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Properties

Label 16T940
16T940 1 5 1->5 8 1->8 15 1->15 2 6 2->6 7 2->7 16 2->16 3 3->5 3->8 13 3->13 4 4->6 4->7 14 4->14 9 5->9 5->9 12 5->12 10 6->10 6->10 11 6->11 7->9 7->12 7->12 8->10 8->11 8->11 9->8 9->13 9->14 10->7 10->13 10->14 11->5 11->15 11->16 12->6 12->15 12->16 13->2 13->3 14->1 14->4 15->1 15->4 16->2 16->3
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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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 940);
 

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 $t$:  $940$
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:  $(1,8,11,15,4,6,10,14)(2,7,12,16,3,5,9,13)$, $(1,15)(2,16)(3,13)(4,14)(5,12,6,11)(7,9,8,10)$, $(1,5,9,14,4,7,12,15)(2,6,10,13,3,8,11,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(C_4^2 : C_2):C_2$ x 2, $((C_8 : C_2):C_2):C_2$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239
$128$:  16T219, 16T222, 16T227, 16T230, 16T234, 16T235, 16T302
$256$:  32T3918, 32T4019, 32T4050

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

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

Low degree siblings

16T940 x 31, 32T10678 x 8, 32T10679 x 8, 32T10680 x 8, 32T10681 x 8, 32T10682 x 8, 32T10683 x 8, 32T10684 x 8, 32T18651 x 4, 32T18656 x 4, 32T18658 x 4, 32T18667 x 4, 32T19251 x 4, 32T19263 x 4, 32T19274 x 4, 32T19276 x 4, 32T19312 x 4, 32T19326 x 4, 32T19327 x 4, 32T19328 x 4, 32T19469 x 4, 32T19689 x 4, 32T19693 x 4, 32T20170 x 4, 32T20208 x 4, 32T20230 x 4, 32T20242 x 4, 32T20307 x 4, 32T20615 x 4

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 $2^{8}$ $1$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)$
2E $2^{8}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,14)(15,16)$
2F $2^{8}$ $2$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2G $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 3)( 2, 4)( 9,11)(10,12)$
2H $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 4)( 2, 3)( 9,12)(10,11)$
2I $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2J $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)$
2K $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)$
2L $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,11)(10,12)(13,15)(14,16)$
2M $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,12)(10,11)(13,16)(14,15)$
2N $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2O $2^{8}$ $4$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2P $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 4)( 2, 3)( 9,11)(10,12)(13,14)(15,16)$
2Q $2^{6},1^{4}$ $8$ $2$ $6$ $( 5, 6)( 7, 8)( 9,12)(10,11)(13,15)(14,16)$
2R $2^{6},1^{4}$ $8$ $2$ $6$ $( 5, 6)( 7, 8)( 9,11)(10,12)(13,16)(14,15)$
2S $2^{8}$ $16$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5, 6)( 7, 8)(13,16)(14,15)$
2T $2^{6},1^{4}$ $16$ $2$ $6$ $( 1,10)( 2, 9)( 3,12)( 4,11)(13,15)(14,16)$
2U $2^{8}$ $16$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,15)( 6,16)( 7,14)( 8,13)$
4A $4^{2},1^{8}$ $8$ $4$ $6$ $( 1, 9, 3,11)( 2,10, 4,12)$
4B $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 9, 3,11)( 2,10, 4,12)( 5, 6)( 7, 8)(13,14)(15,16)$
4C $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 9, 3,11)( 2,10, 4,12)( 5, 7)( 6, 8)(13,16)(14,15)$
4D $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 9, 3,11)( 2,10, 4,12)( 5, 8)( 6, 7)(13,15)(14,16)$
4E $4^{4}$ $8$ $4$ $12$ $( 1, 9, 3,11)( 2,10, 4,12)( 5,16, 8,14)( 6,15, 7,13)$
4F $4^{4}$ $8$ $4$ $12$ $( 1,12, 3,10)( 2,11, 4, 9)( 5,16, 8,14)( 6,15, 7,13)$
4G $4^{4}$ $8$ $4$ $12$ $( 1,11, 4,10)( 2,12, 3, 9)( 5,13, 7,16)( 6,14, 8,15)$
4H $4^{4}$ $8$ $4$ $12$ $( 1,12, 4, 9)( 2,11, 3,10)( 5,15, 7,14)( 6,16, 8,13)$
4I $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 9, 4,12)( 2,10, 3,11)(13,14)(15,16)$
4J $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,10, 2, 9)( 3,12, 4,11)( 5, 6)( 7, 8)(13,15)(14,16)$
4K $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1,10, 2, 9)( 3,12, 4,11)(13,16)(14,15)$
4L $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 9, 4,12)( 2,10, 3,11)( 5, 7)( 6, 8)(13,15)(14,16)$
4M $4^{4}$ $16$ $4$ $12$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,15, 6,16)( 7,14, 8,13)$
4N1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,16,10,15)(11,14,12,13)$
4N-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,15,10,16)(11,13,12,14)$
4O1 $4^{4}$ $16$ $4$ $12$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,13,11,15)(10,14,12,16)$
4O-1 $4^{4}$ $16$ $4$ $12$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,14,11,16)(10,13,12,15)$
4P1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14,10,13)(11,16,12,15)$
4P-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13,10,14)(11,15,12,16)$
4Q1 $4^{4}$ $16$ $4$ $12$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,15,11,13)(10,16,12,14)$
4Q-1 $4^{4}$ $16$ $4$ $12$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,16,11,14)(10,15,12,13)$
8A1 $8^{2}$ $32$ $8$ $14$ $( 1,15,11, 6, 4,14,10, 8)( 2,16,12, 5, 3,13, 9, 7)$
8A-1 $8^{2}$ $32$ $8$ $14$ $( 1, 8,10,14, 4, 6,11,15)( 2, 7, 9,13, 3, 5,12,16)$
8B1 $8^{2}$ $32$ $8$ $14$ $( 1, 6,12,16, 4, 8, 9,13)( 2, 5,11,15, 3, 7,10,14)$
8B-1 $8^{2}$ $32$ $8$ $14$ $( 1,13, 9, 8, 4,16,12, 6)( 2,14,10, 7, 3,15,11, 5)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

47 x 47 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed