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Properties

Label 30T8
Degree $30$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_5$

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Show commands: Magma

magma: G := TransitiveGroup(30, 8);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_5$
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)(3,29,14,10,24,20)(4,30,13,9,23,19)(5,7,16,18,26,28)(6,8,15,17,25,27), (1,9,7,15,14,21,20,27,26,4)(2,10,8,16,13,22,19,28,25,3)(5,23,12,30,18,6,24,11,29,17)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $S_3$

Degree 10: $D_{10}$

Degree 15: $D_5\times S_3$

Low degree siblings

15T7, 30T10, 30T13

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $60=2^{2} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  60.8
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 6A 10A1 10A3 15A1 15A2
Size 1 3 5 15 2 2 2 10 6 6 4 4
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 5A1 5A2 15A2 15A1
3 P 1A 2A 2B 2C 1A 5A2 5A1 2B 10A3 10A1 5A1 5A2
5 P 1A 2A 2B 2C 3A 1A 1A 6A 2A 2A 3A 3A
Type
60.8.1a R 1 1 1 1 1 1 1 1 1 1 1 1
60.8.1b R 1 −1 −1 1 1 1 1 −1 −1 −1 1 1
60.8.1c R 1 −1 1 −1 1 1 1 1 −1 −1 1 1
60.8.1d R 1 1 −1 −1 1 1 1 −1 1 1 1 1
60.8.2a R 2 0 2 0 −1 2 2 −1 0 0 −1 −1
60.8.2b R 2 0 −2 0 −1 2 2 1 0 0 −1 −1
60.8.2c1 R 2 2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52
60.8.2c2 R 2 2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5
60.8.2d1 R 2 −2 0 0 2 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52
60.8.2d2 R 2 −2 0 0 2 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5
60.8.4a1 R 4 0 0 0 −2 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
60.8.4a2 R 4 0 0 0 −2 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5

magma: CharacterTable(G);