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Properties

Label 10T10
10T10 1 6 1->6 2 3 2->3 4 2->4 8 3->8 4->1 4->6 5 10 5->10 6->8 9 6->9 7 7->2 8->7 8->10 9->4 10->2
Degree $10$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^2 : C_4$

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

Group invariants

Abstract group:  $C_5^2 : C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $100=2^{2} \cdot 5^{2}$
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 $t$:  $10$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2[D(5)^{2}]2$
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:  $(2,4,6,8,10)$, $(1,6,9,4)(2,3,8,7)(5,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

10T10, 20T27 x 2, 25T10

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 $2^{4},1^{2}$ $25$ $2$ $4$ $( 1, 7)( 2, 6)( 3, 5)( 8,10)$
4A1 $4^{2},2$ $25$ $4$ $7$ $( 1, 6, 7, 2)( 3, 8, 5,10)( 4, 9)$
4A-1 $4^{2},2$ $25$ $4$ $7$ $( 1, 2, 7, 6)( 3,10, 5, 8)( 4, 9)$
5A $5^{2}$ $4$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2,10, 8, 6, 4)$
5B $5^{2}$ $4$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2,10, 8, 6, 4)$
5C1 $5,1^{5}$ $4$ $5$ $4$ $( 2, 8, 4,10, 6)$
5C2 $5,1^{5}$ $4$ $5$ $4$ $( 2, 4, 6, 8,10)$
5D1 $5^{2}$ $4$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
5D2 $5^{2}$ $4$ $5$ $8$ $( 1, 9, 7, 5, 3)( 2, 4, 6, 8,10)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 4A1 4A-1 5A 5B 5C1 5C2 5D1 5D2
Size 1 25 25 25 4 4 4 4 4 4
2 P 1A 1A 2A 2A 5A 5B 5C2 5C1 5D2 5D1
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A
Type
100.12.1a R 1 1 1 1 1 1 1 1 1 1
100.12.1b R 1 1 −1 −1 1 1 1 1 1 1
100.12.1c1 C 1 −1 −i i 1 1 1 1 1 1
100.12.1c2 C 1 −1 i −i 1 1 1 1 1 1
100.12.4a R 4 0 0 0 −1 −1 4 −1 −1 −1
100.12.4b R 4 0 0 0 4 −1 −1 −1 −1 −1
100.12.4c1 R 4 0 0 0 −1 2ζ5−2+2ζ52 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−1+2ζ5
100.12.4c2 R 4 0 0 0 −1 2ζ5−1+2ζ5 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−2+2ζ52
100.12.4d1 R 4 0 0 0 −1 ζ5−2+2+ζ52 −1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52
100.12.4d2 R 4 0 0 0 −1 −ζ5−2+1−ζ52 −1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $\left(1024 t^{2} + 1024\right) x^{10} + \left(5120 t^{2} + 5120\right) x^{9} + \left(25920 t^{2} + 25920\right) x^{8} + \left(92960 t^{2} + 92960\right) x^{7} + \left(264865 t^{2} - 125760\right) x^{6} + \left(611364 t^{2} - 951136\right) x^{5} + \left(1153860 t^{2} - 408640\right) x^{4} + \left(1676160 t^{2} + 1676160\right) x^{3} + \left(1716480 t^{2} + 1716480\right) x^{2} + \left(1105920 t^{2} + 1105920\right) x + \left(331776 t^{2} + 331776\right)$ Copy content Toggle raw display