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Properties

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

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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

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 15 $ $1$ $15$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
$ 15 $ $1$ $15$ $( 1, 3, 5, 7, 9,11,13,15, 2, 4, 6, 8,10,12,14)$
$ 5, 5, 5 $ $1$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 15 $ $1$ $15$ $( 1, 5, 9,13, 2, 6,10,14, 3, 7,11,15, 4, 8,12)$
$ 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $ $1$ $5$ $( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$
$ 15 $ $1$ $15$ $( 1, 8,15, 7,14, 6,13, 5,12, 4,11, 3,10, 2, 9)$
$ 15 $ $1$ $15$ $( 1, 9, 2,10, 3,11, 4,12, 5,13, 6,14, 7,15, 8)$
$ 5, 5, 5 $ $1$ $5$ $( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
$ 3, 3, 3, 3, 3 $ $1$ $3$ $( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 15 $ $1$ $15$ $( 1,12, 8, 4,15,11, 7, 3,14,10, 6, 2,13, 9, 5)$
$ 5, 5, 5 $ $1$ $5$ $( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
$ 15 $ $1$ $15$ $( 1,14,12,10, 8, 6, 4, 2,15,13,11, 9, 7, 5, 3)$
$ 15 $ $1$ $15$ $( 1,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

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 5A2 5A-2 5A-1 5A1 15A-7 15A4 15A1 15A-1 15A-2 15A7 15A2 15A-4
5 P 1A 1A 1A 5A-2 5A2 5A1 5A-1 5A-1 5A2 5A-2 5A2 5A-1 5A1 5A1 5A-2
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);