# Properties

 Label 30T33 Degree $30$ Order $120$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times S_4$

Show commands: Magma

magma: G := TransitiveGroup(30, 33);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $33$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5\times S_4$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $10$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,29,6,23,9,28,3,21,8,26)(2,30,5,24,10,27,4,22,7,25)(11,14,15,17,19)(12,13,16,18,20), (1,27,13)(2,28,14)(3,30,16)(4,29,15)(5,21,17)(6,22,18)(7,23,19)(8,24,20)(9,25,12)(10,26,11) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$24$:  $S_4$
$30$:  $S_3 \times C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $S_4$

Degree 10: None

Degree 15: $S_3 \times C_5$

## Low degree siblings

20T34, 30T34, 40T64

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $2$ $(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $6$ $2$ $(11,25)(12,26)(13,28)(14,27)(15,30)(16,29)(17,22)(18,21)(19,24)(20,23)$ $4, 4, 4, 4, 4, 2, 2, 2, 2, 2$ $6$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,25,12,26)(13,28,14,27)(15,30,16,29) (17,22,18,21)(19,24,20,23)$ $10, 10, 5, 5$ $3$ $10$ $( 1, 3, 6, 8, 9)( 2, 4, 5, 7,10)(11,13,15,18,19,12,14,16,17,20) (21,24,26,27,29,22,23,25,28,30)$ $5, 5, 5, 5, 5, 5$ $1$ $5$ $( 1, 3, 6, 8, 9)( 2, 4, 5, 7,10)(11,14,15,17,19)(12,13,16,18,20) (21,23,26,28,29)(22,24,25,27,30)$ $10, 10, 5, 5$ $6$ $10$ $( 1, 3, 6, 8, 9)( 2, 4, 5, 7,10)(11,27,15,22,19,25,14,30,17,24) (12,28,16,21,20,26,13,29,18,23)$ $20, 10$ $6$ $20$ $( 1, 4, 6, 7, 9, 2, 3, 5, 8,10)(11,27,16,21,19,25,13,29,17,24,12,28,15,22,20, 26,14,30,18,23)$ $10, 10, 5, 5$ $3$ $10$ $( 1, 5, 9, 4, 8, 2, 6,10, 3, 7)(11,15,19,14,17)(12,16,20,13,18) (21,25,29,24,28,22,26,30,23,27)$ $20, 10$ $6$ $20$ $( 1, 5, 9, 4, 8, 2, 6,10, 3, 7)(11,29,20,27,17,26,16,24,14,21,12,30,19,28,18, 25,15,23,13,22)$ $5, 5, 5, 5, 5, 5$ $1$ $5$ $( 1, 6, 9, 3, 8)( 2, 5,10, 4, 7)(11,15,19,14,17)(12,16,20,13,18) (21,26,29,23,28)(22,25,30,24,27)$ $10, 10, 5, 5$ $6$ $10$ $( 1, 6, 9, 3, 8)( 2, 5,10, 4, 7)(11,29,19,28,17,26,15,23,14,21) (12,30,20,27,18,25,16,24,13,22)$ $10, 10, 5, 5$ $3$ $10$ $( 1, 7, 3,10, 6, 2, 8, 4, 9, 5)(11,17,14,19,15)(12,18,13,20,16) (21,27,23,30,26,22,28,24,29,25)$ $20, 10$ $6$ $20$ $( 1, 7, 3,10, 6, 2, 8, 4, 9, 5)(11,21,13,24,15,26,18,27,19,29,12,22,14,23,16, 25,17,28,20,30)$ $5, 5, 5, 5, 5, 5$ $1$ $5$ $( 1, 8, 3, 9, 6)( 2, 7, 4,10, 5)(11,17,14,19,15)(12,18,13,20,16) (21,28,23,29,26)(22,27,24,30,25)$ $10, 10, 5, 5$ $6$ $10$ $( 1, 8, 3, 9, 6)( 2, 7, 4,10, 5)(11,21,14,23,15,26,17,28,19,29) (12,22,13,24,16,25,18,27,20,30)$ $5, 5, 5, 5, 5, 5$ $1$ $5$ $( 1, 9, 8, 6, 3)( 2,10, 7, 5, 4)(11,19,17,15,14)(12,20,18,16,13) (21,29,28,26,23)(22,30,27,25,24)$ $10, 10, 5, 5$ $3$ $10$ $( 1, 9, 8, 6, 3)( 2,10, 7, 5, 4)(11,20,17,16,14,12,19,18,15,13) (21,30,28,25,23,22,29,27,26,24)$ $10, 10, 5, 5$ $6$ $10$ $( 1, 9, 8, 6, 3)( 2,10, 7, 5, 4)(11,23,17,29,14,26,19,21,15,28) (12,24,18,30,13,25,20,22,16,27)$ $20, 10$ $6$ $20$ $( 1,10, 8, 5, 3, 2, 9, 7, 6, 4)(11,23,18,30,14,26,20,22,15,28,12,24,17,29,13, 25,19,21,16,27)$ $15, 15$ $8$ $15$ $( 1,11,23, 6,15,28, 9,19,21, 3,14,26, 8,17,29)( 2,12,24, 5,16,27,10,20,22, 4, 13,25, 7,18,30)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $8$ $3$ $( 1,13,28)( 2,14,27)( 3,16,29)( 4,15,30)( 5,17,22)( 6,18,21)( 7,19,24) ( 8,20,23)( 9,12,26)(10,11,25)$ $15, 15$ $8$ $15$ $( 1,15,22, 8,11,27, 3,17,24, 9,14,30, 6,19,25)( 2,16,21, 7,12,28, 4,18,23,10, 13,29, 5,20,26)$ $15, 15$ $8$ $15$ $( 1,17,26, 3,19,28, 6,11,29, 8,14,21, 9,15,23)( 2,18,25, 4,20,27, 5,12,30, 7, 13,22,10,16,24)$ $15, 15$ $8$ $15$ $( 1,19,29, 9,17,28, 8,15,26, 6,14,23, 3,11,21)( 2,20,30,10,18,27, 7,16,25, 5, 13,24, 4,12,22)$

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: yes magma: IsSolvable(G); Label: 120.37 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);