# Properties

 Label 30T47 Degree $30$ Order $180$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{15}:C_{12}$

Show commands: Magma

magma: G := TransitiveGroup(30, 47);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $47$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{15}:C_{12}$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $3$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,26,11,30,3,25,10,29,2,27,12,28)(4,18,7,22,6,17,9,24,5,16,8,23)(13,19,14,20,15,21), (1,8,15,4,12,2,9,13,5,10,3,7,14,6,11)(16,23,29,21,25,18,22,28,20,27,17,24,30,19,26) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $S_3$, $C_6$
$12$:  $C_{12}$, $C_3 : C_4$
$18$:  $S_3\times C_3$
$20$:  $F_5$
$36$:  $C_3\times (C_3 : C_4)$
$60$:  $C_{15} : C_4$, $F_5\times C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $F_5$

Degree 6: $S_3\times C_3$

Degree 10: $F_5$

Degree 15: None

## Low degree siblings

45T18

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $180=2^{2} \cdot 3^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 180.21 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);