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Properties

Label 30T32
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(30, 32);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $32$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times F_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,12)(2,21,11)(3,5,20,28,24,26,10,18,14,16,29,7)(4,6,19,27,23,25,9,17,13,15,30,8), (1,27,14,9,26,21,7,4,20,15)(2,28,13,10,25,22,8,3,19,16)(5,11,18,23,29,6,12,17,24,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$20$:  $F_5$
$24$:  $S_3 \times C_4$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $F_5$

Degree 6: $S_3$

Degree 10: $F_{5}\times C_2$

Degree 15: $F_5 \times S_3$

Low degree siblings

15T11, 30T23, 30T24

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.36
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 5A 6A 10A 12A1 12A-1 15A
Size 1 3 5 15 2 5 5 15 15 4 10 12 10 10 8
2 P 1A 1A 1A 1A 3A 2B 2B 2B 2B 5A 3A 5A 6A 6A 15A
3 P 1A 2A 2B 2C 1A 4A-1 4A1 4B-1 4B1 5A 2B 10A 4A1 4A-1 5A
5 P 1A 2A 2B 2C 3A 4A1 4A-1 4B1 4B-1 1A 6A 2A 12A1 12A-1 3A
Type
120.36.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.36.1b R 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1 1
120.36.1c R 1 −1 1 −1 1 1 1 −1 −1 1 1 −1 1 1 1
120.36.1d R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
120.36.1e1 C 1 −1 −1 1 1 −i i i −i 1 −1 −1 i −i 1
120.36.1e2 C 1 −1 −1 1 1 i −i −i i 1 −1 −1 −i i 1
120.36.1f1 C 1 1 −1 −1 1 −i i −i i 1 −1 1 i −i 1
120.36.1f2 C 1 1 −1 −1 1 i −i i −i 1 −1 1 −i i 1
120.36.2a R 2 0 2 0 −1 2 2 0 0 2 −1 0 −1 −1 −1
120.36.2b R 2 0 2 0 −1 −2 −2 0 0 2 −1 0 1 1 −1
120.36.2c1 C 2 0 −2 0 −1 −2i 2i 0 0 2 1 0 −i i −1
120.36.2c2 C 2 0 −2 0 −1 2i −2i 0 0 2 1 0 i −i −1
120.36.4a R 4 4 0 0 4 0 0 0 0 −1 0 −1 0 0 −1
120.36.4b R 4 −4 0 0 4 0 0 0 0 −1 0 1 0 0 −1
120.36.8a R 8 0 0 0 −4 0 0 0 0 −2 0 0 0 0 1

magma: CharacterTable(G);