# Properties

 Label 15T3 Degree $15$ Order $30$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_5\times C_3$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(15, 3);

## Group action invariants

 Degree $n$: $15$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $3$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_5\times C_3$ CHM label: $D(5)[x]3$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $3$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (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)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: $C_3$

Degree 5: $D_{5}$

## Low degree siblings

30T4

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 Representative 1A $1^{15}$ $1$ $1$ $()$ 2A $2^{6},1^{3}$ $5$ $2$ $( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$ 3A1 $3^{5}$ $1$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$ 3A-1 $3^{5}$ $1$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$ 5A1 $5^{3}$ $2$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ 5A2 $5^{3}$ $2$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ 6A1 $6^{2},3$ $5$ $6$ $( 1, 8, 6,13,11, 3)( 2,12, 7)( 4, 5, 9,10,14,15)$ 6A-1 $6^{2},3$ $5$ $6$ $( 1,15,11,10, 6, 5)( 2, 4,12,14, 7, 9)( 3, 8,13)$ 15A1 $15$ $2$ $15$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$ 15A-1 $15$ $2$ $15$ $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$ 15A2 $15$ $2$ $15$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$ 15A-2 $15$ $2$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$

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: 30.2 magma: IdentifyGroup(G); Character table:

 1A 2A 3A1 3A-1 5A1 5A2 6A1 6A-1 15A1 15A-1 15A2 15A-2 Size 1 5 1 1 2 2 5 5 2 2 2 2 2 P 1A 1A 3A-1 3A1 5A2 5A1 3A1 3A-1 15A2 15A-2 15A1 15A-1 3 P 1A 2A 1A 1A 5A2 5A1 2A 2A 5A1 5A1 5A2 5A2 5 P 1A 2A 3A-1 3A1 1A 1A 6A-1 6A1 3A1 3A-1 3A-1 3A1 Type 30.2.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 30.2.1b R $1$ $−1$ $1$ $1$ $1$ $1$ $−1$ $−1$ $1$ $1$ $1$ $1$ 30.2.1c1 C $1$ $1$ $ζ3−1$ $ζ3$ $1$ $1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ 30.2.1c2 C $1$ $1$ $ζ3$ $ζ3−1$ $1$ $1$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ 30.2.1d1 C $1$ $−1$ $ζ3−1$ $ζ3$ $1$ $1$ $−ζ3−1$ $−ζ3$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ 30.2.1d2 C $1$ $−1$ $ζ3$ $ζ3−1$ $1$ $1$ $−ζ3$ $−ζ3−1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ 30.2.2a1 R $2$ $0$ $2$ $2$ $ζ5−2+ζ52$ $ζ5−1+ζ5$ $0$ $0$ $ζ5−1+ζ5$ $ζ5−1+ζ5$ $ζ5−2+ζ52$ $ζ5−2+ζ52$ 30.2.2a2 R $2$ $0$ $2$ $2$ $ζ5−1+ζ5$ $ζ5−2+ζ52$ $0$ $0$ $ζ5−2+ζ52$ $ζ5−2+ζ52$ $ζ5−1+ζ5$ $ζ5−1+ζ5$ 30.2.2b1 C $2$ $0$ $2ζ15−5$ $2ζ155$ $ζ15−6+ζ156$ $ζ15−3+ζ153$ $0$ $0$ $−1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157$ $1−ζ15−ζ154+ζ155$ $ζ15+ζ154$ $1−ζ15−ζ152+ζ153−ζ154−ζ157$ 30.2.2b2 C $2$ $0$ $2ζ155$ $2ζ15−5$ $ζ15−6+ζ156$ $ζ15−3+ζ153$ $0$ $0$ $1−ζ15−ζ154+ζ155$ $−1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157$ $1−ζ15−ζ152+ζ153−ζ154−ζ157$ $ζ15+ζ154$ 30.2.2b3 C $2$ $0$ $2ζ15−5$ $2ζ155$ $ζ15−3+ζ153$ $ζ15−6+ζ156$ $0$ $0$ $1−ζ15−ζ152+ζ153−ζ154−ζ157$ $ζ15+ζ154$ $1−ζ15−ζ154+ζ155$ $−1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157$ 30.2.2b4 C $2$ $0$ $2ζ155$ $2ζ15−5$ $ζ15−3+ζ153$ $ζ15−6+ζ156$ $0$ $0$ $ζ15+ζ154$ $1−ζ15−ζ152+ζ153−ζ154−ζ157$ $−1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157$ $1−ζ15−ζ154+ζ155$

magma: CharacterTable(G);