↑

Properties

Label 10T25
10T25 1 3 1->3 7 1->7 2 4 2->4 2->4 2->7 3->1 5 3->5 6 4->6 8 4->8 5->7 10 5->10 5->10 6->2 6->8 9 7->9 7->9 8->6 8->10 9->1 9->3 10->2
Degree $10$
Order $320$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_2^4 : C_5):C_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(10, 25);
 

Group invariants

Abstract group:  $(C_2^4 : C_5):C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $320=2^{6} \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
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 $t$:  $25$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2[2^{5}]F(5)$
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,7,9,3)(2,4,8,6)(5,10)$, $(1,3,5,7,9)(2,4,6,8,10)$, $(2,7)(5,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$20$:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $F_5$

Low degree siblings

10T24, 16T711, 20T77, 20T78, 20T79, 20T80, 20T83, 20T88, 32T9312, 40T206, 40T207, 40T296, 40T297, 40T298, 40T299, 40T300, 40T301, 40T302, 40T303

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 $2^{4},1^{2}$ $5$ $2$ $4$ $( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B $2^{2},1^{6}$ $10$ $2$ $2$ $(3,8)(4,9)$
2C $2^{4},1^{2}$ $20$ $2$ $4$ $( 2, 5)( 3, 9)( 4, 8)( 7,10)$
4A $4^{2},1^{2}$ $20$ $4$ $6$ $( 2,10, 7, 5)( 3, 9, 8, 4)$
4B $4,2^{3}$ $40$ $4$ $6$ $( 1, 6)( 2,10)( 3, 4, 8, 9)( 5, 7)$
4C1 $4^{2},2$ $40$ $4$ $7$ $( 1, 6)( 2, 9, 5, 3)( 4,10, 8, 7)$
4C-1 $4^{2},2$ $40$ $4$ $7$ $( 1, 6)( 2, 3, 5, 9)( 4, 7, 8,10)$
5A $5^{2}$ $64$ $5$ $8$ $( 1, 9, 7,10, 8)( 2, 5, 3, 6, 4)$
8A1 $8,1^{2}$ $40$ $8$ $7$ $( 2, 8,10, 4, 7, 3, 5, 9)$
8A-1 $8,1^{2}$ $40$ $8$ $7$ $( 2, 9, 5, 3, 7, 4,10, 8)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 4B 4C1 4C-1 5A 8A1 8A-1
Size 1 5 10 20 20 40 40 40 64 40 40
2 P 1A 1A 1A 1A 2A 2B 2C 2C 5A 4A 4A
5 P 1A 2A 2B 2C 4A 4B 4C1 4C-1 1A 8A1 8A-1
Type
320.1635.1a R 1 1 1 1 1 1 1 1 1 1 1
320.1635.1b R 1 1 1 1 1 1 −1 −1 1 −1 −1
320.1635.1c1 C 1 1 1 −1 −1 −1 −i i 1 i −i
320.1635.1c2 C 1 1 1 −1 −1 −1 i −i 1 −i i
320.1635.4a R 4 4 4 0 0 0 0 0 −1 0 0
320.1635.5a R 5 −3 1 1 1 −1 1 1 0 −1 −1
320.1635.5b R 5 −3 1 1 1 −1 −1 −1 0 1 1
320.1635.5c1 C 5 −3 1 −1 −1 1 −i i 0 −i i
320.1635.5c2 C 5 −3 1 −1 −1 1 i −i 0 i −i
320.1635.10a R 10 2 −2 2 −2 0 0 0 0 0 0
320.1635.10b R 10 2 −2 −2 2 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $\left(t^{2} + 4\right) x^{10} + \left(5 t^{2} + 20\right) x^{9} + \left(16 t^{4} + 122 t^{3} + 766 t^{2} + 2007 t + 4248\right) x^{8} + \left(64 t^{4} + 488 t^{3} + 3034 t^{2} + 8028 t + 16872\right) x^{7} + \left(109 t^{4} + 785 t^{3} + 4793 t^{2} + 12254 t + 26068\right) x^{6} + \left(103 t^{4} + 647 t^{3} + 3781 t^{2} + 8664 t + 19236\right) x^{5} + \left(51 t^{4} + 260 t^{3} + 1439 t^{2} + 2559 t + 6380\right) x^{4} + \left(5 t^{4} + 11 t^{3} + 106 t^{2} + 44 t + 344\right) x^{3} + \left(-4 t^{4} - 20 t^{3} - 79 t^{2} - 80 t - 252\right) x^{2} + \left(t^{3} - 6 t^{2} + 4 t - 24\right) x + \left(t^{2} + 4\right)$ Copy content Toggle raw display