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Properties

Label 10T14
Degree $10$
Order $160$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2 \times (C_2^4 : C_5)$

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Show commands: Magma

magma: G := TransitiveGroup(10, 14);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2 \times (C_2^4 : C_5)$
CHM label:  $[2^{5}]5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (5,10), (1,3,5,7,9)(2,4,6,8,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$80$:  $C_2^4 : C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $C_5$

Low degree siblings

10T14 x 2, 20T40 x 12, 20T41 x 6, 20T44 x 3, 20T46 x 3, 32T2133, 40T121 x 6, 40T122 x 6, 40T123 x 12, 40T141, 40T142 x 3

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 4, 9)( 5,10)$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $5$ $2$ $( 3, 8)( 5,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 3, 8)( 4, 9)( 5,10)$
$ 2, 2, 2, 1, 1, 1, 1 $ $5$ $2$ $( 2, 7)( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$
$ 5, 5 $ $16$ $5$ $( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)$
$ 10 $ $16$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 5, 5 $ $16$ $5$ $( 1, 3, 5, 2, 4)( 6, 8,10, 7, 9)$
$ 10 $ $16$ $10$ $( 1, 3, 5, 7, 9, 6, 8,10, 2, 4)$
$ 5, 5 $ $16$ $5$ $( 1, 4, 2, 5, 3)( 6, 9, 7,10, 8)$
$ 10 $ $16$ $10$ $( 1, 4, 2, 5, 8, 6, 9, 7,10, 3)$
$ 5, 5 $ $16$ $5$ $( 1, 5, 4, 3, 2)( 6,10, 9, 8, 7)$
$ 10 $ $16$ $10$ $( 1, 5, 9, 8, 7, 6,10, 4, 3, 2)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $160=2^{5} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  160.235
magma: IdentifyGroup(G);
 
Character table:    
      2  5  5  5  5  5  5  5  1   1  1   1  1   1  1   1  5
      5  1  .  .  .  .  .  .  1   1  1   1  1   1  1   1  1

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

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

A = E(5)^4
B = E(5)^3

magma: CharacterTable(G);