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Properties

Label 16T830
16T830 1 2 1->2 6 1->6 5 2->5 3 4 3->4 8 3->8 7 4->7 9 5->9 10 5->10 15 5->15 6->9 6->10 16 6->16 11 7->11 12 7->12 14 7->14 8->11 8->12 13 8->13 9->4 9->5 9->8 10->3 10->6 10->7 11->2 11->5 11->8 12->1 12->6 12->7 13->9 13->14 13->15 14->10 14->16 15->12 15->14 16->11 16->13
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, 830);
 

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 $t$:  $830$
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:  $(3,4)(5,10,6,9)(7,11,8,12)(13,14)$, $(1,6,16,11,2,5,15,12)(3,8,13,9,4,7,14,10)$, $(1,2)(5,9,8,11)(6,10,7,12)(13,15,14,16)$
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
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239
$128$:  16T208, 16T218, 16T230
$256$:  32T3729

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T830 x 7, 16T835 x 4, 16T848 x 4, 16T887 x 8, 32T9983 x 8, 32T9984 x 4, 32T9985 x 8, 32T9986 x 8, 32T9987 x 4, 32T10025 x 8, 32T10026 x 2, 32T10027 x 2, 32T10028 x 4, 32T10029 x 4, 32T10030 x 4, 32T10031 x 8, 32T10032 x 4, 32T10033 x 4, 32T10034 x 4, 32T10035 x 4, 32T10036 x 4, 32T10037 x 8, 32T10038 x 4, 32T10143 x 4, 32T10144 x 8, 32T10145 x 2, 32T10146 x 4, 32T10147 x 4, 32T10148 x 2, 32T10149 x 4, 32T10398 x 4, 32T10399 x 8, 32T10400 x 8, 32T10401 x 4, 32T19840 x 2, 32T20072 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 $2^{8}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$
2C $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2E $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,14)(15,16)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)$
2G $2^{4},1^{8}$ $4$ $2$ $4$ $(1,4)(2,3)(5,7)(6,8)$
2H $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 8)( 6, 7)( 9,12)(10,11)$
2I $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,15)(14,16)$
2J $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,12)(10,11)(13,16)(14,15)$
2K $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$
2L $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2M $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 4)( 2, 3)( 9,10)(11,12)(13,16)(14,15)$
2N $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,12)(10,11)$
2O $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 3)( 2, 4)( 9,11)(10,12)(13,14)(15,16)$
2P $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,10)(11,12)$
2Q $2^{8}$ $8$ $2$ $8$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5, 9)( 6,10)( 7,11)( 8,12)$
2R $2^{8}$ $8$ $2$ $8$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5,10)( 6, 9)( 7,12)( 8,11)$
2S $2^{6},1^{4}$ $16$ $2$ $6$ $( 1,14)( 2,13)( 3,16)( 4,15)( 5, 6)( 9,10)$
4A $4^{4}$ $8$ $4$ $12$ $( 1,16, 2,15)( 3,13, 4,14)( 5,10, 6, 9)( 7,12, 8,11)$
4B $4^{4}$ $8$ $4$ $12$ $( 1,13, 2,14)( 3,16, 4,15)( 5,10, 6, 9)( 7,12, 8,11)$
4C $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,13)( 2,14)( 3,15)( 4,16)( 5, 7, 6, 8)( 9,11,10,12)$
4D $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1,15, 2,16)( 3,13, 4,14)( 7, 8)(11,12)$
4E $4^{4}$ $16$ $4$ $12$ $( 1,16, 2,15)( 3,14, 4,13)( 5, 8, 6, 7)( 9,12,10,11)$
4F $4^{4}$ $16$ $4$ $12$ $( 1,15, 4,13)( 2,16, 3,14)( 5,10, 7,12)( 6, 9, 8,11)$
4G $4^{4}$ $16$ $4$ $12$ $( 1,16, 4,14)( 2,15, 3,13)( 5,11, 7, 9)( 6,12, 8,10)$
4H1 $4^{4}$ $16$ $4$ $12$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,14, 7,16)( 6,13, 8,15)$
4H-1 $4^{4}$ $16$ $4$ $12$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,16, 7,14)( 6,15, 8,13)$
4I1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,15)( 6,16)( 7,13)( 8,14)$
4I-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
4J1 $4^{4}$ $16$ $4$ $12$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,13, 7,15)( 6,14, 8,16)$
4J-1 $4^{4}$ $16$ $4$ $12$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,15, 7,13)( 6,16, 8,14)$
4K1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,16)( 6,15)( 7,14)( 8,13)$
4K-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,14)( 6,13)( 7,16)( 8,15)$
4L $4^{3},2,1^{2}$ $32$ $4$ $10$ $( 1,16, 4,13)( 2,15, 3,14)( 5, 6)( 9,11,10,12)$
4M $4^{3},2,1^{2}$ $32$ $4$ $10$ $( 1, 3, 2, 4)( 5,11, 7,10)( 6,12, 8, 9)(15,16)$
4N1 $4^{4}$ $32$ $4$ $12$ $( 1, 6,16, 9)( 2, 5,15,10)( 3, 8,13,11)( 4, 7,14,12)$
4N-1 $4^{4}$ $32$ $4$ $12$ $( 1, 9,16, 6)( 2,10,15, 5)( 3,11,13, 8)( 4,12,14, 7)$
8A1 $8^{2}$ $32$ $8$ $14$ $( 1, 7,13,12, 2, 8,14,11)( 3, 5,16,10, 4, 6,15, 9)$
8A-1 $8^{2}$ $32$ $8$ $14$ $( 1,11,14, 8, 2,12,13, 7)( 3, 9,15, 6, 4,10,16, 5)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

41 x 41 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed