Properties

Label 10T34
10T34 1 3 1->3 2 4 2->4 2->4 7 2->7 5 3->5 6 4->6 10 4->10 5->7 5->7 5->10 8 6->8 9 7->9 7->9 8->10 9->1 9->5 10->2 10->2
Degree 1010
Order 960960
Cyclic no
Abelian no
Solvable no
Primitive no
pp-group no
Group: C24:A5C_2^4 : A_5

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Copy content magma:G := TransitiveGroup(10, 34);
 

Group invariants

Abstract group:  C24:A5C_2^4 : A_5
Copy content magma:IdentifyGroup(G);
 
Order:  960=2635960=2^{6} \cdot 3 \cdot 5
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 nn:  1010
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  3434
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   [24]A(5)[2^{4}]A(5)
Parity:  11
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
#Aut(F/K)\card{\Aut(F/K)}:  22
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (2,4,10)(5,7,9)(2,4,10)(5,7,9), (1,3,5,7,9)(2,4,6,8,10)(1,3,5,7,9)(2,4,6,8,10), (2,7)(5,10)(2,7)(5,10)
Copy content magma:Generators(G);
 

Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
6060A5A_5

Resolvents shown for degrees 47\leq 47

Subfields

Degree 2: None

Degree 5: A5A_5

Low degree siblings

16T1081, 20T172, 20T177, 30T214, 30T217, 40T888, 40T889, 40T932, 40T942, 40T944, 40T945

Siblings are shown with degree 47\leq 47

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A 1101^{10} 11 11 00 ()()
2A 24,122^{4},1^{2} 55 22 44 (1,6)(2,7)(3,8)(5,10)( 1, 6)( 2, 7)( 3, 8)( 5,10)
2B 22,162^{2},1^{6} 1010 22 22 (4,9)(5,10)( 4, 9)( 5,10)
2C 24,122^{4},1^{2} 6060 22 44 (1,8)(2,5)(3,6)(7,10)( 1, 8)( 2, 5)( 3, 6)( 7,10)
3A 32,143^{2},1^{4} 8080 33 44 (2,3,5)(7,8,10)( 2, 3, 5)( 7, 8,10)
4A 42,124^{2},1^{2} 6060 44 66 (1,9,6,4)(3,5,8,10)( 1, 9, 6, 4)( 3, 5, 8,10)
4B 4,234,2^{3} 120120 44 66 (1,2)(3,8)(4,10,9,5)(6,7)( 1, 2)( 3, 8)( 4,10, 9, 5)( 6, 7)
5A1 525^{2} 192192 55 88 (1,8,5,7,4)(2,9,6,3,10)( 1, 8, 5, 7, 4)( 2, 9, 6, 3,10)
5A2 525^{2} 192192 55 88 (1,5,4,8,7)(2,6,10,9,3)( 1, 5, 4, 8, 7)( 2, 6,10, 9, 3)
6A 32,223^{2},2^{2} 8080 66 66 (1,5,3)(2,7)(4,9)(6,10,8)( 1, 5, 3)( 2, 7)( 4, 9)( 6,10, 8)
6B1 6,2,126,2,1^{2} 8080 66 66 (1,6)(2,10,3,7,5,8)( 1, 6)( 2,10, 3, 7, 5, 8)
6B-1 6,2,126,2,1^{2} 8080 66 66 (1,6)(2,8,5,7,3,10)( 1, 6)( 2, 8, 5, 7, 3,10)

Malle's constant a(G)a(G):     1/21/2

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A 4B 5A1 5A2 6A 6B1 6B-1
Size 1 5 10 60 80 60 120 192 192 80 80 80
2 P 1A 1A 1A 1A 3A 2A 2B 5A2 5A1 3A 3A 3A
3 P 1A 2A 2B 2C 1A 4A 4B 5A2 5A1 2B 2A 2A
5 P 1A 2A 2B 2C 3A 4A 4B 1A 1A 6A 6B-1 6B1
Type
960.11358.1a R 1 1 1 1 1 1 1 1 1 1 1 1
960.11358.3a1 R 3 3 3 1 0 1 1 ζ51ζ5 ζ52ζ52 0 0 0
960.11358.3a2 R 3 3 3 1 0 1 1 ζ52ζ52 ζ51ζ5 0 0 0
960.11358.4a R 4 4 4 0 1 0 0 1 1 1 1 1
960.11358.5a R 5 5 5 1 1 1 1 0 0 1 1 1
960.11358.5b R 5 3 1 1 2 1 1 0 0 2 0 0
960.11358.5c1 C 5 3 1 1 1 1 1 0 0 1 12ζ3 1+2ζ3
960.11358.5c2 C 5 3 1 1 1 1 1 0 0 1 1+2ζ3 12ζ3
960.11358.10a R 10 2 2 2 1 2 0 0 0 1 1 1
960.11358.10b R 10 2 2 2 1 2 0 0 0 1 1 1
960.11358.15a R 15 9 3 1 0 1 1 0 0 0 0 0
960.11358.20a R 20 4 4 0 1 0 0 0 0 1 1 1

Copy content magma:CharacterTable(G);
 

Regular extensions

f1=f_{ 1 } = x10+75x63t2x49t2x^{10} + 75 x^{6} - 3 t^{2} x^{4} - 9 t^{2} Copy content Toggle raw display