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Properties

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

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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);
 
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 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 $ $2$ $3$ $(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
$ 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);
 
Nilpotency class:   not nilpotent
Label:  180.21
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);