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Properties

Label 16T969
16T969 1 3 1->3 12 1->12 13 1->13 2 4 2->4 11 2->11 14 2->14 3->2 9 3->9 3->11 4->1 10 4->10 4->12 5 7 5->7 5->9 16 5->16 6 8 6->8 6->10 15 6->15 7->6 7->13 7->15 8->5 8->14 8->16 9->4 9->8 9->15 10->3 10->7 10->16 11->1 11->9 12->2 12->10 13->4 13->8 13->11 14->3 14->7 14->12 15->5 15->13 16->6 16->14
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^6:D_4$

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Copy content magma:G := TransitiveGroup(16, 969);
 

Group invariants

Abstract group:  $C_2^6:D_4$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $969$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3,2,4)(5,7,6,8)(9,15,13,11)(10,16,14,12)$, $(1,13,8,16,6,10,3,11)(2,14,7,15,5,9,4,12)$, $(1,12)(2,11)(3,9,8,14)(4,10,7,13)(5,16)(6,15)$
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 12, $C_2^3$
$16$:  $D_4\times C_2$ x 6, $Q_8:C_2$
$32$:  $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 6, 32T320
$128$:  $C_2 \wr C_2\wr C_2$ x 2, 16T342 x 2, 16T350 x 3
$256$:  32T5807 x 2, 32T6030

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$, $C_2 \wr C_2\wr C_2$ x 2

Low degree siblings

16T969 x 63, 32T10844 x 32, 32T10845 x 32, 32T10846 x 16, 32T10847 x 16, 32T10848 x 16, 32T19406 x 8, 32T19412 x 8, 32T19429 x 16, 32T20944 x 16, 32T21627 x 8, 32T21633 x 16, 32T21675 x 16, 32T21677 x 8

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 $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)$
2E $2^{8}$ $2$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12)(13,14)(15,16)$
2F $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,14)(10,13)(11,16)(12,15)$
2G $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2H $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,13)(10,14)(11,15)(12,16)$
2I $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,10)(11,12)(13,14)(15,16)$
2J $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2K $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,15)(12,16)(13,14)$
2L $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,16)(10,15)(11,14)(12,13)$
2M $2^{4},1^{8}$ $4$ $2$ $4$ $( 3, 8)( 4, 7)( 9,14)(10,13)$
2N $2^{4},1^{8}$ $8$ $2$ $4$ $( 9,15)(10,16)(11,13)(12,14)$
2O $2^{6},1^{4}$ $8$ $2$ $6$ $( 3, 8)( 4, 7)( 9,13)(10,14)(11,12)(15,16)$
2P $2^{4},1^{8}$ $8$ $2$ $4$ $( 3, 7)( 4, 8)(11,12)(15,16)$
2Q $2^{6},1^{4}$ $8$ $2$ $6$ $( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
2R $2^{6},1^{4}$ $8$ $2$ $6$ $( 3, 4)( 7, 8)( 9,14)(10,13)(11,12)(15,16)$
2S $2^{8}$ $8$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15)(10,16)(11,13)(12,14)$
2T $2^{8}$ $8$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,14)(10,13)(11,12)(15,16)$
2U $2^{8}$ $8$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15)(10,16)(11,13)(12,14)$
2V $2^{8}$ $8$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,15)(10,16)(11,13)(12,14)$
2W $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)$
2X $2^{8}$ $16$ $2$ $8$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7,16)( 8,15)$
4A $4^{4}$ $8$ $4$ $12$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,15,14,12)(10,16,13,11)$
4B $4^{4}$ $8$ $4$ $12$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,12,14,15)(10,11,13,16)$
4C $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 3, 8)( 4, 7)( 9,15,14,12)(10,16,13,11)$
4D $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 3, 7)( 4, 8)( 9,15,10,16)(11,14,12,13)$
4E $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 3, 4)( 7, 8)( 9,15,13,11)(10,16,14,12)$
4F $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,15,13,11)(10,16,14,12)$
4G $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 4)( 7, 8)( 9,15,10,16)(11,14,12,13)$
4H $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 5)( 2, 6)( 3, 4)( 7, 8)( 9,15,14,12)(10,16,13,11)$
4I $4^{4}$ $16$ $4$ $12$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,15,10,16)(11,14,12,13)$
4J $4^{4}$ $16$ $4$ $12$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,12,10,11)(13,16,14,15)$
4K $4^{4}$ $16$ $4$ $12$ $( 1,13, 6,10)( 2,14, 5, 9)( 3,12, 8,15)( 4,11, 7,16)$
4L $4^{4}$ $16$ $4$ $12$ $( 1,13, 5, 9)( 2,14, 6,10)( 3,12, 7,16)( 4,11, 8,15)$
4M $4^{4}$ $16$ $4$ $12$ $( 1,13, 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7,16, 8,15)$
4N $4^{4}$ $32$ $4$ $12$ $( 1,16, 3,14)( 2,15, 4,13)( 5,12, 7,10)( 6,11, 8, 9)$
4O $4^{4}$ $32$ $4$ $12$ $( 1,13, 2,14)( 3,16, 7,12)( 4,15, 8,11)( 5, 9, 6,10)$
4P $4^{4}$ $32$ $4$ $12$ $( 1,11, 3,14)( 2,12, 4,13)( 5,15, 7,10)( 6,16, 8, 9)$
4Q $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,12)( 2,11)( 3, 9, 8,14)( 4,10, 7,13)( 5,16)( 6,15)$
8A1 $8^{2}$ $32$ $8$ $14$ $( 1,10, 3,11, 6,13, 8,16)( 2, 9, 4,12, 5,14, 7,15)$
8A-1 $8^{2}$ $32$ $8$ $14$ $( 1,16, 8,13, 6,11, 3,10)( 2,15, 7,14, 5,12, 4, 9)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

44 x 44 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed