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Properties

Label 30T2
Degree $30$
Order $30$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times S_3$

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Show commands: Magma

magma: G := TransitiveGroup(30, 2);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times S_3$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $30$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,22,12)(2,21,11)(3,24,14)(4,23,13)(5,26,16)(6,25,15)(7,28,18)(8,27,17)(9,30,19)(10,29,20), (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)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $S_3$

Degree 10: $C_{10}$

Degree 15: $S_3 \times C_5$

Low degree siblings

15T4

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $30=2 \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  30.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 15A1 15A-1 15A2 15A-2
Size 1 3 2 1 1 1 1 3 3 3 3 2 2 2 2
2 P 1A 1A 3A 5A2 5A-1 5A1 5A-2 5A1 5A2 5A-1 5A-2 15A-1 15A2 15A-2 15A1
3 P 1A 2A 1A 5A-2 5A1 5A-1 5A2 10A3 10A1 10A-3 10A-1 5A2 5A1 5A-1 5A-2
5 P 1A 2A 3A 1A 1A 1A 1A 2A 2A 2A 2A 3A 3A 3A 3A
Type
30.1.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
30.1.1b R 1 −1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
30.1.1c1 C 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2
30.1.1c2 C 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52
30.1.1c3 C 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1
30.1.1c4 C 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5
30.1.1d1 C 1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2
30.1.1d2 C 1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5−1 ζ5 ζ5−2 ζ52
30.1.1d3 C 1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1
30.1.1d4 C 1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ52 ζ5−2 ζ5−1 ζ5
30.1.2a R 2 0 −1 2 2 2 2 0 0 0 0 −1 −1 −1 −1
30.1.2b1 C 2 0 −1 2ζ5−2 2ζ52 2ζ5 2ζ5−1 0 0 0 0 −ζ5 −ζ5−1 −ζ52 −ζ5−2
30.1.2b2 C 2 0 −1 2ζ52 2ζ5−2 2ζ5−1 2ζ5 0 0 0 0 −ζ5−1 −ζ5 −ζ5−2 −ζ52
30.1.2b3 C 2 0 −1 2ζ5−1 2ζ5 2ζ5−2 2ζ52 0 0 0 0 −ζ5−2 −ζ52 −ζ5 −ζ5−1
30.1.2b4 C 2 0 −1 2ζ5 2ζ5−1 2ζ52 2ζ5−2 0 0 0 0 −ζ52 −ζ5−2 −ζ5−1 −ζ5

magma: CharacterTable(G);