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Properties

Label 10T10
Degree $10$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2 : C_4$

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Show commands: Magma

magma: G := TransitiveGroup(10, 10);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $10$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^2 : C_4$
CHM label:  $1/2[D(5)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,6,8,10), (1,6,9,4)(2,3,8,7)(5,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T10, 20T27 x 2, 25T10

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, 2, 2, 1, 1 $ $25$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)$
$ 5, 1, 1, 1, 1, 1 $ $4$ $5$ $( 2, 4, 6, 8,10)$
$ 5, 1, 1, 1, 1, 1 $ $4$ $5$ $( 2, 6,10, 4, 8)$
$ 4, 4, 2 $ $25$ $4$ $( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$
$ 4, 4, 2 $ $25$ $4$ $( 1, 2)( 3,10, 9, 4)( 5, 8, 7, 6)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 8, 4,10, 6)$
$ 5, 5 $ $4$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  100.12
magma: IdentifyGroup(G);
 
Character table:    
      2  2  2  .  .  2  2  .  .  .  .
      5  2  .  2  2  .  .  2  2  2  2

        1a 2a 5a 5b 4a 4b 5c 5d 5e 5f
     2P 1a 1a 5b 5a 2a 2a 5f 5d 5e 5c
     3P 1a 2a 5b 5a 4b 4a 5f 5d 5e 5c
     5P 1a 2a 1a 1a 4a 4b 1a 1a 1a 1a

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

A = 2*E(5)^2+2*E(5)^3
  = -1-Sqrt(5) = -1-r5
B = -E(5)-2*E(5)^2-2*E(5)^3-E(5)^4
  = (3+Sqrt(5))/2 = 2+b5
C = -E(4)
  = -Sqrt(-1) = -i

magma: CharacterTable(G);