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Properties

Label 16T1455
16T1455 1 6 1->6 12 1->12 2 5 2->5 11 2->11 3 7 3->7 10 3->10 4 8 4->8 9 4->9 5->6 5->12 6->11 7->8 7->10 8->9 9->3 9->12 15 9->15 10->4 10->11 16 10->16 11->1 11->9 14 11->14 12->2 12->10 13 12->13 13->4 13->16 13->16 14->3 14->15 14->15 15->2 15->13 16->1 16->14
Degree $16$
Order $2048$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5.C_2\wr C_2^2$

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Copy content magma:G := TransitiveGroup(16, 1455);
 

Group invariants

Abstract group:  $C_2^5.C_2\wr C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $2048=2^{11}$
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 $t$:  $1455$
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:  $(1,6,11,14,3,7,10,16)(2,5,12,13,4,8,9,15)$, $(5,6)(9,12,10,11)(13,16)(14,15)$, $(1,12,2,11)(3,10,4,9)(7,8)(13,16,14,15)$
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, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $Z_8 : Z_8^\times$, $C_2^2 \wr C_2$, 16T44
$64$:  $(C_4^2 : C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 32T329
$128$:  16T336, 16T345, 16T389
$256$:  32T5717
$512$:  16T981
$1024$:  32T67457

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(C_4^2 : C_2):C_2$

Low degree siblings

16T1444 x 4, 16T1455 x 3, 32T99046 x 4, 32T99047 x 2, 32T99048 x 2, 32T99049 x 2, 32T99050 x 2, 32T99051 x 2, 32T99052 x 2, 32T99149 x 2, 32T99150 x 2, 32T99151 x 2, 32T99152 x 2, 32T99153 x 2, 32T99154 x 2, 32T122208 x 2, 32T122212 x 2, 32T122236 x 2, 32T122237 x 2, 32T122242 x 2, 32T122243 x 2, 32T145654, 32T145678, 32T145682, 32T145692 x 2, 32T145702, 32T180476, 32T180521, 32T180547, 32T180557 x 2, 32T180558, 32T181406, 32T181450, 32T181476, 32T181477, 32T181485 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2C $2^{2},1^{12}$ $4$ $2$ $2$ $(1,2)(3,4)$
2D $2^{4},1^{8}$ $4$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2E $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2F $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2G $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
2H $2^{4},1^{8}$ $16$ $2$ $4$ $( 1, 2)( 5, 6)( 9,10)(15,16)$
2I $2^{6},1^{4}$ $32$ $2$ $6$ $( 1, 3)( 2, 4)( 5,13)( 6,14)( 7,16)( 8,15)$
2J $2^{6},1^{4}$ $32$ $2$ $6$ $( 1, 3)( 2, 4)( 5, 6)( 9,11)(10,12)(15,16)$
2K $2^{8}$ $32$ $2$ $8$ $( 1, 2)( 3, 4)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
2L $2^{8}$ $32$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,16)(10,15)(11,13)(12,14)$
4A $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)(13,15,14,16)$
4B $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,10)(11,12)(13,15,14,16)$
4C $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)(13,15,14,16)$
4D $4^{2},1^{8}$ $8$ $4$ $6$ $( 1, 3, 2, 4)( 9,11,10,12)$
4E $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
4F $4^{2},2^{2},1^{4}$ $32$ $4$ $8$ $( 1, 3)( 2, 4)( 5,14, 6,13)( 7,15, 8,16)$
4G $4^{2},2^{4}$ $32$ $4$ $10$ $( 1, 2)( 3, 4)( 5,15, 6,16)( 7,14, 8,13)( 9,11)(10,12)$
4H $4^{4}$ $32$ $4$ $12$ $( 1, 8, 2, 7)( 3, 5, 4, 6)( 9,16,10,15)(11,13,12,14)$
4I1 $4,2^{3},1^{6}$ $32$ $4$ $6$ $( 1, 4, 2, 3)( 5, 6)(13,16)(14,15)$
4I-1 $4,2^{3},1^{6}$ $32$ $4$ $6$ $( 1, 3, 2, 4)( 5, 6)(13,16)(14,15)$
4J1 $4,2^{5},1^{2}$ $32$ $4$ $8$ $( 3, 4)( 5, 7, 6, 8)( 9,11)(10,12)(13,14)(15,16)$
4J-1 $4,2^{5},1^{2}$ $32$ $4$ $8$ $( 1, 2)( 5, 7, 6, 8)( 9,11)(10,12)(13,14)(15,16)$
4K $4^{4}$ $64$ $4$ $12$ $( 1,11, 4, 9)( 2,12, 3,10)( 5,15, 7,13)( 6,16, 8,14)$
4L $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13)(10,14)(11,15)(12,16)$
4M $4^{4}$ $64$ $4$ $12$ $( 1, 9, 3,11)( 2,10, 4,12)( 5,16, 8,13)( 6,15, 7,14)$
4N $4,2^{5},1^{2}$ $64$ $4$ $8$ $( 1, 2)( 5,13)( 6,14)( 7,16)( 8,15)( 9,11,10,12)$
4O $4^{3},2,1^{2}$ $64$ $4$ $10$ $( 1,12, 2,11)( 3, 9, 4,10)( 5, 8, 6, 7)(13,14)$
4P1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 9, 2,10)( 3,12)( 4,11)( 5,15, 6,16)( 7,13)( 8,14)$
4P-1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1,10, 2, 9)( 3,12)( 4,11)( 5,16, 6,15)( 7,13)( 8,14)$
4Q1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 7, 2, 8)( 3, 5)( 4, 6)( 9,14)(10,13)(11,15,12,16)$
4Q-1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 8, 2, 7)( 3, 5)( 4, 6)( 9,14)(10,13)(11,16,12,15)$
4R $4^{4}$ $128$ $4$ $12$ $( 1,16, 3,13)( 2,15, 4,14)( 5, 9, 8,12)( 6,10, 7,11)$
8A $8,2,1^{6}$ $64$ $8$ $8$ $( 1,11, 3,10, 2,12, 4, 9)( 5, 6)$
8B $8,2^{3},1^{2}$ $64$ $8$ $10$ $( 1, 9, 3,11, 2,10, 4,12)( 7, 8)(13,14)(15,16)$
8C1 $8,4,2^{2}$ $64$ $8$ $12$ $( 1, 3, 2, 4)( 5,15, 8,14, 6,16, 7,13)( 9,12)(10,11)$
8C-1 $8,4,2^{2}$ $64$ $8$ $12$ $( 1, 4, 2, 3)( 5,13, 7,16, 6,14, 8,15)( 9,12)(10,11)$
8D $8^{2}$ $128$ $8$ $14$ $( 1, 7, 3, 6, 2, 8, 4, 5)( 9,16,11,13,10,15,12,14)$
8E1 $8^{2}$ $128$ $8$ $14$ $( 1,15,11, 7, 4,13, 9, 5)( 2,16,12, 8, 3,14,10, 6)$
8E-1 $8^{2}$ $128$ $8$ $14$ $( 1, 5, 9,13, 4, 7,11,15)( 2, 6,10,14, 3, 8,12,16)$
8F1 $8^{2}$ $128$ $8$ $14$ $( 1, 5, 9,16, 3, 8,11,13)( 2, 6,10,15, 4, 7,12,14)$
8F3 $8^{2}$ $128$ $8$ $14$ $( 1,16,11, 5, 3,13, 9, 8)( 2,15,12, 6, 4,14,10, 7)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

44 x 44 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed