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Properties

Label 10T8
Degree $10$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4 : C_5$

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Show commands: Magma

magma: G := TransitiveGroup(10, 8);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4 : C_5$
CHM label:  $[2^{4}]5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $80=2^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  80.49
magma: IdentifyGroup(G);
 
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

magma: CharacterTable(G);