# Properties

 Label 15T4 Degree $15$ Order $30$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3 \times C_5$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(15, 4);

## Group action invariants

 Degree $n$: $15$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $4$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3 \times C_5$ CHM label: $5[x]S(3)$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $5$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,11)(2,7)(4,14)(5,10)(8,13), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) 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 3: $S_3$

Degree 5: $C_5$

## Low degree siblings

30T2

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Label Cycle Type Size Order Index Representative 1A $1^{15}$ $1$ $1$ $0$ $()$ 2A $2^{5},1^{5}$ $3$ $2$ $5$ $( 2,12)( 3, 8)( 5,15)( 6,11)( 9,14)$ 3A $3^{5}$ $2$ $3$ $10$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ 5A1 $5^{3}$ $1$ $5$ $12$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ 5A-1 $5^{3}$ $1$ $5$ $12$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ 5A2 $5^{3}$ $1$ $5$ $12$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ 5A-2 $5^{3}$ $1$ $5$ $12$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ 10A1 $10,5$ $3$ $10$ $13$ $( 1,13,10, 7, 4)( 2, 9,11, 3, 5,12,14, 6, 8,15)$ 10A-1 $10,5$ $3$ $10$ $13$ $( 1, 7,13, 4,10)( 2, 3,14,15,11,12, 8, 9, 5, 6)$ 10A3 $10,5$ $3$ $10$ $13$ $( 1, 4, 7,10,13)( 2,15, 8, 6,14,12, 5, 3,11, 9)$ 10A-3 $10,5$ $3$ $10$ $13$ $( 1,10, 4,13, 7)( 2, 6, 5, 9, 8,12,11,15,14, 3)$ 15A1 $15$ $2$ $15$ $14$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$ 15A-1 $15$ $2$ $15$ $14$ $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$ 15A2 $15$ $2$ $15$ $14$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$ 15A-2 $15$ $2$ $15$ $14$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/5$

## 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-2 5A-1 5A1 5A-1 5A2 5A1 5A-2 15A1 15A-1 15A-2 15A2 3 P 1A 2A 1A 5A-2 5A2 5A1 5A-1 10A1 10A3 10A-1 10A-3 5A-2 5A2 5A-1 5A1 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);