Properties

Label 15T21
15T21 1 2 1->2 4 1->4 1->4 9 1->9 5 2->5 6 2->6 15 2->15 2->15 3 7 3->7 3->7 14 3->14 4->5 4->5 10 4->10 5->6 8 5->8 11 5->11 5->15 6->2 6->4 6->9 7->11 12 7->12 8->2 8->4 8->9 8->9 9->10 9->10 9->15 10->1 10->3 10->8 11->3 13 11->13 12->6 12->13 12->13 12->14 13->8 13->12 13->14 14->1 14->12 14->13 15->1 15->6 15->7
Degree 1515
Order 360360
Cyclic no
Abelian no
Solvable no
Primitive no
pp-group no
Group: GL(2,4):C2\GL(2,4):C_2

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Copy content magma:G := TransitiveGroup(15, 21);
 

Group invariants

Abstract group:  GL(2,4):C2\GL(2,4):C_2
Copy content magma:IdentifyGroup(G);
 
Order:  360=23325360=2^{3} \cdot 3^{2} \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:  1515
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number tt:  2121
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
CHM label:   3S5(15)3S_{5}(15)
Parity:  11
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
#Aut(F/K)\card{\Aut(F/K)}:  11
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8)(1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,2,15)(4,5,6)(8,9,10)(12,13,14)(1,2,15)(4,5,6)(8,9,10)(12,13,14), (1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13)(1,4,10)(2,5,8)(3,7,11)(6,9,15)(12,14,13), (1,4)(2,6)(3,7)(5,15)(8,9)(12,13)(1,4)(2,6)(3,7)(5,15)(8,9)(12,13)
Copy content magma:Generators(G);
 

Low degree resolvents

#(G/N)\card{(G/N)}Galois groups for stem field(s)
22C2C_2
66S3S_3
120120S5S_5

Resolvents shown for degrees 47\leq 47

Subfields

Degree 3: None

Degree 5: S5S_5

Low degree siblings

15T21, 15T22, 18T146, 30T89, 30T93 x 2, 30T101, 36T554, 45T45

Siblings are shown with degree 47\leq 47

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A 1151^{15} 11 11 00 ()()
2A 26,132^{6},1^{3} 1515 22 66 (1,11)(2,13)(3,6)(4,14)(7,8)(9,12)( 1,11)( 2,13)( 3, 6)( 4,14)( 7, 8)( 9,12)
2B 26,132^{6},1^{3} 3030 22 66 (2,12)(3,7)(4,14)(6,8)(9,13)(10,15)( 2,12)( 3, 7)( 4,14)( 6, 8)( 9,13)(10,15)
3A 353^{5} 22 33 1010 (1,6,8)(2,4,9)(3,7,11)(5,10,15)(12,13,14)( 1, 6, 8)( 2, 4, 9)( 3, 7,11)( 5,10,15)(12,13,14)
3B 353^{5} 2020 33 1010 (1,11,5)(2,9,4)(3,10,6)(7,15,8)(12,13,14)( 1,11, 5)( 2, 9, 4)( 3,10, 6)( 7,15, 8)(12,13,14)
3C 34,133^{4},1^{3} 4040 33 88 (1,5,9)(2,6,10)(3,7,11)(4,8,15)( 1, 5, 9)( 2, 6,10)( 3, 7,11)( 4, 8,15)
4A 43,2,14^{3},2,1 9090 44 1010 (1,2,11,13)(3,12,6,9)(4,7,14,8)(5,15)( 1, 2,11,13)( 3,12, 6, 9)( 4, 7,14, 8)( 5,15)
5A 535^{3} 2424 55 1212 (1,13,15,7,4)(2,8,12,10,3)(5,11,9,6,14)( 1,13,15, 7, 4)( 2, 8,12,10, 3)( 5,11, 9, 6,14)
6A 62,36^{2},3 3030 66 1212 (1,8,6)(2,7,4,11,9,3)(5,14,10,12,15,13)( 1, 8, 6)( 2, 7, 4,11, 9, 3)( 5,14,10,12,15,13)
6B 62,36^{2},3 6060 66 1212 (1,5,11)(2,14,9,12,4,13)(3,8,10,7,6,15)( 1, 5,11)( 2,14, 9,12, 4,13)( 3, 8,10, 7, 6,15)
15A1 1515 2424 1515 1414 (1,10,9,13,3,6,15,2,14,7,8,5,4,12,11)( 1,10, 9,13, 3, 6,15, 2,14, 7, 8, 5, 4,12,11)
15A-1 1515 2424 1515 1414 (1,11,12,4,5,8,7,14,2,15,6,3,13,9,10)( 1,11,12, 4, 5, 8, 7,14, 2,15, 6, 3,13, 9,10)

Malle's constant a(G)a(G):     1/61/6

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 3B 3C 4A 5A 6A 6B 15A1 15A-1
Size 1 15 30 2 20 40 90 24 30 60 24 24
2 P 1A 1A 1A 3A 3B 3C 2A 5A 3A 3B 15A1 15A-1
3 P 1A 2A 2B 1A 1A 1A 4A 5A 2A 2B 5A 5A
5 P 1A 2A 2B 3A 3B 3C 4A 1A 6A 6B 3A 3A
Type
360.120.1a R 1 1 1 1 1 1 1 1 1 1 1 1
360.120.1b R 1 1 1 1 1 1 1 1 1 1 1 1
360.120.2a R 2 2 0 1 2 1 0 2 1 0 1 1
360.120.4a R 4 0 2 4 1 1 0 1 0 1 1 1
360.120.4b R 4 0 2 4 1 1 0 1 0 1 1 1
360.120.5a R 5 1 1 5 1 1 1 0 1 1 0 0
360.120.5b R 5 1 1 5 1 1 1 0 1 1 0 0
360.120.6a R 6 2 0 6 0 0 0 1 2 0 1 1
360.120.6b1 C 6 2 0 3 0 0 0 1 1 0 12ζ15ζ152+ζ1532ζ154+ζ155ζ157 2+2ζ15+ζ152ζ153+2ζ154ζ155+ζ157
360.120.6b2 C 6 2 0 3 0 0 0 1 1 0 2+2ζ15+ζ152ζ153+2ζ154ζ155+ζ157 12ζ15ζ152+ζ1532ζ154+ζ155ζ157
360.120.8a R 8 0 0 4 2 1 0 2 0 0 1 1
360.120.10a R 10 2 0 5 2 1 0 0 1 0 0 0

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Regular extensions

Data not computed