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Properties

Label 16T972
16T972 1 2 1->2 8 1->8 13 1->13 1->13 7 2->7 14 2->14 2->14 3 4 3->4 5 3->5 11 3->11 3->11 6 4->6 12 4->12 4->12 5->6 10 5->10 5->10 9 6->9 6->9 7->8 16 7->16 7->16 15 8->15 8->15 9->2 9->5 9->13 10->1 10->6 10->14 11->4 11->15 12->3 12->16 13->2 13->5 13->10 14->1 14->6 14->9 15->7 15->7 15->12 16->8 16->8 16->11
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8\wr C_2$

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Copy content magma:G := TransitiveGroup(16, 972);
 

Group invariants

Abstract group:  $D_8\wr C_2$
Copy content magma:IdentifyGroup(G);
 
Order:  $512=2^{9}$
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:  $6$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $972$
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:  $(1,13,2,14)(3,11,4,12)(5,10,6,9)(7,16,8,15)$, $(1,2)(3,4)(5,6)(7,8)$, $(1,8)(2,7)(3,5)(4,6)(9,13,10,14)(11,15,12,16)$, $(1,13,5,10)(2,14,6,9)(3,11)(4,12)(7,16,8,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$
$128$:  $C_2 \wr C_2\wr C_2$
$256$:  16T659

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2 \wr C_2\wr C_2$

Low degree siblings

16T972 x 7, 32T10866 x 4, 32T10867 x 4, 32T10868 x 4, 32T10869 x 4, 32T10870 x 4, 32T10871 x 4, 32T10872 x 4, 32T21596 x 2, 32T21635 x 2, 32T21959 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $8$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,10)(11,12)(13,14)(15,16)$
2D $2^{4},1^{8}$ $8$ $2$ $4$ $(1,7)(2,8)(3,6)(4,5)$
2E $2^{3},1^{10}$ $8$ $2$ $3$ $( 9,14)(10,13)(15,16)$
2F $2^{7},1^{2}$ $8$ $2$ $7$ $( 1, 2)( 3, 8)( 4, 7)( 9,10)(11,12)(13,14)(15,16)$
2G $2^{8}$ $16$ $2$ $8$ $( 1,16)( 2,15)( 3, 9)( 4,10)( 5,12)( 6,11)( 7,13)( 8,14)$
2H $2^{8}$ $16$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,15)(10,16)(11,13)(12,14)$
2I $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 2)( 3, 8)( 4, 7)( 9,10)(11,16)(12,15)$
2J $2^{7},1^{2}$ $32$ $2$ $7$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13)(10,14)(11,12)$
4A $4^{2},1^{8}$ $4$ $4$ $6$ $( 9,14,10,13)(11,16,12,15)$
4B $4^{4}$ $4$ $4$ $12$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
4C $4^{2},2^{4}$ $4$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,10,14)(11,15,12,16)$
4D $4^{4}$ $16$ $4$ $12$ $( 1,15, 2,16)( 3,10, 4, 9)( 5,11, 6,12)( 7,14, 8,13)$
4E $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13,10,14)(11,15,12,16)$
4F $4^{2},2^{3},1^{2}$ $16$ $4$ $9$ $( 1, 2)( 3, 8)( 4, 7)( 9,14,10,13)(11,16,12,15)$
4G $4^{4}$ $64$ $4$ $12$ $( 1,13, 3,11)( 2,14, 4,12)( 5,10, 7,16)( 6, 9, 8,15)$
4H $4^{3},2^{2}$ $64$ $4$ $11$ $( 1,10, 2, 9)( 3,15, 8,12)( 4,16, 7,11)( 5,14)( 6,13)$
8A1 $8,2^{4}$ $4$ $8$ $11$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11,14,16,10,12,13,15)$
8A3 $8,2^{4}$ $4$ $8$ $11$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,14,15,10,11,13,16)$
8B1 $8,1^{8}$ $4$ $8$ $7$ $( 9,12,14,15,10,11,13,16)$
8B3 $8,1^{8}$ $4$ $8$ $7$ $( 9,11,14,16,10,12,13,15)$
8C1 $8^{2}$ $4$ $8$ $14$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,16,13,11,10,15,14,12)$
8C3 $8^{2}$ $4$ $8$ $14$ $( 1, 8, 6, 3, 2, 7, 5, 4)( 9,11,14,16,10,12,13,15)$
8D $8^{2}$ $8$ $8$ $14$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,14,16,10,12,13,15)$
8E1 $8,4^{2}$ $8$ $8$ $13$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,16,13,11,10,15,14,12)$
8E3 $8,4^{2}$ $8$ $8$ $13$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,13,10,14)(11,15,12,16)$
8F1 $8,2^{4}$ $16$ $8$ $11$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,12,14,15,10,11,13,16)$
8F3 $8,2^{4}$ $16$ $8$ $11$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,11,14,16,10,12,13,15)$
8G1 $8,2^{3},1^{2}$ $16$ $8$ $10$ $( 3, 7)( 4, 8)( 5, 6)( 9,11,14,16,10,12,13,15)$
8G3 $8,2^{3},1^{2}$ $16$ $8$ $10$ $( 3, 7)( 4, 8)( 5, 6)( 9,12,14,15,10,11,13,16)$
8H $8^{2}$ $32$ $8$ $14$ $( 1,13, 6,10, 2,14, 5, 9)( 3,16, 7,11, 4,15, 8,12)$
16A1 $16$ $32$ $16$ $15$ $( 1,11, 3,10, 5,15, 8,14, 2,12, 4, 9, 6,16, 7,13)$
16A3 $16$ $32$ $16$ $15$ $( 1,10, 8,12, 6,13, 3,15, 2, 9, 7,11, 5,14, 4,16)$

Malle's constant $a(G)$:     $1/3$

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Character table

35 x 35 character table

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Regular extensions

Data not computed