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Properties

Label 16T1291
16T1291 1 12 1->12 2 11 2->11 3 5 3->5 9 3->9 4 6 4->6 10 4->10 8 5->8 14 5->14 7 6->7 13 6->13 7->4 16 7->16 8->3 15 8->15 9->7 9->15 10->8 10->16 11->1 11->13 12->2 12->14 13->3 13->12 14->4 14->11 15->6 15->10 16->5 16->9
Degree $16$
Order $1152$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $\SL(2,3)\wr C_2$

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Copy content magma:G := TransitiveGroup(16, 1291);
 

Group invariants

Abstract group:  $\SL(2,3)\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $1152=2^{7} \cdot 3^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1291$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,12,2,11)(3,9,7,16,5,14,4,10,8,15,6,13)$, $(3,5,8)(4,6,7)(9,15,10,16)(11,13,12,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $S_3$, $C_6$
$18$:  $S_3\times C_3$
$288$:  $A_4\wr C_2$
$576$:  24T1482

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 8: $A_4\wr C_2$

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 $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{4},1^{8}$ $2$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2C $2^{8}$ $24$ $2$ $8$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,14)( 6,13)( 7,16)( 8,15)$
3A1 $3^{2},1^{10}$ $8$ $3$ $4$ $(1,4,5)(2,3,6)$
3A-1 $3^{2},1^{10}$ $8$ $3$ $4$ $(1,5,4)(2,6,3)$
3B1 $3^{4},1^{4}$ $16$ $3$ $8$ $( 1, 4, 5)( 2, 3, 6)(11,14,16)(12,13,15)$
3B-1 $3^{4},1^{4}$ $16$ $3$ $8$ $( 1, 5, 4)( 2, 6, 3)(11,16,14)(12,15,13)$
3C $3^{4},1^{4}$ $32$ $3$ $8$ $( 1, 5, 4)( 2, 6, 3)(11,14,16)(12,13,15)$
4A $4^{2},1^{8}$ $12$ $4$ $6$ $( 9,16,10,15)(11,14,12,13)$
4B $4^{2},2^{4}$ $12$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,16,10,15)(11,14,12,13)$
4C $4^{4}$ $24$ $4$ $12$ $( 1,12, 2,11)( 3,15, 4,16)( 5, 9, 6,10)( 7,13, 8,14)$
4D $4^{4}$ $36$ $4$ $12$ $( 1, 7, 2, 8)( 3, 5, 4, 6)( 9,16,10,15)(11,14,12,13)$
6A1 $6,2,1^{8}$ $8$ $6$ $6$ $(1,6,4,2,5,3)(7,8)$
6A-1 $6,2,1^{8}$ $8$ $6$ $6$ $(1,3,5,2,4,6)(7,8)$
6B1 $6,2^{5}$ $8$ $6$ $10$ $( 1, 6, 4, 2, 5, 3)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
6B-1 $6,2^{5}$ $8$ $6$ $10$ $( 1, 3, 5, 2, 4, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
6C1 $3^{2},2^{4},1^{2}$ $8$ $6$ $8$ $( 3, 8, 5)( 4, 7, 6)( 9,10)(11,12)(13,14)(15,16)$
6C-1 $3^{2},2^{4},1^{2}$ $8$ $6$ $8$ $( 3, 5, 8)( 4, 6, 7)( 9,10)(11,12)(13,14)(15,16)$
6D1 $6^{2},2^{2}$ $16$ $6$ $12$ $( 1, 6, 4, 2, 5, 3)( 7, 8)( 9,15,12,10,16,11)(13,14)$
6D-1 $6^{2},2^{2}$ $16$ $6$ $12$ $( 1, 3, 5, 2, 4, 6)( 7, 8)( 9,11,16,10,12,15)(13,14)$
6E $6^{2},2^{2}$ $32$ $6$ $12$ $( 1, 3, 5, 2, 4, 6)( 7, 8)( 9,10)(11,15,14,12,16,13)$
6F1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1, 6, 4, 2, 5, 3)( 7, 8)(11,16,14)(12,15,13)$
6F-1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1, 3, 5, 2, 4, 6)( 7, 8)(11,14,16)(12,13,15)$
6G1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1, 3, 5, 2, 4, 6)( 7, 8)(11,16,14)(12,15,13)$
6G-1 $6,3^{2},2,1^{2}$ $32$ $6$ $10$ $( 1, 6, 4, 2, 5, 3)( 7, 8)(11,14,16)(12,13,15)$
6H1 $6^{2},2^{2}$ $96$ $6$ $12$ $( 1,15, 6,12, 8,13)( 2,16, 5,11, 7,14)( 3, 9)( 4,10)$
6H-1 $6^{2},2^{2}$ $96$ $6$ $12$ $( 1,13, 8,12, 6,15)( 2,14, 7,11, 5,16)( 3, 9)( 4,10)$
8A $8^{2}$ $144$ $8$ $14$ $( 1,12, 7,13, 2,11, 8,14)( 3,15, 5, 9, 4,16, 6,10)$
12A1 $4^{2},3^{2},1^{2}$ $48$ $12$ $10$ $( 3, 5, 8)( 4, 6, 7)( 9,15,10,16)(11,13,12,14)$
12A-1 $4^{2},3^{2},1^{2}$ $48$ $12$ $10$ $( 3, 8, 5)( 4, 7, 6)( 9,16,10,15)(11,14,12,13)$
12B1 $6,4^{2},2$ $48$ $12$ $12$ $( 1, 3, 5, 2, 4, 6)( 7, 8)( 9,15,10,16)(11,13,12,14)$
12B-1 $6,4^{2},2$ $48$ $12$ $12$ $( 1, 6, 4, 2, 5, 3)( 7, 8)( 9,16,10,15)(11,14,12,13)$
12C1 $12,4$ $96$ $12$ $14$ $( 1,15, 6,12, 4,10, 2,16, 5,11, 3, 9)( 7,14, 8,13)$
12C-1 $12,4$ $96$ $12$ $14$ $( 1, 9, 3,11, 5,16, 2,10, 4,12, 6,15)( 7,13, 8,14)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

35 x 35 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed