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Properties

Label 17T7
17T7 1 6 1->6 7 1->7 16 1->16 2 3 2->3 2->3 13 2->13 15 2->15 9 3->9 10 3->10 4 4->2 5 4->5 4->9 11 4->11 5->4 5->7 12 5->12 6->7 8 6->8 6->13 7->11 8->1 8->9 14 8->14 9->7 10->11 10->14 10->14 11->8 11->13 12->3 12->13 12->15 13->5 14->12 14->15 15->10 17 16->17
Degree $17$
Order $8160$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(2,16):C_2$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(17, 7);
 

Group invariants

Abstract group:  $\PSL(2,16):C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $8160=2^{5} \cdot 3 \cdot 5 \cdot 17$
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 $n$:  $17$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $7$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  yes
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(2,3)(4,9)(5,7)(6,8)(10,14)(11,13)(12,15)(16,17)$, $(1,16)(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)$, $(1,6,13,5,4,2,15,10,14,12,3,9,7,11,8)$, $(1,7)(2,13)(3,10)(4,11)(5,12)(8,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{17}$ $1$ $1$ $0$ $()$
2A $2^{6},1^{5}$ $68$ $2$ $6$ $( 1, 7)( 2, 6)( 3, 9)(11,16)(12,15)(14,17)$
2B $2^{8},1$ $255$ $2$ $8$ $( 1, 7)( 2,16)( 3,15)( 4,13)( 5,10)( 6,11)( 9,12)(14,17)$
3A $3^{5},1^{2}$ $272$ $3$ $10$ $( 2,10,12)( 3,11, 5)( 4, 9, 6)( 7,17,14)(13,15,16)$
4A $4^{4},1$ $1020$ $4$ $12$ $( 1, 4, 7,13)( 2,12,16, 9)( 3,11,15, 6)( 5,14,10,17)$
5A1 $5^{3},1^{2}$ $272$ $5$ $12$ $( 2,17,11, 4,13)( 3, 6,16,12, 7)( 5, 9,15,10,14)$
5A2 $5^{3},1^{2}$ $272$ $5$ $12$ $( 2,11,13,17, 4)( 3,16, 7, 6,12)( 5,15,14, 9,10)$
6A $6^{2},3,1^{2}$ $1360$ $6$ $12$ $( 1, 6,14, 7, 2,17)( 3,11,15, 9,16,12)( 8,13,10)$
10A1 $10,5,2$ $816$ $10$ $14$ $( 1,12, 7, 4,13, 8, 2,17, 6,16)( 3,11)( 5,14,15,10, 9)$
10A3 $10,5,2$ $816$ $10$ $14$ $( 1, 4, 2,16, 7, 8, 6,12,13,17)( 3,11)( 5,10,14, 9,15)$
15A1 $15,1^{2}$ $544$ $15$ $14$ $( 2, 3,15,17, 6,10,11,16,14, 4,12, 5,13, 7, 9)$
15A2 $15,1^{2}$ $544$ $15$ $14$ $( 2,15, 6,11,14,12,13, 9, 3,17,10,16, 4, 5, 7)$
17A1 $17$ $480$ $17$ $16$ $( 1,11, 3, 2,10,16, 5,14,13, 9, 6,17, 7, 8,12,15, 4)$
17A2 $17$ $480$ $17$ $16$ $( 1, 3,10, 5,13, 6, 7,12, 4,11, 2,16,14, 9,17, 8,15)$
17A3 $17$ $480$ $17$ $16$ $( 1, 2, 5, 9, 7,15,11,10,14, 6, 8, 4, 3,16,13,17,12)$
17A6 $17$ $480$ $17$ $16$ $( 1, 5, 7,11,14, 8, 3,13,12, 2, 9,15,10, 6, 4,16,17)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 4A 5A1 5A2 6A 10A1 10A3 15A1 15A2 17A1 17A2 17A3 17A6
Size 1 68 255 272 1020 272 272 1360 816 816 544 544 480 480 480 480
2 P 1A 1A 1A 3A 2B 5A2 5A1 3A 5A1 5A2 15A2 15A1 17A2 17A1 17A6 17A3
3 P 1A 2A 2B 1A 4A 5A2 5A1 2A 10A3 10A1 5A1 5A2 17A3 17A6 17A2 17A1
5 P 1A 2A 2B 3A 4A 1A 1A 6A 2A 2A 3A 3A 17A3 17A6 17A2 17A1
17 P 1A 2A 2B 3A 4A 5A2 5A1 6A 10A3 10A1 15A2 15A1 1A 1A 1A 1A
Type
8160.a.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
8160.a.1b R 1 −1 1 1 −1 1 1 −1 −1 −1 1 1 1 1 1 1
8160.a.16a R 16 4 0 1 0 1 1 1 −1 −1 1 1 −1 −1 −1 −1
8160.a.16b R 16 −4 0 1 0 1 1 −1 1 1 1 1 −1 −1 −1 −1
8160.a.17a R 17 5 1 −1 1 2 2 −1 0 0 −1 −1 0 0 0 0
8160.a.17b R 17 −5 1 −1 −1 2 2 1 0 0 −1 −1 0 0 0 0
8160.a.17c1 R 17 3 1 2 −1 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0
8160.a.17c2 R 17 3 1 2 −1 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0
8160.a.17d1 R 17 −3 1 2 1 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 0
8160.a.17d2 R 17 −3 1 2 1 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 0
8160.a.30a1 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−4−ζ17−1−ζ17−ζ174 −ζ17−8−ζ17−2−ζ172−ζ178 −ζ17−5−ζ17−3−ζ173−ζ175 −ζ17−7−ζ17−6−ζ176−ζ177
8160.a.30a2 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−5−ζ17−3−ζ173−ζ175 −ζ17−7−ζ17−6−ζ176−ζ177 −ζ17−8−ζ17−2−ζ172−ζ178 −ζ17−4−ζ17−1−ζ17−ζ174
8160.a.30a3 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−8−ζ17−2−ζ172−ζ178 −ζ17−4−ζ17−1−ζ17−ζ174 −ζ17−7−ζ17−6−ζ176−ζ177 −ζ17−5−ζ17−3−ζ173−ζ175
8160.a.30a4 R 30 0 −2 0 0 0 0 0 0 0 0 0 −ζ17−7−ζ17−6−ζ176−ζ177 −ζ17−5−ζ17−3−ζ173−ζ175 −ζ17−4−ζ17−1−ζ17−ζ174 −ζ17−8−ζ17−2−ζ172−ζ178
8160.a.34a1 R 34 0 2 −2 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0
8160.a.34a2 R 34 0 2 −2 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed