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Properties

Label 10T42
10T42 1 6 1->6 2 4 2->4 5 2->5 3 8 3->8 4->6 9 4->9 10 5->10 6->8 7 7->2 8->10 10->2 10->7
Degree $10$
Order $14400$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $A_5^2 : C_4$

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Copy content magma:G := TransitiveGroup(10, 42);
 

Group invariants

Abstract group:  $A_5^2 : C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $14400=2^{6} \cdot 3^{2} \cdot 5^{2}$
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 $t$:  $42$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $1/2[S(5)^{2}]2$
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:  $(2,4,6,8,10)$, $(1,6)(2,5,10,7)(3,8)(4,9)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

12T278, 20T457, 20T461, 24T12116, 25T100, 30T817, 36T9861, 40T10509, 40T10510, 40T10511

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 $2^{2},1^{6}$ $30$ $2$ $2$ $( 4, 6)( 8,10)$
2B $2^{2},1^{6}$ $100$ $2$ $2$ $( 1, 7)( 4,10)$
2C $2^{4},1^{2}$ $225$ $2$ $4$ $( 1, 9)( 2, 8)( 3, 7)( 4,10)$
3A $3,1^{7}$ $40$ $3$ $2$ $(1,9,5)$
3B $3^{2},1^{4}$ $400$ $3$ $4$ $(2,6,8)(3,5,9)$
4A $4,2,1^{4}$ $600$ $4$ $4$ $(2,8,6,4)(3,7)$
4B1 $4,2^{3}$ $600$ $4$ $6$ $( 1, 4, 7,10)( 2, 5)( 3, 8)( 6, 9)$
4B-1 $4,2^{3}$ $600$ $4$ $6$ $( 1,10, 7, 4)( 2, 5)( 3, 8)( 6, 9)$
4C $4^{2},1^{2}$ $900$ $4$ $6$ $( 1, 7, 9, 3)( 2, 4, 8,10)$
5A $5,1^{5}$ $48$ $5$ $4$ $(1,3,9,7,5)$
5B $5^{2}$ $576$ $5$ $8$ $( 1, 5, 9, 3, 7)( 2,10, 4, 6, 8)$
6A $3^{2},2^{2}$ $400$ $6$ $6$ $( 1, 7)( 2, 8, 6)( 3, 9, 5)( 4,10)$
6B $3,2^{2},1^{3}$ $400$ $6$ $4$ $(1,5,3)(2,6)(7,9)$
6C $3,2^{2},1^{3}$ $600$ $6$ $4$ $(1,5,9)(2,6)(4,8)$
8A1 $8,2$ $1800$ $8$ $8$ $( 1, 2, 7, 4, 9, 8, 3,10)( 5, 6)$
8A-1 $8,2$ $1800$ $8$ $8$ $( 1,10, 3, 8, 9, 4, 7, 2)( 5, 6)$
10A $5,2^{2},1$ $720$ $10$ $6$ $( 1, 7, 3, 5, 9)( 4, 6)( 8,10)$
12A $4,3,2,1$ $1200$ $12$ $6$ $(1,9,5)(2,4,6,8)(3,7)$
12B1 $6,4$ $1200$ $12$ $8$ $( 1,10, 7, 4)( 2, 9, 8, 5, 6, 3)$
12B-1 $6,4$ $1200$ $12$ $8$ $( 1, 4, 7,10)( 2, 3, 6, 5, 8, 9)$
15A $5,3,1^{2}$ $960$ $15$ $6$ $(1,3,9,5,7)(2,4,8)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 3B 4A 4B1 4B-1 4C 5A 5B 6A 6B 6C 8A1 8A-1 10A 12A 12B1 12B-1 15A
Size 1 30 100 225 40 400 600 600 600 900 48 576 400 400 600 1800 1800 720 1200 1200 1200 960
2 P 1A 1A 1A 1A 3A 3B 2A 2B 2B 2C 5A 5B 3B 3A 3A 4C 4C 5A 6C 6A 6A 15A
3 P 1A 2A 2B 2C 1A 1A 4A 4B-1 4B1 4C 5A 5B 2B 2B 2A 8A-1 8A1 10A 4A 4B1 4B-1 5A
5 P 1A 2A 2B 2C 3A 3B 4A 4B1 4B-1 4C 1A 1A 6A 6B 6C 8A1 8A-1 2A 12A 12B1 12B-1 3A
Type
14400.bd.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
14400.bd.1b R 1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 1 −1 −1 1
14400.bd.1c1 C 1 1 −1 1 1 1 −1 −i i −1 1 1 −1 −1 1 −i i 1 −1 i −i 1
14400.bd.1c2 C 1 1 −1 1 1 1 −1 i −i −1 1 1 −1 −1 1 i −i 1 −1 −i i 1
14400.bd.8a R 8 4 4 0 5 2 2 0 0 0 3 −2 −2 1 1 0 0 −1 −1 0 0 0
14400.bd.8b R 8 4 −4 0 5 2 −2 0 0 0 3 −2 2 −1 1 0 0 −1 1 0 0 0
14400.bd.10a R 10 6 −2 2 4 −2 0 0 0 2 5 0 −2 −2 0 0 0 1 0 0 0 −1
14400.bd.10b R 10 6 2 2 4 −2 0 0 0 −2 5 0 2 2 0 0 0 1 0 0 0 −1
14400.bd.12a R 12 4 0 −4 6 0 0 0 0 0 7 2 0 0 −2 0 0 −1 0 0 0 1
14400.bd.16a R 16 0 4 0 4 1 0 2 2 0 −4 1 1 −2 0 0 0 0 0 −1 −1 −1
14400.bd.16b R 16 0 4 0 4 1 0 −2 −2 0 −4 1 1 −2 0 0 0 0 0 1 1 −1
14400.bd.16c1 C 16 0 −4 0 4 1 0 −2i 2i 0 −4 1 −1 2 0 0 0 0 0 −i i −1
14400.bd.16c2 C 16 0 −4 0 4 1 0 2i −2i 0 −4 1 −1 2 0 0 0 0 0 i −i −1
14400.bd.25a R 25 5 1 1 −5 1 −1 −1 −1 1 0 0 1 1 −1 1 1 0 −1 −1 −1 0
14400.bd.25b R 25 5 1 1 −5 1 −1 1 1 1 0 0 1 1 −1 −1 −1 0 −1 1 1 0
14400.bd.25c1 C 25 5 −1 1 −5 1 1 −i i −1 0 0 −1 −1 −1 i −i 0 1 i −i 0
14400.bd.25c2 C 25 5 −1 1 −5 1 1 i −i −1 0 0 −1 −1 −1 −i i 0 1 −i i 0
14400.bd.36a R 36 −12 0 4 0 0 0 0 0 0 6 1 0 0 0 0 0 −2 0 0 0 0
14400.bd.40a R 40 4 −4 0 1 −2 2 0 0 0 −5 0 2 −1 1 0 0 −1 −1 0 0 1
14400.bd.40b R 40 4 4 0 1 −2 −2 0 0 0 −5 0 −2 1 1 0 0 −1 1 0 0 1
14400.bd.48a R 48 −8 0 0 6 0 0 0 0 0 −2 −2 0 0 −2 0 0 2 0 0 0 1
14400.bd.60a R 60 −4 0 −4 −6 0 0 0 0 0 5 0 0 0 2 0 0 1 0 0 0 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $16 x^{10} + 40 x^{9} + 25 x^{8} + \left(-t^{2} - 1\right)$ Copy content Toggle raw display