↑

Properties

Label 30T10
Degree $30$
Order $60$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_5$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(30, 10);
 

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$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: $D_{6}$

Degree 10: $D_5$

Degree 15: $D_5\times S_3$

Low degree siblings

15T7, 30T8, 30T13

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{30}$ $1$ $1$ $0$ $()$
2A $2^{10},1^{10}$ $3$ $2$ $10$ $( 3,24)( 4,23)( 5,15)( 6,16)( 9,29)(10,30)(11,21)(12,22)(17,28)(18,27)$
2B $2^{15}$ $5$ $2$ $15$ $( 1,20)( 2,19)( 3,27)( 4,28)( 5, 6)( 7,14)( 8,13)( 9,22)(10,21)(11,30)(12,29)(15,16)(17,23)(18,24)(25,26)$
2C $2^{15}$ $15$ $2$ $15$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,14)( 8,13)( 9,12)(10,11)(21,30)(22,29)(23,28)(24,27)(25,26)$
3A $3^{10}$ $2$ $3$ $20$ $( 1,21,11)( 2,22,12)( 3,24,13)( 4,23,14)( 5,26,15)( 6,25,16)( 7,28,17)( 8,27,18)( 9,29,19)(10,30,20)$
5A1 $5^{6}$ $2$ $5$ $24$ $( 1,19, 7,26,13)( 2,20, 8,25,14)( 3,21, 9,28,15)( 4,22,10,27,16)( 5,24,11,29,17)( 6,23,12,30,18)$
5A2 $5^{6}$ $2$ $5$ $24$ $( 1, 7,13,19,26)( 2, 8,14,20,25)( 3, 9,15,21,28)( 4,10,16,22,27)( 5,11,17,24,29)( 6,12,18,23,30)$
6A $6^{5}$ $10$ $6$ $25$ $( 1, 4,11,14,21,23)( 2, 3,12,13,22,24)( 5,20,15,30,26,10)( 6,19,16,29,25, 9)( 7,27,17, 8,28,18)$
10A1 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1,19, 7,26,13)( 2,20, 8,25,14)( 3,11, 9,17,15,24,21,29,28, 5)( 4,12,10,18,16,23,22,30,27, 6)$
10A3 $10^{2},5^{2}$ $6$ $10$ $26$ $( 1, 7,13,19,26)( 2, 8,14,20,25)( 3,29,15,11,28,24, 9, 5,21,17)( 4,30,16,12,27,23,10, 6,22,18)$
15A1 $15^{2}$ $4$ $15$ $28$ $( 1,28,24,19,15,11, 7, 3,29,26,21,17,13, 9, 5)( 2,27,23,20,16,12, 8, 4,30,25,22,18,14,10, 6)$
15A2 $15^{2}$ $4$ $15$ $28$ $( 1, 3, 5, 7, 9,11,13,15,17,19,21,24,26,28,29)( 2, 4, 6, 8,10,12,14,16,18,20,22,23,25,27,30)$

Malle's constant $a(G)$:     $1/10$

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 5A2 5A1 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);