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Properties

Label 16T834
16T834 1 8 1->8 1->8 13 1->13 2 7 2->7 2->7 14 2->14 3 5 3->5 6 3->6 12 3->12 4 4->5 4->6 11 4->11 5->3 15 5->15 6->4 16 6->16 7->1 9 7->9 8->2 10 8->10 9->8 9->11 9->11 10->7 10->12 10->12 11->3 11->10 12->4 12->9 13->15 13->15 14->16 14->16 15->14 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, 834);
 

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:  $5$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $834$
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:  $(1,8,2,7)(3,5)(4,6)(9,11)(10,12)(13,15)(14,16)$, $(1,8)(2,7)(3,6,4,5)(9,11,10,12)(13,15,14,16)$, $(1,13)(2,14)(3,12,4,11)(5,15)(6,16)(7,9,8,10)$
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 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:  $Z_8 : Z_8^\times$, $C_4\wr C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$, 16T32
$64$:  $((C_8 : C_2):C_2):C_2$ x 2, 16T76, 16T111, 32T264
$128$:  16T227, 16T234, 16T254
$256$:  32T4016

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$

Low degree siblings

16T834 x 3, 16T862 x 4, 16T897 x 8, 16T900 x 4, 16T921 x 4, 32T10002 x 4, 32T10003 x 4, 32T10004 x 4, 32T10005 x 2, 32T10006 x 4, 32T10007 x 4, 32T10008 x 2, 32T10009 x 2, 32T10010 x 2, 32T10011 x 2, 32T10012 x 2, 32T10013 x 4, 32T10014 x 2, 32T10015 x 2, 32T10016 x 4, 32T10017 x 8, 32T10018 x 2, 32T10019 x 2, 32T10020 x 2, 32T10021 x 2, 32T10022 x 4, 32T10023 x 4, 32T10024 x 4, 32T10262 x 2, 32T10263 x 2, 32T10264 x 2, 32T10265 x 2, 32T10266 x 2, 32T10267 x 2, 32T10448 x 4, 32T10449 x 4, 32T10450 x 4, 32T10451 x 4, 32T10452 x 4, 32T10453 x 4, 32T10462 x 4, 32T10463 x 4, 32T10464 x 4, 32T10584 x 4, 32T10585 x 4, 32T10586 x 4, 32T22167 x 2, 32T22180 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 $2^{4},1^{8}$ $2$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2C $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(15,16)$
2D $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(15,16)$
2E $2^{2},1^{12}$ $4$ $2$ $2$ $(5,6)(7,8)$
2F $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(15,16)$
2G $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 5, 6)( 9,10)(11,12)(13,14)(15,16)$
2H $2^{2},1^{12}$ $4$ $2$ $2$ $(1,2)(5,6)$
2I $2^{4},1^{8}$ $4$ $2$ $4$ $( 3, 4)( 5, 6)( 9,10)(11,12)$
2J $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 7, 8)( 9,10)(11,12)$
2K $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2L $2^{2},1^{12}$ $4$ $2$ $2$ $(1,2)(7,8)$
2M $2^{4},1^{8}$ $4$ $2$ $4$ $( 3, 4)( 5, 6)(11,12)(15,16)$
2N $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 5, 6)(13,14)(15,16)$
2O $2^{4},1^{8}$ $8$ $2$ $4$ $( 1, 2)( 7, 8)(11,12)(13,14)$
2P $2^{4},1^{8}$ $8$ $2$ $4$ $( 1, 2)( 5, 6)(11,12)(13,14)$
2Q $2^{8}$ $16$ $2$ $8$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5,15)( 6,16)( 7, 9)( 8,10)$
4A1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,16,10,15)(11,13)(12,14)$
4A-1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 8, 6, 7)( 9,15,10,16)(11,13)(12,14)$
4B1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,15)(10,16)(11,14,12,13)$
4B-1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 8, 6, 7)( 9,15)(10,16)(11,13,12,14)$
4C $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 4)( 2, 3)( 5, 8, 6, 7)( 9,16,10,15)(11,13)(12,14)$
4D $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 4, 2, 3)( 5, 8)( 6, 7)( 9,15,10,16)(11,14)(12,13)$
4E $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,14)( 2,13)( 3,11)( 4,12)( 5,16, 6,15)( 7,10, 8, 9)$
4F $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,13, 2,14)( 3,12, 4,11)( 5,15)( 6,16)( 7, 9)( 8,10)$
4G $4^{4}$ $16$ $4$ $12$ $( 1,13, 2,14)( 3,12, 4,11)( 5,16, 6,15)( 7,10, 8, 9)$
4H1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1, 8, 2, 7)( 3, 6)( 4, 5)( 9,12,10,11)(13,16,14,15)$
4H-1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1, 7, 2, 8)( 3, 6)( 4, 5)( 9,11,10,12)(13,15,14,16)$
4I1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1, 8)( 2, 7)( 3, 6, 4, 5)( 9,12,10,11)(13,16,14,15)$
4I-1 $4^{3},2^{2}$ $16$ $4$ $11$ $( 1, 7)( 2, 8)( 3, 6, 4, 5)( 9,11,10,12)(13,15,14,16)$
4J1 $4,2^{6}$ $16$ $4$ $9$ $( 1, 8, 2, 7)( 3, 6)( 4, 5)( 9,11)(10,12)(13,16)(14,15)$
4J-1 $4,2^{6}$ $16$ $4$ $9$ $( 1, 7, 2, 8)( 3, 6)( 4, 5)( 9,12)(10,11)(13,15)(14,16)$
4K1 $4,2^{6}$ $16$ $4$ $9$ $( 1, 8)( 2, 7)( 3, 6, 4, 5)( 9,11)(10,12)(13,16)(14,15)$
4K-1 $4,2^{6}$ $16$ $4$ $9$ $( 1, 7)( 2, 8)( 3, 6, 4, 5)( 9,12)(10,11)(13,15)(14,16)$
4L $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,13)( 2,14)( 3,11, 4,12)( 5,15, 6,16)( 7,10)( 8, 9)$
4M $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,13, 2,14)( 3,12)( 4,11)( 5,16, 6,15)( 7,10)( 8, 9)$
8A1 $8,4^{2}$ $32$ $8$ $13$ $( 1,10, 4,16)( 2, 9, 3,15)( 5,11, 7,14, 6,12, 8,13)$
8A-1 $8,4^{2}$ $32$ $8$ $13$ $( 1,16, 4,10)( 2,15, 3, 9)( 5,13, 8,12, 6,14, 7,11)$
8B1 $8,4^{2}$ $32$ $8$ $13$ $( 1,10, 4,16)( 2, 9, 3,15)( 5,11, 8,13, 6,12, 7,14)$
8B-1 $8,4^{2}$ $32$ $8$ $13$ $( 1,16, 4,10)( 2,15, 3, 9)( 5,14, 7,12, 6,13, 8,11)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

41 x 41 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed