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Properties

Label 10T41
10T41 1 6 1->6 2 4 2->4 7 2->7 10 2->10 3 8 3->8 4->6 9 4->9 5 5->7 5->10 6->8 8->10 10->2
Degree $10$
Order $14400$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $(A_5^2 : C_2):C_2$

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Copy content magma:G := TransitiveGroup(10, 41);
 

Group invariants

Abstract group:  $(A_5^2 : C_2):C_2$
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 $t$:  $41$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   $[1/2.S(5)^{2}]2=[A(5):2]2$
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:  $(2,10)(5,7)$, $(2,4,6,8,10)$, $(1,6)(2,7)(3,8)(4,9)(5,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: None

Low degree siblings

12T279, 20T456, 20T459, 24T12117, 25T101, 30T819, 36T9862, 40T10506, 40T10507, 40T10508

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 $2^{2},1^{6}$ $30$ $2$ $2$ $( 4, 8)( 6,10)$
2B $2^{5}$ $60$ $2$ $5$ $( 1, 8)( 2, 9)( 3, 4)( 5, 6)( 7,10)$
2C $2^{5}$ $60$ $2$ $5$ $( 1,10)( 2, 7)( 3, 4)( 5, 8)( 6, 9)$
2D $2^{2},1^{6}$ $100$ $2$ $2$ $( 1, 7)( 6,10)$
2E $2^{4},1^{2}$ $225$ $2$ $4$ $( 2, 6)( 3, 5)( 7, 9)( 8,10)$
3A $3,1^{7}$ $40$ $3$ $2$ $(1,3,9)$
3B $3^{2},1^{4}$ $400$ $3$ $4$ $( 1, 9, 7)( 2,10, 8)$
4A $4,2,1^{4}$ $600$ $4$ $4$ $( 1, 7)( 4,10, 8, 6)$
4B $4^{2},1^{2}$ $900$ $4$ $6$ $( 2,10, 6, 8)( 3, 7, 5, 9)$
4C $4^{2},2$ $900$ $4$ $7$ $( 1, 2, 5, 6)( 3, 8)( 4, 9,10, 7)$
4D $4^{2},2$ $900$ $4$ $7$ $( 1,10)( 2, 5, 6, 7)( 3, 4, 9, 8)$
5A $5,1^{5}$ $48$ $5$ $4$ $( 2, 8, 6, 4,10)$
5B $5^{2}$ $288$ $5$ $8$ $( 1, 5, 3, 9, 7)( 2, 6, 4,10, 8)$
5C $5^{2}$ $288$ $5$ $8$ $( 1, 3, 9, 7, 5)( 2, 4, 8,10, 6)$
6A $3^{2},2^{2}$ $400$ $6$ $6$ $( 1, 7)( 2, 8, 4)( 3, 9, 5)( 6,10)$
6B $3,2^{2},1^{3}$ $400$ $6$ $4$ $(1,9,7)(2,6)(3,5)$
6C $3,2^{2},1^{3}$ $600$ $6$ $4$ $( 3, 5, 9)( 4, 8)( 6,10)$
6D $6,2^{2}$ $1200$ $6$ $7$ $( 1,10, 9, 8, 7, 2)( 3, 4)( 5, 6)$
6E $6,2^{2}$ $1200$ $6$ $7$ $( 1, 4, 9,10, 3, 6)( 2, 7)( 5, 8)$
10A $5,2^{2},1$ $720$ $10$ $6$ $( 1, 7)( 2,10, 8, 4, 6)( 3, 5)$
10B $10$ $1440$ $10$ $9$ $( 1, 2, 5, 6, 3, 4, 9,10, 7, 8)$
10C $10$ $1440$ $10$ $9$ $( 1, 4, 3, 8, 9,10, 7, 6, 5, 2)$
12A $4,3,2,1$ $1200$ $12$ $6$ $( 1, 7)( 3, 9, 5)( 4, 6, 8,10)$
15A $5,3,1^{2}$ $960$ $15$ $6$ $( 1, 9, 3)( 2, 6,10, 8, 4)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 5C 6A 6B 6C 6D 6E 10A 10B 10C 12A 15A
