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Properties

Label 10T5
Degree $10$
Order $40$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{5}\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 5);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_{5}\times C_2$
CHM label:  $F(5)[x]2$
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:  (1,7,9,3)(2,4,8,6), (1,2,3,4,5,6,7,8,9,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

10T5, 20T9, 20T13, 40T14

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $40=2^{3} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  40.12
magma: IdentifyGroup(G);
 
Character table:    
      2  3  3  3  3  3   1  3  3  1  3
      5  1  .  .  .  .   1  .  .  1  1

        1a 4a 4b 2a 2b 10a 4c 4d 5a 2c
     2P 1a 2a 2a 1a 1a  5a 2a 2a 5a 1a
     3P 1a 4b 4a 2a 2b 10a 4d 4c 5a 2c
     5P 1a 4a 4b 2a 2b  2c 4c 4d 1a 2c
     7P 1a 4b 4a 2a 2b 10a 4d 4c 5a 2c

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 -1 -1  1  1   1 -1 -1  1  1
X.4      1  1  1  1 -1  -1 -1 -1  1 -1
X.5      1  A -A -1 -1   1  A -A  1  1
X.6      1 -A  A -1 -1   1 -A  A  1  1
X.7      1  A -A -1  1  -1 -A  A  1 -1
X.8      1 -A  A -1  1  -1  A -A  1 -1
X.9      4  .  .  .  .   1  .  . -1 -4
X.10     4  .  .  .  .  -1  .  . -1  4

A = -E(4)
  = -Sqrt(-1) = -i

magma: CharacterTable(G);