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Properties

Label 10T9
Degree $10$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5^2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 9);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5^2$
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)(2,7)(3,8)(4,9)(5,10), (1,9)(2,8)(3,7)(4,6)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$ x 2
$20$:  $D_{10}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T9, 20T28 x 2, 25T12

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

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:  100.13
magma: IdentifyGroup(G);
 
Character table:    
      2  2  2  .  .  2  2   1   1   1   1  1  .  .  1  1  1
      5  2  .  2  2  1  1   1   1   1   1  2  2  2  2  2  2

        1a 2a 5a 5b 2b 2c 10a 10b 10c 10d 5c 5d 5e 5f 5g 5h
     2P 1a 1a 5b 5a 1a 1a  5c  5f  5g  5h 5g 5e 5d 5h 5c 5f
     3P 1a 2a 5b 5a 2b 2c 10c 10d 10a 10b 5g 5e 5d 5h 5c 5f
     5P 1a 2a 1a 1a 2b 2c  2b  2c  2b  2c 1a 1a 1a 1a 1a 1a
     7P 1a 2a 5b 5a 2b 2c 10c 10d 10a 10b 5g 5e 5d 5h 5c 5f

X.1      1  1  1  1  1  1   1   1   1   1  1  1  1  1  1  1
X.2      1 -1  1  1 -1  1  -1   1  -1   1  1  1  1  1  1  1
X.3      1 -1  1  1  1 -1   1  -1   1  -1  1  1  1  1  1  1
X.4      1  1  1  1 -1 -1  -1  -1  -1  -1  1  1  1  1  1  1
X.5      2  .  A *A -2  .  -A   . -*A   . *A *A  A  2  A  2
X.6      2  . *A  A -2  . -*A   .  -A   .  A  A *A  2 *A  2
X.7      2  .  A *A  . -2   .  -A   . -*A  2  A *A *A  2  A
X.8      2  . *A  A  . -2   . -*A   .  -A  2 *A  A  A  2 *A
X.9      2  .  A *A  .  2   .   A   .  *A  2  A *A *A  2  A
X.10     2  . *A  A  .  2   .  *A   .   A  2 *A  A  A  2 *A
X.11     2  .  A *A  2  .   A   .  *A   . *A *A  A  2  A  2
X.12     2  . *A  A  2  .  *A   .   A   .  A  A *A  2 *A  2
X.13     4  .  B *B  .  .   .   .   .   .  C -1 -1  C *C *C
X.14     4  . *B  B  .  .   .   .   .   . *C -1 -1 *C  C  C
X.15     4  . -1 -1  .  .   .   .   .   .  C *B  B *C *C  C
X.16     4  . -1 -1  .  .   .   .   .   . *C  B *B  C  C *C

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

magma: CharacterTable(G);