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Properties

Label 16T1255
16T1255 1 5 1->5 15 1->15 2 6 2->6 16 2->16 3 7 3->7 14 3->14 4 8 4->8 13 4->13 5->7 5->16 6->8 6->15 7->13 8->14 9 9->4 10 10->3 11 11->2 12 12->1 13->2 13->10 14->1 14->9 15->4 15->12 16->3 16->11
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(D_4\times C_2^3).Q_{16}$

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Copy content magma:G := TransitiveGroup(16, 1255);
 

Group invariants

Abstract group:  $(D_4\times C_2^3).Q_{16}$
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 $t$:  $1255$
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:  $(1,5,16,11,2,6,15,12)(3,7,13,10)(4,8,14,9)$, $(1,15,4,13,2,16,3,14)(5,7)(6,8)$
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
$512$:  32T26909

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2^3 : C_4 $

Low degree siblings

16T1255 x 3, 16T1258 x 4, 32T36662 x 4, 32T36663 x 4, 32T36664 x 2, 32T36665 x 4, 32T36666 x 2, 32T36678 x 4, 32T36679 x 2, 32T36680 x 4, 32T36681 x 2, 32T55451 x 2, 32T55452 x 2, 32T71146 x 2, 32T90080 x 2, 32T90323 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
2C $2^{2},1^{12}$ $4$ $2$ $2$ $(13,14)(15,16)$
2D $2^{4},1^{8}$ $4$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2E $2^{6},1^{4}$ $4$ $2$ $6$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2F $2^{4},1^{8}$ $8$ $2$ $4$ $( 3, 4)( 5, 6)( 9,10)(13,14)$
2G $2^{4},1^{8}$ $8$ $2$ $4$ $( 3, 4)( 7, 8)( 9,10)(13,14)$
2H $2^{8}$ $16$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2I $2^{6},1^{4}$ $32$ $2$ $6$ $( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(15,16)$
4A $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
4B $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
4C $4^{2},1^{8}$ $8$ $4$ $6$ $( 5, 7, 6, 8)( 9,11,10,12)$
4D $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 5, 7, 6, 8)( 9,12,10,11)(13,14)(15,16)$
4E $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
4F1 $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,16)( 2,15)( 3,14, 4,13)( 5,11)( 6,12)( 7, 9, 8,10)$
4F-1 $4^{2},2^{4}$ $32$ $4$ $10$ $( 1,16)( 2,15)( 3,13, 4,14)( 5,11)( 6,12)( 7,10, 8, 9)$
4G $4^{2},2^{4}$ $64$ $4$ $10$ $( 1,15)( 2,16)( 3,14, 4,13)( 5,10, 6, 9)( 7,12)( 8,11)$
4H $4^{2},2^{4}$ $64$ $4$ $10$ $( 1,10)( 2, 9)( 3,11, 4,12)( 5,16, 6,15)( 7,13)( 8,14)$
4I $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 8, 2, 7)( 3, 6)( 4, 5)( 9,14,10,13)(11,15)(12,16)$
4J1 $4^{4}$ $64$ $4$ $12$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,14,11,15)(10,13,12,16)$
4J-1 $4^{4}$ $64$ $4$ $12$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,15,11,14)(10,16,12,13)$
8A1 $8,2^{2},1^{4}$ $32$ $8$ $9$ $( 5, 9, 7,11, 6,10, 8,12)(13,15)(14,16)$
8A-1 $8,2^{2},1^{4}$ $32$ $8$ $9$ $( 5,11, 7,10, 6,12, 8, 9)(13,15)(14,16)$
8B1 $8,4,2,1^{2}$ $32$ $8$ $11$ $( 3, 4)( 5, 9, 7,12, 6,10, 8,11)(13,15,14,16)$
8B-1 $8,4,2,1^{2}$ $32$ $8$ $11$ $( 3, 4)( 5,11, 8,10, 6,12, 7, 9)(13,16,14,15)$
8C1 $8,4,2,1^{2}$ $32$ $8$ $11$ $( 3, 4)( 5, 9, 7,12, 6,10, 8,11)(13,16,14,15)$
8C-1 $8,4,2,1^{2}$ $32$ $8$ $11$ $( 3, 4)( 5,11, 8,10, 6,12, 7, 9)(13,15,14,16)$
8D1 $8,2^{4}$ $32$ $8$ $11$ $( 1, 2)( 3, 4)( 5, 9, 7,11, 6,10, 8,12)(13,15)(14,16)$
8D-1 $8,2^{4}$ $32$ $8$ $11$ $( 1, 2)( 3, 4)( 5,11, 7,10, 6,12, 8, 9)(13,15)(14,16)$
8E1 $8,4^{2}$ $64$ $8$ $13$ $( 1,11,16, 5)( 2,12,15, 6)( 3, 9,14, 8, 4,10,13, 7)$
8E-1 $8,4^{2}$ $64$ $8$ $13$ $( 1, 5,16,11)( 2, 6,15,12)( 3, 7,13,10, 4, 8,14, 9)$
8E3 $8,4^{2}$ $64$ $8$ $13$ $( 1, 5,16,11)( 2, 6,15,12)( 3, 8,13, 9, 4, 7,14,10)$
8E-3 $8,4^{2}$ $64$ $8$ $13$ $( 1,11,16, 5)( 2,12,15, 6)( 3,10,14, 7, 4, 9,13, 8)$

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