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Properties

Label 10T38
Degree $10$
Order $1920$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $(C_2^4:A_5) : C_2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 38);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_2^4:A_5) : C_2$
CHM label:  $1/2[2^{5}]S(5)$
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:  (2,4)(5,10)(7,9), (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $S_5$

Low degree siblings

10T37, 16T1328, 20T218, 20T219, 20T222, 20T223, 20T226, 30T329, 30T332, 30T333, 30T341, 32T97736, 40T1581, 40T1582, 40T1583, 40T1584, 40T1587, 40T1588, 40T1595, 40T1596, 40T1658, 40T1659, 40T1676, 40T1677, 40T1678

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 $ $10$ $2$ $( 4, 9)( 5,10)$
$ 2, 2, 2, 2, 1, 1 $ $5$ $2$ $( 1, 6)( 2, 7)( 4, 9)( 5,10)$
$ 4, 1, 1, 1, 1, 1, 1 $ $20$ $4$ $( 4,10, 9, 5)$
$ 2, 2, 2, 1, 1, 1, 1 $ $60$ $2$ $( 2, 7)( 4, 5)( 9,10)$
$ 4, 2, 2, 1, 1 $ $60$ $4$ $( 1, 6)( 2, 7)( 4,10, 9, 5)$
$ 2, 2, 2, 2, 2 $ $20$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 5)( 9,10)$
$ 3, 3, 1, 1, 1, 1 $ $80$ $3$ $( 3, 4,10)( 5, 8, 9)$
$ 6, 2, 1, 1 $ $160$ $6$ $( 2, 7)( 3, 4, 5, 8, 9,10)$
$ 3, 3, 2, 2 $ $80$ $6$ $( 1, 6)( 2, 7)( 3, 4,10)( 5, 8, 9)$
$ 2, 2, 2, 2, 1, 1 $ $60$ $2$ $( 2, 3)( 4, 5)( 7, 8)( 9,10)$
$ 4, 4, 1, 1 $ $60$ $4$ $( 2, 3, 7, 8)( 4,10, 9, 5)$
$ 4, 2, 2, 2 $ $120$ $4$ $( 1, 6)( 2, 3)( 4,10, 9, 5)( 7, 8)$
$ 8, 1, 1 $ $240$ $8$ $( 2, 3, 9,10, 7, 8, 4, 5)$
$ 4, 4, 2 $ $240$ $4$ $( 1, 6)( 2, 3, 9, 5)( 4,10, 7, 8)$
$ 4, 3, 3 $ $160$ $12$ $( 1, 7, 6, 2)( 3, 9,10)( 4, 5, 8)$
$ 6, 2, 2 $ $160$ $6$ $( 1, 2)( 3, 9, 5, 8, 4,10)( 6, 7)$
$ 5, 5 $ $384$ $5$ $( 1, 7, 3, 9,10)( 2, 8, 4, 5, 6)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1920=2^{7} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1920.240996
magma: IdentifyGroup(G);
 
Character table:    
      2  7  6  7  5  5  5  5  3  2  3  5  5  4  3  3   2  2  .
      3  1  1  1  1  .  .  1  1  1  1  .  .  .  .  .   1  1  .
      5  1  .  .  .  .  .  .  .  .  .  .  .  .  .  .   .  .  1

        1a 2a 2b 4a 2c 4b 2d 3a 6a 6b 2e 4c 4d 8a 4e 12a 6c 5a
     2P 1a 1a 1a 2a 1a 2a 1a 3a 3a 3a 1a 2b 2a 4c 2e  6b 3a 5a
     3P 1a 2a 2b 4a 2c 4b 2d 1a 2b 2a 2e 4c 4d 8a 4e  4a 2d 5a
     5P 1a 2a 2b 4a 2c 4b 2d 3a 6a 6b 2e 4c 4d 8a 4e 12a 6c 1a
     7P 1a 2a 2b 4a 2c 4b 2d 3a 6a 6b 2e 4c 4d 8a 4e 12a 6c 5a
    11P 1a 2a 2b 4a 2c 4b 2d 3a 6a 6b 2e 4c 4d 8a 4e 12a 6c 5a

X.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
X.3      4  4  4 -2 -2 -2 -2  1  1  1  .  .  .  .  .   1  1 -1
X.4      4  4  4  2  2  2  2  1  1  1  .  .  .  .  .  -1 -1 -1
X.5      5  5  5 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  -1 -1  .
X.6      5  5  5  1  1  1  1 -1 -1 -1  1  1  1 -1 -1   1  1  .
X.7      5  1 -3  3  1 -1 -3  2  . -2  1  1 -1  1 -1   .  .  .
X.8      5  1 -3 -3 -1  1  3  2  . -2  1  1 -1 -1  1   .  .  .
X.9      6  6  6  .  .  .  .  .  .  . -2 -2 -2  .  .   .  .  1
X.10    10 -2  2 -4  2  . -2  1 -1  1 -2  2  .  .  .  -1  1  .
X.11    10 -2  2 -2  .  2 -4  1 -1  1  2 -2  .  .  .   1 -1  .
X.12    10 -2  2  4 -2  .  2  1 -1  1 -2  2  .  .  .   1 -1  .
X.13    10 -2  2  2  . -2  4  1 -1  1  2 -2  .  .  .  -1  1  .
X.14    10  2 -6  .  .  .  . -2  .  2  2  2 -2  .  .   .  .  .
X.15    15  3 -9 -3 -1  1  3  .  .  . -1 -1  1  1 -1   .  .  .
X.16    15  3 -9  3  1 -1 -3  .  .  . -1 -1  1 -1  1   .  .  .
X.17    20 -4  4 -2  2 -2  2 -1  1 -1  .  .  .  .  .   1 -1  .
X.18    20 -4  4  2 -2  2 -2 -1  1 -1  .  .  .  .  .  -1  1  .

magma: CharacterTable(G);