Properties

Label 10T1
Order \(10\)
n \(10\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive No
$p$-group No
Group: $C_{10}$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $1$
Group :  $C_{10}$
CHM label :  $C(10)=5[x]2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,2,3,4,5,6,7,8,9,10)
$|\Aut(F/K)|$:  $10$

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

Group invariants

Order:  $10=2 \cdot 5$
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [10, 2]
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)