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Properties

Label 10T21
Degree $10$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5^2 : C_2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 21);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5^2 : C_2$
CHM label:  $[D(5)^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,6,8,10), (2,8)(4,6), (1,6)(2,7)(3,8)(4,9)(5,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T19, 10T21, 20T48 x 2, 20T50 x 2, 20T55, 20T57 x 2, 25T21, 40T167 x 2, 40T170

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 $ $10$ $2$ $( 4,10)( 6, 8)$
$ 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, 2, 2, 1 $ $20$ $10$ $( 2, 4, 6, 8,10)( 3, 9)( 5, 7)$
$ 5, 1, 1, 1, 1, 1 $ $4$ $5$ $( 2, 6,10, 4, 8)$
$ 5, 2, 2, 1 $ $20$ $10$ $( 2, 6,10, 4, 8)( 3, 9)( 5, 7)$
$ 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)$
$ 4, 4, 2 $ $50$ $4$ $( 1, 2)( 3, 4, 9,10)( 5, 6, 7, 8)$
$ 10 $ $20$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 10 $ $20$ $10$ $( 1, 2, 5, 6, 9,10, 3, 4, 7, 8)$
$ 5, 5 $ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 5, 5 $ $8$ $5$ $( 1, 3, 5, 7, 9)( 2, 6,10, 4, 8)$
$ 5, 5 $ $4$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  200.43
magma: IdentifyGroup(G);
 
Character table:    
      2  3  2  3  1   1  1   1  2  2   1   1  1  .  1
      5  2  1  .  2   1  2   1  1  .   1   1  2  2  2

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

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

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

magma: CharacterTable(G);