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Properties

Label 10T34
Degree $10$
Order $960$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2^4 : A_5$

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Show commands: Magma

magma: G := TransitiveGroup(10, 34);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4 : A_5$
CHM label:   $[2^{4}]A(5)$
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:  (2,4,10)(5,7,9), (1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$60$:  $A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $A_5$

Low degree siblings

16T1081, 20T172, 20T177, 30T214, 30T217, 40T888, 40T889, 40T932, 40T942, 40T944, 40T945

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{4},1^{2}$ $5$ $2$ $4$ $( 1, 6)( 2, 7)( 3, 8)( 5,10)$
2B $2^{2},1^{6}$ $10$ $2$ $2$ $( 1, 6)( 5,10)$
2C $2^{4},1^{2}$ $60$ $2$ $4$ $( 1, 7)( 2, 6)( 3,10)( 5, 8)$
3A $3^{2},1^{4}$ $80$ $3$ $4$ $( 2, 5, 8)( 3, 7,10)$
4A $4^{2},1^{2}$ $60$ $4$ $6$ $( 1, 2, 6, 7)( 3,10, 8, 5)$
4B $4,2^{3}$ $120$ $4$ $6$ $( 1,10, 6, 5)( 2, 4)( 3, 8)( 7, 9)$
5A1 $5^{2}$ $192$ $5$ $8$ $( 1,10, 4, 2, 8)( 3, 6, 5, 9, 7)$
5A2 $5^{2}$ $192$ $5$ $8$ $( 1, 4, 8,10, 2)( 3, 5, 7, 6, 9)$
6A $3^{2},2^{2}$ $80$ $6$ $6$ $( 1, 6)( 2, 3, 5)( 4, 9)( 7, 8,10)$
6B1 $6,2,1^{2}$ $80$ $6$ $6$ $( 1, 6)( 2,10, 8, 7, 5, 3)$
6B-1 $6,2,1^{2}$ $80$ $6$ $6$ $( 1, 6)( 2, 3, 5, 7, 8,10)$

Malle's constant $a(G)$:     $1/2$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $960=2^{6} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  960.11358
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 4A 4B 5A1 5A2 6A 6B1 6B-1
Size 1 5 10 60 80 60 120 192 192 80 80 80
2 P 1A 1A 1A 1A 3A 2A 2B 5A2 5A1 3A 3A 3A
3 P 1A 2A 2B 2C 1A 4A 4B 5A2 5A1 2B 2A 2A
5 P 1A 2A 2B 2C 3A 4A 4B 1A 1A 6A 6B-1 6B1
Type
960.11358.1a R 1 1 1 1 1 1 1 1 1 1 1 1
960.11358.3a1 R 3 3 3 −1 0 −1 −1 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0
960.11358.3a2 R 3 3 3 −1 0 −1 −1 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0
960.11358.4a R 4 4 4 0 1 0 0 −1 −1 1 1 1
960.11358.5a R 5 5 5 1 −1 1 1 0 0 −1 −1 −1
960.11358.5b R 5 −3 1 1 2 1 −1 0 0 −2 0 0
960.11358.5c1 C 5 −3 1 1 −1 1 −1 0 0 1 −1−2ζ3 1+2ζ3
960.11358.5c2 C 5 −3 1 1 −1 1 −1 0 0 1 1+2ζ3 −1−2ζ3
960.11358.10a R 10 2 −2 2 1 −2 0 0 0 1 −1 −1
960.11358.10b R 10 2 −2 −2 1 2 0 0 0 1 −1 −1
960.11358.15a R 15 −9 3 −1 0 −1 1 0 0 0 0 0
960.11358.20a R 20 4 −4 0 −1 0 0 0 0 −1 1 1

magma: CharacterTable(G);