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Properties

Label 16T815
16T815 1 2 1->2 7 1->7 9 1->9 8 2->8 10 2->10 3 3->2 4 3->4 12 3->12 4->1 11 4->11 5 5->3 13 5->13 6 6->4 14 6->14 7->6 15 7->15 8->5 16 8->16 9->5 9->10 9->15 10->6 10->16 11->8 11->10 12->7 12->9 13->2 13->11 13->14 14->1 14->12 15->4 15->14 16->3 16->13
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^6:(C_2\times C_4)$

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Copy content magma:G := TransitiveGroup(16, 815);
 

Group invariants

Abstract group:  $C_2^6:(C_2\times C_4)$
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 $t$:  $815$
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:  $(1,2)(3,4)$, $(1,9,5,13,2,10,6,14)(3,12,7,15,4,11,8,16)$, $(9,10)(13,14)$, $(1,7,6,4)(2,8,5,3)(9,15,14,12)(10,16,13,11)$
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: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$

Low degree siblings

16T815 x 3, 16T821 x 4, 16T863 x 8, 16T865 x 8, 32T9851 x 4, 32T9852 x 8, 32T9853 x 4, 32T9854 x 2, 32T9855 x 4, 32T9856 x 8, 32T9857 x 4, 32T9858 x 4, 32T9859 x 8, 32T9860 x 4, 32T9861 x 4, 32T9862 x 2, 32T9863 x 8, 32T9864 x 4, 32T9917 x 4, 32T9918 x 8, 32T9919 x 2, 32T9920 x 4, 32T9921 x 8, 32T9922 x 2, 32T9923 x 4, 32T10268 x 8, 32T10269 x 4, 32T10270 x 4, 32T10271 x 8, 32T10277 x 4, 32T10278 x 4, 32T19785 x 2, 32T20109 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 5, 6)( 9,10)(13,14)$
2C $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2D $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 5, 6)(11,12)(15,16)$
2E $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$
2F $2^{2},1^{12}$ $4$ $2$ $2$ $( 9,10)(13,14)$
2G $2^{8}$ $4$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
2H $2^{8}$ $4$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,14)(10,13)(11,16)(12,15)$
2I $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 2)( 3, 4)(11,12)(13,14)$
2J $2^{4},1^{8}$ $4$ $2$ $4$ $( 5, 6)( 7, 8)(11,12)(13,14)$
2K $2^{4},1^{8}$ $8$ $2$ $4$ $( 1, 2)( 5, 6)(13,14)(15,16)$
2L $2^{4},1^{8}$ $8$ $2$ $4$ $( 3, 4)( 7, 8)(13,14)(15,16)$
2M $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2N $2^{2},1^{12}$ $8$ $2$ $2$ $(11,12)(13,14)$
2O $2^{4},1^{8}$ $8$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2P $2^{8}$ $8$ $2$ $8$ $( 1,16)( 2,15)( 3,10)( 4, 9)( 5,12)( 6,11)( 7,13)( 8,14)$
2Q $2^{8}$ $8$ $2$ $8$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,16)( 6,15)( 7,10)( 8, 9)$
2R $2^{8}$ $8$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
4A $4^{4}$ $8$ $4$ $12$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
4B $4^{4}$ $8$ $4$ $12$ $( 1,16, 2,15)( 3,10, 4, 9)( 5,12, 6,11)( 7,13, 8,14)$
4C $4^{4}$ $8$ $4$ $12$ $( 1,12, 2,11)( 3,13, 4,14)( 5,16, 6,15)( 7,10, 8, 9)$
4D $4^{4}$ $8$ $4$ $12$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$
4E $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,13,10,14)(11,16,12,15)$
4F $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14,10,13)(11,15,12,16)$
4G $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,15, 2,16)( 3, 9)( 4,10)( 5,11, 6,12)( 7,14)( 8,13)$
4H $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,11, 2,12)( 3,14)( 4,13)( 5,15, 6,16)( 7, 9)( 8,10)$
4I1 $4^{4}$ $16$ $4$ $12$ $( 1, 3, 5, 7)( 2, 4, 6, 8)( 9,11,14,15)(10,12,13,16)$
4I-1 $4^{4}$ $16$ $4$ $12$ $( 1, 7, 5, 3)( 2, 8, 6, 4)( 9,15,14,11)(10,16,13,12)$
4J1 $4^{4}$ $16$ $4$ $12$ $( 1, 3, 6, 7)( 2, 4, 5, 8)( 9,12,14,15)(10,11,13,16)$
4J-1 $4^{4}$ $16$ $4$ $12$ $( 1, 8, 5, 3)( 2, 7, 6, 4)( 9,15,13,11)(10,16,14,12)$
4K1 $4^{4}$ $16$ $4$ $12$ $( 1,10, 5,13)( 2, 9, 6,14)( 3,12, 7,16)( 4,11, 8,15)$
4K-1 $4^{4}$ $16$ $4$ $12$ $( 1,13, 5,10)( 2,14, 6, 9)( 3,16, 7,12)( 4,15, 8,11)$
4L1 $4^{4}$ $16$ $4$ $12$ $( 1, 9, 6,13)( 2,10, 5,14)( 3,11, 7,15)( 4,12, 8,16)$
4L-1 $4^{4}$ $16$ $4$ $12$ $( 1,14, 6,10)( 2,13, 5, 9)( 3,15, 7,11)( 4,16, 8,12)$
4M $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,12, 2,11)( 3,14, 4,13)( 5,16)( 6,15)( 7, 9)( 8,10)$
4N $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,15)( 2,16)( 3, 9)( 4,10)( 5,12, 6,11)( 7,13, 8,14)$
4O1 $4^{4}$ $32$ $4$ $12$ $( 1, 3, 6, 7)( 2, 4, 5, 8)( 9,11,14,15)(10,12,13,16)$
4O-1 $4^{4}$ $32$ $4$ $12$ $( 1, 7, 6, 3)( 2, 8, 5, 4)( 9,15,14,11)(10,16,13,12)$
8A1 $8^{2}$ $32$ $8$ $14$ $( 1,10, 6,14, 2, 9, 5,13)( 3,12, 8,15, 4,11, 7,16)$
8A-1 $8^{2}$ $32$ $8$ $14$ $( 1,13, 5, 9, 2,14, 6,10)( 3,16, 7,11, 4,15, 8,12)$

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