# Properties

 Label 30T43 Degree $30$ Order $180$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_{15}:D_6$

Show commands: Magma

magma: G := TransitiveGroup(30, 43);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $43$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{15}:D_6$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $5$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,30)(2,28)(3,29)(4,27)(5,25)(6,26)(7,24)(8,22)(9,23)(10,21)(11,19)(12,20)(13,18)(14,16)(15,17), (1,13,10,7,4)(2,15,11,9,5,3,14,12,8,6)(16,28,25,22,19)(17,30,26,24,20,18,29,27,23,21), (1,2,3)(4,5,6)(7,8,9)(10,11,12)(13,14,15) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$ x 2
$10$:  $D_{5}$
$12$:  $D_{6}$ x 2
$20$:  $D_{10}$
$36$:  $S_3^2$
$60$:  $D_5\times S_3$ x 2

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $S_3^2$

Degree 10: $D_5$

Degree 15: None

## Low degree siblings

45T21

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

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: 180.30 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);