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Properties

Label 16T1276
16T1276 1 6 1->6 10 1->10 16 1->16 2 5 2->5 9 2->9 15 2->15 3 8 3->8 11 3->11 14 3->14 4 7 4->7 12 4->12 13 4->13 5->6 5->15 6->16 7->3 7->8 7->14 8->4 8->13 9->4 9->12 9->13 10->3 10->11 10->14 11->1 11->16 12->2 12->15 13->3 13->6 13->12 14->4 14->5 14->11 15->1 15->7 15->10 16->2 16->8 16->9
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2^2\times C_4^2):D_8$

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Copy content magma:G := TransitiveGroup(16, 1276);
 

Group invariants

Abstract group:  $(C_2^2\times C_4^2):D_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $1024=2^{10}$
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:  $6$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1276$
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:  $(1,16,2,15)(3,14,4,13)(5,6)(7,8)(9,12)(10,11)$, $(1,6)(2,5)(3,8,4,7)(9,13,12,15,10,14,11,16)$, $(1,10,3,11)(2,9,4,12)(5,15,7,14)(6,16,8,13)$
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 6, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$ x 3
$32$:  $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T29
$64$:  $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T126
$128$:  $C_2 \wr C_2\wr C_2$ x 2, 16T409
$256$:  16T689
$512$:  16T962

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2 \wr C_2\wr C_2$

Low degree siblings

16T1271 x 8, 16T1276 x 15, 16T1282 x 8, 32T36786 x 8, 32T36787 x 8, 32T36788 x 8, 32T36789 x 8, 32T36790 x 4, 32T36791 x 8, 32T36792 x 16, 32T36793 x 4, 32T36794 x 8, 32T36795 x 8, 32T36796 x 8, 32T36797 x 4, 32T36798 x 8, 32T36799 x 4, 32T36800 x 8, 32T36801 x 8, 32T36802 x 4, 32T36803 x 8, 32T36804 x 4, 32T36826 x 8, 32T36827 x 8, 32T36828 x 8, 32T36829 x 8, 32T36830 x 8, 32T36831 x 8, 32T36862 x 16, 32T36863 x 8, 32T36864 x 4, 32T36865 x 8, 32T36866 x 8, 32T36867 x 4, 32T36868 x 8, 32T56379 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 $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 3)( 2, 4)(13,16)(14,15)$
2D $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,16)(14,15)$
2E $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2G $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2H $2^{6},1^{4}$ $8$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)$
2I $2^{4},1^{8}$ $16$ $2$ $4$ $( 1, 3)( 2, 4)( 5, 6)( 9,10)$
2J $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 3)( 2, 4)( 5,12)( 6,11)( 7,10)( 8, 9)$
2K $2^{8}$ $16$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(13,15)(14,16)$
2L $2^{6},1^{4}$ $16$ $2$ $6$ $( 5, 9)( 6,10)( 7,12)( 8,11)(13,14)(15,16)$
2M $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 3)( 2, 4)( 7, 8)( 9,10)(13,14)(15,16)$
2N $2^{8}$ $16$ $2$ $8$ $( 1, 3)( 2, 4)( 5,10)( 6, 9)( 7,12)( 8,11)(13,14)(15,16)$
2O $2^{8}$ $16$ $2$ $8$ $( 1,16)( 2,15)( 3,13)( 4,14)( 5, 9)( 6,10)( 7,12)( 8,11)$
2P $2^{8}$ $32$ $2$ $8$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,13)( 6,14)( 7,15)( 8,16)$
4A $4^{2},1^{8}$ $8$ $4$ $6$ $( 5,10, 6, 9)( 7,11, 8,12)$
4B $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 3)( 2, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,16)(14,15)$
4C $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 3)( 2, 4)( 5,12, 6,11)( 7, 9, 8,10)(13,16)(14,15)$
4D $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 2)( 3, 4)( 5,10, 6, 9)( 7,11, 8,12)(13,14)(15,16)$
4E $4^{4}$ $8$ $4$ $12$ $( 1,14, 2,13)( 3,15, 4,16)( 5, 9, 6,10)( 7,12, 8,11)$
4F $4^{4}$ $8$ $4$ $12$ $( 1,16, 2,15)( 3,13, 4,14)( 5,11, 6,12)( 7,10, 8, 9)$
4G $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 3)( 2, 4)( 5, 8, 6, 7)( 9,11,10,12)$
4H $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 1, 3)( 2, 4)( 5, 9, 6,10)( 7,11, 8,12)$
4I $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 3)( 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
4J $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 3)( 2, 4)( 5,11, 6,12)( 7, 9, 8,10)(13,14)(15,16)$
4K $4^{2},2^{2},1^{4}$ $32$ $4$ $8$ $( 1,13, 3,16)( 2,14, 4,15)( 7, 8)( 9,10)$
4L $4^{4}$ $32$ $4$ $12$ $( 1,13, 3,16)( 2,14, 4,15)( 5, 7, 6, 8)( 9,11,10,12)$
4M $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,13, 3,16)( 2,14, 4,15)( 5,10)( 6, 9)( 7,12)( 8,11)$
4N $4^{4}$ $32$ $4$ $12$ $( 1,13, 3,16)( 2,14, 4,15)( 5,11, 6,12)( 7, 9, 8,10)$
4O $4^{4}$ $32$ $4$ $12$ $( 1, 9, 3,11)( 2,10, 4,12)( 5,13, 8,16)( 6,14, 7,15)$
4P $4^{4}$ $32$ $4$ $12$ $( 1,11, 4,10)( 2,12, 3, 9)( 5,15, 7,13)( 6,16, 8,14)$
4Q $4^{4}$ $32$ $4$ $12$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,15, 6,16)( 7,13, 8,14)$
4R $4^{4}$ $32$ $4$ $12$ $( 1,15, 3,13)( 2,16, 4,14)( 5,10, 7,12)( 6, 9, 8,11)$
4S $4^{2},2^{2},1^{4}$ $32$ $4$ $8$ $( 3, 4)( 5, 6)( 9,11,10,12)(13,15,14,16)$
4T $4^{3},2,1^{2}$ $64$ $4$ $10$ $( 1, 4, 2, 3)( 5,11, 7, 9)( 6,12, 8,10)(15,16)$
8A $8^{2}$ $64$ $8$ $14$ $( 1, 9,14, 6, 2,10,13, 5)( 3,12,15, 8, 4,11,16, 7)$
8B $8^{2}$ $64$ $8$ $14$ $( 1, 7,16,10, 2, 8,15, 9)( 3, 5,13,11, 4, 6,14,12)$
8C1 $8^{2}$ $64$ $8$ $14$ $( 1, 9,15, 8, 3,11,13, 6)( 2,10,16, 7, 4,12,14, 5)$
8C3 $8^{2}$ $64$ $8$ $14$ $( 1, 8,13, 9, 3, 6,15,11)( 2, 7,14,10, 4, 5,16,12)$
8D1 $8,4,2^{2}$ $64$ $8$ $12$ $( 1, 7)( 2, 8)( 3, 5, 4, 6)( 9,16,11,13,10,15,12,14)$
8D3 $8,4,2^{2}$ $64$ $8$ $12$ $( 1, 7)( 2, 8)( 3, 6, 4, 5)( 9,13,12,16,10,14,11,15)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

43 x 43 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed