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Properties

Label 10T1
Degree $10$
Order $10$
Cyclic yes
Abelian yes
Solvable yes
Primitive no
$p$-group no
Group: $C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(10, 1);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{10}$
CHM label:  $C(10)=5[x]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $1$
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8,9,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 10 $ $1$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 5, 5 $ $1$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 10 $ $1$ $10$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$
$ 5, 5 $ $1$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 5, 5 $ $1$ $5$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
$ 10 $ $1$ $10$ $( 1, 8, 5, 2, 9, 6, 3,10, 7, 4)$
$ 5, 5 $ $1$ $5$ $( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$
$ 10 $ $1$ $10$ $( 1,10, 9, 8, 7, 6, 5, 4, 3, 2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $10=2 \cdot 5$
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  10.2
magma: IdentifyGroup(G);
 
Character table:    
      2  1   1  1   1  1  1  1   1  1   1
      5  1   1  1   1  1  1  1   1  1   1

        1a 10a 5a 10b 5b 2a 5c 10c 5d 10d

X.1      1   1  1   1  1  1  1   1  1   1
X.2      1  -1  1  -1  1 -1  1  -1  1  -1
X.3      1   A  B  /B /A  1  A   B /B  /A
X.4      1  -A  B -/B /A -1  A  -B /B -/A
X.5      1   B /A   A /B  1  B  /A  A  /B
X.6      1  -B /A  -A /B -1  B -/A  A -/B
X.7      1  /B  A  /A  B  1 /B   A /A   B
X.8      1 -/B  A -/A  B -1 /B  -A /A  -B
X.9      1  /A /B   B  A  1 /A  /B  B   A
X.10     1 -/A /B  -B  A -1 /A -/B  B  -A

A = E(5)^3
B = E(5)

magma: CharacterTable(G);