Size 1 30 60 60 100 225 40 400 600 900 900 900 48 288 288 400 400 600 1200 1200 720 1440 1440 1200 960
2 P 1A 1A 1A 1A 1A 1A 3A 3B 2A 2E 2E 2E 5A 5B 5C 3B 3A 3A 3B 3B 5A 5B 5C 6C 15A
3 P 1A 2A 2B 2C 2D 2E 1A 1A 4A 4B 4C 4D 5A 5B 5C 2D 2D 2A 2B 2C 10A 10B 10C 4A 5A
5 P 1A 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 1A 1A 1A 6A 6B 6C 6D 6E 2A 2B 2C 12A 3A
Type
14400.c.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
14400.c.1b R 1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 1 −1 −1 1 1
14400.c.1c R 1 1 −1 1 −1 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 1 −1 1 1 −1 1 −1 1
14400.c.1d R 1 1 1 −1 −1 1 1 1 −1 −1 −1 1 1 1 1 −1 −1 1 1 −1 1 1 −1 −1 1
14400.c.8a R 8 4 0 0 4 0 5 2 2 0 0 0 3 −2 −2 −2 1 1 0 0 −1 0 0 −1 0
14400.c.8b R 8 4 0 0 −4 0 5 2 −2 0 0 0 3 −2 −2 2 −1 1 0 0 −1 0 0 1 0
14400.c.10a R 10 6 0 0 −2 2 4 −2 0 2 0 0 5 0 0 −2 −2 0 0 0 1 0 0 0 −1
14400.c.10b R 10 6 0 0 2 2 4 −2 0 −2 0 0 5 0 0 2 2 0 0 0 1 0 0 0 −1
14400.c.12a R 12 4 0 0 0 −4 6 0 0 0 0 0 7 2 2 0 0 −2 0 0 −1 0 0 0 1
14400.c.16a R 16 0 4 4 4 0 4 1 0 0 0 0 −4 1 1 1 −2 0 1 1 0 −1 −1 0 −1
14400.c.16b R 16 0 −4 −4 4 0 4 1 0 0 0 0 −4 1 1 1 −2 0 −1 −1 0 1 1 0 −1
14400.c.16c R 16 0 −4 4 −4 0 4 1 0 0 0 0 −4 1 1 −1 2 0 −1 1 0 1 −1 0 −1
14400.c.16d R 16 0 4 −4 −4 0 4 1 0 0 0 0 −4 1 1 −1 2 0 1 −1 0 −1 1 0 −1
14400.c.18a R 18 −6 0 6 0 2 0 0 0 0 −2 0 3 −2 3 0 0 0 0 0 −1 0 1 0 0
14400.c.18b R 18 −6 6 0 0 2 0 0 0 0 0 −2 3 3 −2 0 0 0 0 0 −1 1 0 0 0
14400.c.18c R 18 −6 −6 0 0 2 0 0 0 0 0 2 3 3 −2 0 0 0 0 0 −1 −1 0 0 0
14400.c.18d R 18 −6 0 −6 0 2 0 0 0 0 2 0 3 −2 3 0 0 0 0 0 −1 0 −1 0 0
14400.c.25a R 25 5 5 5 1 1 −5 1 −1 1 1 1 0 0 0 1 1 −1 −1 −1 0 0 0 −1 0
14400.c.25b R 25 5 −5 −5 1 1 −5 1 −1 1 −1 −1 0 0 0 1 1 −1 1 1 0 0 0 −1 0
14400.c.25c R 25 5 −5 5 −1 1 −5 1 1 −1 1 −1 0 0 0 −1 −1 −1 1 −1 0 0 0 1 0
14400.c.25d R 25 5 5 −5 −1 1 −5 1 1 −1 −1 1 0 0 0 −1 −1 −1 −1 1 0 0 0 1 0
14400.c.40a R 40 4 0 0 −4 0 1 −2 2 0 0 0 −5 0 0 2 −1 1 0 0 −1 0 0 −1 1
14400.c.40b R 40 4 0 0 4 0 1 −2 −2 0 0 0 −5 0 0 −2 1 1 0 0 −1 0 0 1 1
14400.c.48a R 48 −8 0 0 0 0 6 0 0 0 0 0 −2 −2 −2 0 0 −2 0 0 2 0 0 0 1
14400.c.60a R 60 −4 0 0 0 −4 −6 0 0 0 0 0 5 0 0 0 0 2 0 0 1 0 0 0 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $50000 x^{10} - 25000 x^{9} + 3125 x^{8} + 4 x^{5} - x^{4} + t$ Copy content Toggle raw display