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Properties

Label 30T16
Degree $30$
Order $90$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_3\times D_{15}$

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Show commands: Magma

magma: G := TransitiveGroup(30, 16);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_3\times D_{15}$
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:  (1,15,12,9,5,3,14,11,8,4,2,13,10,7,6)(16,29,25,22,20,17,30,26,23,21,18,28,27,24,19), (1,21,2,19,3,20)(4,16,5,17,6,18)(7,28,8,29,9,30)(10,27,11,25,12,26)(13,24,14,22,15,23)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$10$:  $D_{5}$
$18$:  $S_3\times C_3$
$30$:  $D_{15}$, $D_5\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $D_{5}$

Degree 6: $S_3\times C_3$

Degree 10: $D_5$

Degree 15: None

Low degree siblings

45T5

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $90=2 \cdot 3^{2} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  90.7
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);