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Properties

Label 10T6
Degree $10$
Order $50$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times C_5$

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Show commands: Magma

magma: G := TransitiveGroup(10, 6);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\times C_5$
CHM label:  $[5^{2}]2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,6,8,10), (1,6)(2,7)(3,8)(4,9)(5,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T6, 25T3

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $50=2 \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  50.3
magma: IdentifyGroup(G);
 
Character table:    
      2  1  .  .  .  .  1   1   1   1   1  1  .  .  .  1  .  .  1  .  1
      5  2  2  2  2  2  1   1   1   1   1  2  2  2  2  2  2  2  2  2  2

        1a 5a 5b 5c 5d 2a 10a 10b 10c 10d 5e 5f 5g 5h 5i 5j 5k 5l 5m 5n
     2P 1a 5b 5d 5a 5c 1a  5e  5i  5l  5n 5i 5k 5f 5j 5n 5h 5m 5e 5g 5l
     3P 1a 5c 5a 5d 5b 2a 10c 10a 10d 10b 5l 5g 5m 5j 5e 5h 5f 5n 5k 5i
     5P 1a 1a 1a 1a 1a 2a  2a  2a  2a  2a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a
     7P 1a 5b 5d 5a 5c 2a 10b 10d 10a 10c 5i 5k 5f 5j 5n 5h 5m 5e 5g 5l

X.1      1  1  1  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  1  1  1  1
X.3      1  A  B /B /A -1  -A  -B -/B -/A  B /B /A  1 /A  1  A  A  B /B
X.4      1  B /A  A /B -1  -B -/A  -A -/B /A  A /B  1 /B  1  B  B /A  A
X.5      1 /B  A /A  B -1 -/B  -A -/A  -B  A /A  B  1  B  1 /B /B  A /A
X.6      1 /A /B  B  A -1 -/A -/B  -B  -A /B  B  A  1  A  1 /A /A /B  B
X.7      1  A  B /B /A  1   A   B  /B  /A  B /B /A  1 /A  1  A  A  B /B
X.8      1  B /A  A /B  1   B  /A   A  /B /A  A /B  1 /B  1  B  B /A  A
X.9      1 /B  A /A  B  1  /B   A  /A   B  A /A  B  1  B  1 /B /B  A /A
X.10     1 /A /B  B  A  1  /A  /B   B   A /B  B  A  1  A  1 /A /A /B  B
X.11     2  C *C *C  C  .   .   .   .   .  2  C *C *C  2  C *C  2  C  2
X.12     2 *C  C  C *C  .   .   .   .   .  2 *C  C  C  2 *C  C  2 *C  2
X.13     2  D  E /E /D  .   .   .   .   .  H /F /G *C  I  C  G /I  F /H
X.14     2 /D /E  E  D  .   .   .   .   . /H  F  G *C /I  C /G  I /F  H
X.15     2  E /D  D /E  .   .   .   .   .  I  G /F  C /H *C  F  H /G /I
X.16     2 /E  D /D  E  .   .   .   .   . /I /G  F  C  H *C /F /H  G  I
X.17     2  F /G  G /F  .   .   .   .   .  I  D /E *C /H  C  E  H /D /I
X.18     2 /F  G /G  F  .   .   .   .   . /I /D  E *C  H  C /E /H  D  I
X.19     2  G  F /F /G  .   .   .   .   .  H /E /D  C  I *C  D /I  E /H
X.20     2 /G /F  F  G  .   .   .   .   . /H  E  D  C /I *C /D  I /E  H

A = E(5)^4
B = E(5)^3
C = E(5)+E(5)^4
  = (-1+Sqrt(5))/2 = b5
D = E(5)+E(5)^3
E = E(5)+E(5)^2
F = -E(5)-E(5)^2-E(5)^4
G = -E(5)-E(5)^2-E(5)^3
H = 2*E(5)^4
I = 2*E(5)^3

magma: CharacterTable(G);