 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 5: $S_5$

Degree 6: None

Degree 10: $S_5$

Degree 15: $S_5$

## Low degree siblings

5T5, 6T14, 10T12, 10T13, 12T74, 15T10, 20T30, 20T32, 20T35, 24T202, 30T22, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $15$ $2$ $( 3,13)( 4,14)( 5, 9)( 6,10)( 7,22)( 8,21)(11,30)(12,29)(17,26)(18,25)(19,27) (20,28)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $10$ $2$ $( 1, 2)( 3,12)( 4,11)( 5, 7)( 6, 8)( 9,10)(13,25)(14,26)(15,23)(16,24)(17,19) (18,20)(21,22)(27,30)(28,29)$ $4, 4, 4, 4, 4, 4, 2, 2, 2$ $30$ $4$ $( 1, 2)( 3,25,20,29)( 4,26,19,30)( 5,10, 8,22)( 6, 9, 7,21)(11,27,17,14) (12,28,18,13)(15,23)(16,24)$ $5, 5, 5, 5, 5, 5$ $24$ $5$ $( 1, 3, 9, 5,13)( 2, 4,10, 6,14)( 7,15,22,18,25)( 8,16,21,17,26) (11,20,28,30,23)(12,19,27,29,24)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $20$ $3$ $( 1, 3,28)( 2, 4,27)( 5,16,21)( 6,15,22)( 7,14,29)( 8,13,30)( 9,17,20) (10,18,19)(11,26,23)(12,25,24)$ $6, 6, 6, 6, 6$ $20$ $6$ $( 1, 4,23,22,16,12)( 2, 3,24,21,15,11)( 5,18,30,27,20, 7)( 6,17,29,28,19, 8) ( 9,14,26,10,13,25)$

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 2a 2b 4a 5a 3a 6a 2P 1a 1a 1a 2a 5a 3a 3a 3P 1a 2a 2b 4a 5a 1a 2b 5P 1a 2a 2b 4a 1a 3a 6a X.1 1 1 1 1 1 1 1 X.2 1 1 -1 -1 1 1 -1 X.3 4 . -2 . -1 1 1 X.4 4 . 2 . -1 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);