# Properties

 Label 30T36 Degree $30$ Order $150$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5\times D_{15}$

Show commands: Magma

magma: G := TransitiveGroup(30, 36);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $36$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5\times D_{15}$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $15$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,9,14,19,27,3,8,13,21,26,2,7,15,20,25)(4,28,22,17,12,6,30,24,16,11,5,29,23,18,10), (1,22,26,16,19,10,15,6,8,29)(2,24,27,18,20,12,13,5,9,28)(3,23,25,17,21,11,14,4,7,30) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $D_{5}$, $C_{10}$
$30$:  $D_{15}$, $S_3 \times C_5$
$50$:  $D_5\times C_5$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_3$

Degree 10: $D_5\times C_5$

Degree 15: None

## Low degree siblings

30T36

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $150=2 \cdot 3 \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 150.11 magma: IdentifyGroup(G);
 Character table: not available.

magma: CharacterTable(G);