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Properties

Label 16T1243
16T1243 1 4 1->4 16 1->16 2 3 2->3 15 2->15 3->1 14 3->14 4->2 13 4->13 5 5->2 7 5->7 6 6->1 8 6->8 7->3 8->4 9 9->8 10 10->7 11 11->5 12 12->6 13->11 13->14 14->12 15->9 16->10
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2^2\times C_4^2).\OD_{16}$

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Copy content magma:G := TransitiveGroup(16, 1243);
 

Group invariants

Abstract group:  $(C_2^2\times C_4^2).\OD_{16}$
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 $t$:  $1243$
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:  $(1,16,10,7,3,14,12,6)(2,15,9,8,4,13,11,5)$, $(1,4,2,3)(5,7)(6,8)(13,14)$
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, $C_8$ x 2, $C_4\times C_2$
$16$:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
$32$:  $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$
$64$:  16T84, 16T140 x 2
$128$:  32T1650
$256$:  16T495
$512$:  32T26977

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $(C_8:C_2):C_2$

Low degree siblings

16T1138 x 4, 16T1243 x 3, 32T35730 x 4, 32T35731 x 4, 32T35732 x 2, 32T35733 x 4, 32T35734 x 2, 32T36573 x 4, 32T36574 x 4, 32T36575 x 2, 32T36576 x 2, 32T47040 x 2, 32T47044 x 2, 32T50822 x 2, 32T50824 x 2, 32T71017 x 2, 32T88062 x 2, 32T89296 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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2C $2^{4},1^{8}$ $4$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2D $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
2E $2^{2},1^{12}$ $4$ $2$ $2$ $(5,6)(7,8)$
2F $2^{8}$ $8$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
2G $2^{8}$ $8$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
2H $2^{4},1^{8}$ $16$ $2$ $4$ $( 3, 4)( 7, 8)(11,12)(13,14)$
2I $2^{6},1^{4}$ $32$ $2$ $6$ $( 1, 2)( 5, 7)( 6, 8)( 9,10)(13,16)(14,15)$
4A $4^{2},1^{8}$ $8$ $4$ $6$ $( 1, 3, 2, 4)( 9,12,10,11)$
4B $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 3, 2, 4)( 5, 6)( 7, 8)( 9,12,10,11)$
4C $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 4, 2, 3)( 5, 6)( 7, 8)( 9,12,10,11)$
4D $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 3, 2, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,14)(15,16)$
4E $4^{4}$ $16$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$
4F1 $4,2^{3},1^{6}$ $16$ $4$ $6$ $( 1, 2)( 5, 7, 6, 8)( 9,12)(10,11)$
4F-1 $4,2^{3},1^{6}$ $16$ $4$ $6$ $( 3, 4)( 5, 7, 6, 8)( 9,12)(10,11)$
4G1 $4,2^{3},1^{6}$ $16$ $4$ $6$ $( 1, 3)( 2, 4)( 5, 7, 6, 8)( 9,10)$
4G-1 $4,2^{3},1^{6}$ $16$ $4$ $6$ $( 1, 4)( 2, 3)( 5, 7, 6, 8)( 9,10)$
4H1 $4,2^{5},1^{2}$ $16$ $4$ $8$ $( 1, 3, 2, 4)( 5, 6)( 9,10)(11,12)(13,16)(14,15)$
4H-1 $4,2^{5},1^{2}$ $16$ $4$ $8$ $( 1, 4, 2, 3)( 5, 6)( 9,10)(11,12)(13,16)(14,15)$
4I1 $4,2^{5},1^{2}$ $16$ $4$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 9,12,10,11)(13,16)(14,15)$
4I-1 $4,2^{5},1^{2}$ $16$ $4$ $8$ $( 1, 2)( 3, 4)( 7, 8)( 9,12,10,11)(13,16)(14,15)$
4J1 $4^{4}$ $32$ $4$ $12$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,15, 8,14)( 6,16, 7,13)$
4J-1 $4^{4}$ $32$ $4$ $12$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,14, 8,15)( 6,13, 7,16)$
4K1 $4^{4}$ $32$ $4$ $12$ $( 1,11, 4,10)( 2,12, 3, 9)( 5,15, 8,14)( 6,16, 7,13)$
4K-1 $4^{4}$ $32$ $4$ $12$ $( 1,10, 4,11)( 2, 9, 3,12)( 5,14, 8,15)( 6,13, 7,16)$
4L1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 9)( 2,10)( 3,12, 4,11)( 5,16)( 6,15)( 7,13, 8,14)$
4L-1 $4^{2},2^{4}$ $64$ $4$ $10$ $( 1, 9)( 2,10)( 3,11, 4,12)( 5,16)( 6,15)( 7,14, 8,13)$
8A1 $8^{2}$ $64$ $8$ $14$ $( 1, 5,10,15, 3, 8,11,14)( 2, 6, 9,16, 4, 7,12,13)$
8A-1 $8^{2}$ $64$ $8$ $14$ $( 1,14,11, 8, 3,15,10, 5)( 2,13,12, 7, 4,16, 9, 6)$
8A3 $8^{2}$ $64$ $8$ $14$ $( 1,15,11, 5, 3,14,10, 8)( 2,16,12, 6, 4,13, 9, 7)$
8A-3 $8^{2}$ $64$ $8$ $14$ $( 1, 8,10,14, 3, 5,11,15)( 2, 7, 9,13, 4, 6,12,16)$
8B1 $8^{2}$ $64$ $8$ $14$ $( 1,14,11, 5, 4,15,10, 8)( 2,13,12, 6, 3,16, 9, 7)$
8B-1 $8^{2}$ $64$ $8$ $14$ $( 1, 8,10,15, 4, 5,11,14)( 2, 7, 9,16, 3, 6,12,13)$
8B3 $8^{2}$ $64$ $8$ $14$ $( 1, 5,10,14, 4, 8,11,15)( 2, 6, 9,13, 3, 7,12,16)$
8B-3 $8^{2}$ $64$ $8$ $14$ $( 1,15,11, 8, 4,14,10, 5)( 2,16,12, 7, 3,13, 9, 6)$

Malle's constant $a(G)$:     $1/2$

Copy content magma:ConjugacyClasses(G);
 

Character table

37 x 37 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed