Properties

Label 10T8
Order \(80\)
n \(10\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^4 : C_5$

Related objects

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Group action invariants

Degree $n$ :  $10$
Transitive number $t$ :  $8$
Group :  $C_2^4 : C_5$
CHM label :  $[2^{4}]5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $C_5$

Low degree siblings

10T8 x 2, 16T178, 20T17 x 6, 20T23, 40T57 x 3

Siblings are shown 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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 4, 9)( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 8)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 5, 5 $ $16$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
$ 5, 5 $ $16$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$
$ 5, 5 $ $16$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)$
$ 5, 5 $ $16$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)$

Group invariants

Order:  $80=2^{4} \cdot 5$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [80, 49]
Character table:   
     2  4  4  4  4  .  .  .  .
     5  1  .  .  .  1  1  1  1

       1a 2a 2b 2c 5a 5b 5c 5d
    2P 1a 1a 1a 1a 5b 5d 5a 5c
    3P 1a 2a 2b 2c 5c 5a 5d 5b
    5P 1a 2a 2b 2c 1a 1a 1a 1a

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

A = E(5)^4
B = E(5)^3