↑

Properties

Label 10T40
10T40 1 6 1->6 2 4 2->4 2->4 7 2->7 3 8 3->8 4->6 9 4->9 10 4->10 5 5->10 6->8 8->10 10->2 10->2
Degree $10$
Order $7200$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5 \wr C_2$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(10, 40);
 

Group invariants

Abstract group:  $A_5 \wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $7200=2^{5} \cdot 3^{2} \cdot 5^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $10$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $40$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[A(5)^{2}]2$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,4,6,8,10)$, $(1,6)(2,7)(3,8)(4,9)(5,10)$, $(2,4,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

12T269, 20T363, 24T9631, 25T88, 30T652, 36T7075, 40T5410

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 $2^{2},1^{6}$ $30$ $2$ $2$ $(2,8)(4,6)$
2B $2^{5}$ $60$ $2$ $5$ $( 1, 8)( 2, 5)( 3, 6)( 4, 9)( 7,10)$
2C $2^{4},1^{2}$ $225$ $2$ $4$ $( 1, 9)( 2,10)( 3, 5)( 6, 8)$
3A $3,1^{7}$ $40$ $3$ $2$ $(3,9,7)$
3B $3^{2},1^{4}$ $400$ $3$ $4$ $( 1, 9, 7)( 2, 4,10)$
4A $4^{2},2$ $900$ $4$ $7$ $( 1, 2, 9,10)( 3, 6, 5, 8)( 4, 7)$
5A1 $5,1^{5}$ $24$ $5$ $4$ $( 2, 4, 6,10, 8)$
5A2 $5,1^{5}$ $24$ $5$ $4$ $( 2, 6, 8, 4,10)$
5B1 $5^{2}$ $144$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2, 8,10, 6, 4)$
5B2 $5^{2}$ $144$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2,10, 4, 8, 6)$
5C $5^{2}$ $288$ $5$ $8$ $( 1, 3, 5, 7, 9)( 2, 6, 8,10, 4)$
6A $3,2^{2},1^{3}$ $600$ $6$ $4$ $( 1, 7, 9)( 4,10)( 6, 8)$
6B $6,2^{2}$ $1200$ $6$ $7$ $( 1, 4, 9,10, 7, 2)( 3, 8)( 5, 6)$
10A1 $5,2^{2},1$ $360$ $10$ $6$ $(1,7,3,5,9)(2,8)(4,6)$
10A3 $5,2^{2},1$ $360$ $10$ $6$ $(1,5,7,9,3)(2,8)(4,6)$
10B1 $10$ $720$ $10$ $9$ $( 1, 4, 7, 2, 3, 8, 9,10, 5, 6)$
10B3 $10$ $720$ $10$ $9$ $( 1, 2, 9, 6, 7, 8, 5, 4, 3,10)$
15A1 $5,3,1^{2}$ $480$ $15$ $6$ $( 2, 6, 8, 4,10)( 3, 7, 9)$
15A2 $5,3,1^{2}$ $480$ $15$ $6$ $( 2, 8,10, 6, 4)( 3, 9, 7)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 4A 5A1 5A2 5B1 5B2 5C 6A 6B 10A1 10A3 10B1 10B3 15A1 15A2
Size 1 30 60 225 40 400 900 24 24 144 144 288 600 1200 360 360 720 720 480 480
2 P 1A 1A 1A 1A 3A 3B 2C 5A2 5A1 5B2 5B1 5C 3A 3B 5A1 5A2 5B1 5B2 15A2 15A1
3 P 1A 2A 2B 2C 1A 1A 4A 5A2 5A1 5B2 5B1 5C 2A 2B 10A3 10A1 10B3 10B1 5A1 5A2
5 P 1A 2A 2B 2C 3A 3B 4A 1A 1A 1A 1A 1A 6A 6B 2A 2A 2B 2B 3A 3A
Type
7200.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
7200.a.1b R 1 1 −1 1 1 1 −1 1 1 1 1 1 1 −1 1 1 −1 −1 1 1
7200.a.6a1 R 6 2 0 −2 3 0 0 ζ5−2+4+ζ52 −ζ5−2+3−ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 1 −1 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
7200.a.6a2 R 6 2 0 −2 3 0 0 −ζ5−2+3−ζ52 ζ5−2+4+ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 1 −1 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
7200.a.8a R 8 4 0 0 5 2 0 3 3 −2 −2 −2 1 0 −1 −1 0 0 0 0
7200.a.9a1 R 9 −3 3 1 0 0 −1 −3ζ5−1−3ζ5 −3ζ5−2−3ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
7200.a.9a2 R 9 −3 3 1 0 0 −1 −3ζ5−2−3ζ52 −3ζ5−1−3ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
7200.a.9b1 R 9 −3 −3 1 0 0 1 −3ζ5−1−3ζ5 −3ζ5−2−3ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0
7200.a.9b2 R 9 −3 −3 1 0 0 1 −3ζ5−2−3ζ52 −3ζ5−1−3ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0
7200.a.10a R 10 6 0 2 4 −2 0 5 5 0 0 0 0 0 1 1 0 0 −1 −1
7200.a.16a R 16 0 4 0 4 1 0 −4 −4 1 1 1 0 1 0 0 −1 −1 −1 −1
7200.a.16b R 16 0 −4 0 4 1 0 −4 −4 1 1 1 0 −1 0 0 1 1 −1 −1
7200.a.18a R 18 −6 0 2 0 0 0 3 3 −2 −2 3 0 0 −1 −1 0 0 0 0
7200.a.24a1 R 24 −4 0 0 3 0 0 4ζ5−2+1+4ζ52 −4ζ5−2−3−4ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −1 0 1 1 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
7200.a.24a2 R 24 −4 0 0 3 0 0 −4ζ5−2−3−4ζ52 4ζ5−2+1+4ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −1 0 1 1 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
7200.a.25a R 25 5 5 1 −5 1 1 0 0 0 0 0 −1 −1 0 0 0 0 0 0
7200.a.25b R 25 5 −5 1 −5 1 −1 0 0 0 0 0 −1 1 0 0 0 0 0 0
7200.a.30a1 R 30 −2 0 −2 −3 0 0 −5ζ5−1−5ζ5 −5ζ5−2−5ζ52 0 0 0 1 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5
7200.a.30a2 R 30 −2 0 −2 −3 0 0 −5ζ5−2−5ζ52 −5ζ5−1−5ζ5 0 0 0 1 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52
7200.a.40a R 40 4 0 0 1 −2 0 −5 −5 0 0 0 1 0 −1 −1 0 0 1 1

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{10} + 150 x^{8} + t x^{7} + 5625 x^{6} + 78 t x^{5} + x^{4} + 225 t x^{3} + 6 x^{2} + 9$ Copy content Toggle raw display