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Properties

Label 16T1250
16T1250 1 4 1->4 10 1->10 2 3 2->3 9 2->9 3->1 11 3->11 4->2 12 4->12 5 7 5->7 14 5->14 6 8 6->8 13 6->13 15 7->15 16 8->16 9->3 9->14 10->4 10->13 11->6 12->5 13->7 13->9 14->8 14->10 15->1 15->16 16->2
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^6.\SD_{16}$

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Copy content magma:G := TransitiveGroup(16, 1250);
 

Group invariants

Abstract group:  $C_2^6.\SD_{16}$
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:  $7$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1250$
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:  $(1,4,2,3)(5,7)(6,8)(9,14,10,13)(15,16)$, $(1,10,4,12,5,14,8,16,2,9,3,11,6,13,7,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $QD_{16}$, $C_2^2:C_4$, $Q_{16}$
$32$:  $C_4\wr C_2$, $C_2^3 : C_4 $, 32T50
$64$:  $((C_8 : C_2):C_2):C_2$, 16T154, 16T161
$128$:  32T1744
$256$:  16T684
$512$:  32T16784

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $QD_{16}$

Low degree siblings

16T1250 x 3, 16T1264 x 4, 32T36613 x 2, 32T36614 x 8, 32T36615 x 4, 32T36616 x 2, 32T36729 x 4, 32T36730 x 2, 32T36731 x 4, 32T36732 x 2, 32T56626 x 2, 32T56629 x 2, 32T71138 x 2, 32T90322 x 2, 32T94985 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{2},1^{12}$ $4$ $2$ $2$ $(11,12)(15,16)$
2D $2^{4},1^{8}$ $4$ $2$ $4$ $( 3, 4)( 7, 8)(11,12)(15,16)$
2E $2^{6},1^{4}$ $4$ $2$ $6$ $( 3, 4)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2F $2^{4},1^{8}$ $8$ $2$ $4$ $( 5, 6)( 7, 8)(11,12)(15,16)$
2G $2^{6},1^{4}$ $8$ $2$ $6$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2H $2^{2},1^{12}$ $8$ $2$ $2$ $(13,14)(15,16)$
2I $2^{4},1^{8}$ $8$ $2$ $4$ $( 3, 4)( 7, 8)(13,14)(15,16)$
2J $2^{4},1^{8}$ $16$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2K $2^{8}$ $16$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
4A $4^{4}$ $8$ $4$ $12$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$
4B $4^{4}$ $8$ $4$ $12$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
4C $4^{2},2^{4}$ $32$ $4$ $10$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13,10,14)(11,15,12,16)$
4D1 $4,2^{4},1^{4}$ $32$ $4$ $7$ $( 3, 7)( 4, 8)( 9,11)(10,12)(13,15,14,16)$
4D-1 $4,2^{4},1^{4}$ $32$ $4$ $7$ $( 3, 7)( 4, 8)( 9,11)(10,12)(13,16,14,15)$
4E1 $4^{2},2^{3},1^{2}$ $32$ $4$ $9$ $( 3, 7, 4, 8)( 5, 6)( 9,11)(10,12)(13,15,14,16)$
4E-1 $4^{2},2^{3},1^{2}$ $32$ $4$ $9$ $( 3, 7, 4, 8)( 5, 6)( 9,11)(10,12)(13,16,14,15)$
4F1 $4^{2},2^{3},1^{2}$ $32$ $4$ $9$ $( 3, 7, 4, 8)( 5, 6)( 9,11,10,12)(13,15)(14,16)$
4F-1 $4^{2},2^{3},1^{2}$ $32$ $4$ $9$ $( 3, 7, 4, 8)( 5, 6)( 9,11,10,12)(13,16)(14,15)$
4G1 $4,2^{6}$ $32$ $4$ $9$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,11)(10,12)(13,15,14,16)$
4G-1 $4,2^{6}$ $32$ $4$ $9$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,11)(10,12)(13,16,14,15)$
4H1 $4^{4}$ $64$ $4$ $12$ $( 1,10, 6,13)( 2, 9, 5,14)( 3,16, 8,11)( 4,15, 7,12)$
4H-1 $4^{4}$ $64$ $4$ $12$ $( 1,13, 6,10)( 2,14, 5, 9)( 3,11, 8,16)( 4,12, 7,15)$
8A1 $8^{2}$ $32$ $8$ $14$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,15,14,12,10,16,13,11)$
8A-1 $8^{2}$ $32$ $8$ $14$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,13,16,10,12,14,15)$
8B $8^{2}$ $64$ $8$ $14$ $( 1, 3, 6, 8, 2, 4, 5, 7)( 9,11,14,15,10,12,13,16)$
8C $8^{2}$ $64$ $8$ $14$ $( 1,13, 6,10, 2,14, 5, 9)( 3,12, 8,15, 4,11, 7,16)$
8D $8^{2}$ $64$ $8$ $14$ $( 1,12, 6,16, 2,11, 5,15)( 3,10, 8,13, 4, 9, 7,14)$
16A1 $16$ $64$ $16$ $15$ $( 1,15, 7,14, 6,12, 4,10, 2,16, 8,13, 5,11, 3, 9)$
16A-1 $16$ $64$ $16$ $15$ $( 1, 9, 3,11, 5,13, 8,16, 2,10, 4,12, 6,14, 7,15)$
16A3 $16$ $64$ $16$ $15$ $( 1,14, 4,16, 5, 9, 7,12, 2,13, 3,15, 6,10, 8,11)$
16A-3 $16$ $64$ $16$ $15$ $( 1,11, 8,10, 6,15, 3,13, 2,12, 7, 9, 5,16, 4,14)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

34 x 34 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed