Properties

 Label 15T1 Degree $15$ Order $15$ Cyclic yes Abelian yes Solvable yes Primitive no $p$-group no Group: $C_{15}$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(15, 1);

Group action invariants

 Degree $n$: $15$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $1$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_{15}$ CHM label: $C(15)=5[x]3$ Parity: $1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $15$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (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)
$3$:  $C_3$
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $C_5$

Low degree siblings

There are no siblings 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 $1^{15}$ $1$ $1$ $()$ 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 $5^{3}$ $1$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$ 5A-1 $5^{3}$ $1$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$ 5A2 $5^{3}$ $1$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$ 5A-2 $5^{3}$ $1$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$ 15A1 $15$ $1$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$ 15A-1 $15$ $1$ $15$ $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$ 15A2 $15$ $1$ $15$ $( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$ 15A-2 $15$ $1$ $15$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$ 15A4 $15$ $1$ $15$ $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$ 15A-4 $15$ $1$ $15$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$ 15A7 $15$ $1$ $15$ $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$ 15A-7 $15$ $1$ $15$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$

magma: ConjugacyClasses(G);

Group invariants

 Order: $15=3 \cdot 5$ magma: Order(G); Cyclic: yes magma: IsCyclic(G); Abelian: yes magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: $1$ Label: 15.1 magma: IdentifyGroup(G); Character table:

 1A 3A1 3A-1 5A1 5A-1 5A2 5A-2 15A1 15A-1 15A2 15A-2 15A4 15A-4 15A7 15A-7 Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 P 1A 3A-1 3A1 5A-1 5A-2 5A1 5A2 15A2 15A-2 15A4 15A-4 15A-7 15A-1 15A1 15A7 5 P 1A 1A 1A 5A1 5A2 5A-1 5A-2 5A1 5A-1 5A2 5A-2 5A-1 5A2 5A-2 5A1 Type 15.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ 15.1.1b1 C $1$ $ζ3−1$ $ζ3$ $1$ $1$ $1$ $1$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $ζ3$ $ζ3−1$ 15.1.1b2 C $1$ $ζ3$ $ζ3−1$ $1$ $1$ $1$ $1$ $ζ3−1$ $ζ3$ $ζ3$ $ζ3−1$ $ζ3−1$ $ζ3$ $ζ3−1$ $ζ3$ 15.1.1c1 C $1$ $1$ $1$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ52$ $ζ5−2$ 15.1.1c2 C $1$ $1$ $1$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ5−2$ $ζ52$ 15.1.1c3 C $1$ $1$ $1$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ5−2$ $ζ52$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ5$ $ζ5−1$ 15.1.1c4 C $1$ $1$ $1$ $ζ5$ $ζ5−1$ $ζ52$ $ζ5−2$ $ζ52$ $ζ5−2$ $ζ5−1$ $ζ5$ $ζ5−2$ $ζ52$ $ζ5−1$ $ζ5$ 15.1.1d1 C $1$ $ζ15−5$ $ζ155$ $ζ15−6$ $ζ156$ $ζ153$ $ζ15−3$ $ζ15−7$ $ζ157$ $ζ15$ $ζ15−1$ $ζ152$ $ζ15−2$ $ζ15−4$ $ζ154$ 15.1.1d2 C $1$ $ζ155$ $ζ15−5$ $ζ156$ $ζ15−6$ $ζ15−3$ $ζ153$ $ζ157$ $ζ15−7$ $ζ15−1$ $ζ15$ $ζ15−2$ $ζ152$ $ζ154$ $ζ15−4$ 15.1.1d3 C $1$ $ζ15−5$ $ζ155$ $ζ156$ $ζ15−6$ $ζ15−3$ $ζ153$ $ζ152$ $ζ15−2$ $ζ154$ $ζ15−4$ $ζ15−7$ $ζ157$ $ζ15−1$ $ζ15$ 15.1.1d4 C $1$ $ζ155$ $ζ15−5$ $ζ15−6$ $ζ156$ $ζ153$ $ζ15−3$ $ζ15−2$ $ζ152$ $ζ15−4$ $ζ154$ $ζ157$ $ζ15−7$ $ζ15$ $ζ15−1$ 15.1.1d5 C $1$ $ζ15−5$ $ζ155$ $ζ15−3$ $ζ153$ $ζ15−6$ $ζ156$ $ζ15−1$ $ζ15$ $ζ15−2$ $ζ152$ $ζ15−4$ $ζ154$ $ζ15−7$ $ζ157$ 15.1.1d6 C $1$ $ζ155$ $ζ15−5$ $ζ153$ $ζ15−3$ $ζ156$ $ζ15−6$ $ζ15$ $ζ15−1$ $ζ152$ $ζ15−2$ $ζ154$ $ζ15−4$ $ζ157$ $ζ15−7$ 15.1.1d7 C $1$ $ζ15−5$ $ζ155$ $ζ153$ $ζ15−3$ $ζ156$ $ζ15−6$ $ζ15−4$ $ζ154$ $ζ157$ $ζ15−7$ $ζ15−1$ $ζ15$ $ζ152$ $ζ15−2$ 15.1.1d8 C $1$ $ζ155$ $ζ15−5$ $ζ15−3$ $ζ153$ $ζ15−6$ $ζ156$ $ζ154$ $ζ15−4$ $ζ15−7$ $ζ157$ $ζ15$ $ζ15−1$ $ζ15−2$ $ζ152$

magma: CharacterTable(G);