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Properties

Label 16T892
16T892 1 2 1->2 14 1->14 13 2->13 3 15 3->15 4 16 4->16 5 5->3 6 5->6 5->15 6->4 6->16 7 7->1 7->14 8 8->2 8->13 9 9->7 12 9->12 10 10->8 11 10->11 11->5 12->6 13->5 13->12 13->15 14->6 14->11 14->16 15->7 15->10 16->8 16->9
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_8^2:(C_2\times C_4)$

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Copy content magma:G := TransitiveGroup(16, 892);
 

Group invariants

Abstract group:  $C_8^2:(C_2\times C_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $512=2^{9}$
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:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $892$
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:  $(5,15,7,14,6,16,8,13)$, $(1,2)(5,6)(9,12)(10,11)(13,15)(14,16)$, $(1,14,11,5,3,15,10,8,2,13,12,6,4,16,9,7)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(C_4^2 : C_2):C_2$ x 2, 32T239
$128$:  16T221 x 2, 16T302
$256$:  32T3905

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

16T892 x 7, 32T10424 x 4, 32T10425 x 4, 32T10426 x 4, 32T10427 x 4, 32T10428 x 4, 32T10429 x 4, 32T10430 x 4, 32T19720 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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 $2^{4},1^{8}$ $4$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2D $2^{6},1^{4}$ $16$ $2$ $6$ $( 3, 4)( 7, 8)( 9,12)(10,11)(13,15)(14,16)$
2E $2^{8}$ $16$ $2$ $8$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,14)( 6,13)( 7,15)( 8,16)$
2F $2^{6},1^{4}$ $16$ $2$ $6$ $( 1, 4)( 2, 3)( 5, 6)(11,12)(13,15)(14,16)$
2G $2^{7},1^{2}$ $32$ $2$ $7$ $( 3, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
4C $4^{2},1^{8}$ $4$ $4$ $6$ $( 1, 3, 2, 4)( 9,11,10,12)$
4D $4^{2},2^{4}$ $4$ $4$ $10$ $( 1, 4, 2, 3)( 5, 6)( 7, 8)( 9,12,10,11)(13,14)(15,16)$
4E $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
4F $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$
4G $4^{4}$ $16$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,15, 6,16)( 7,13, 8,14)$
4H1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13)(10,14)(11,16)(12,15)$
4H-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,12, 6,11)( 7,10, 8, 9)$
4I1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,15,10,16)(11,14,12,13)$
4I-1 $4^{2},2^{4}$ $16$ $4$ $10$ $( 1,16, 2,15)( 3,13, 4,14)( 5,11)( 6,12)( 7,10)( 8, 9)$
4J $4^{2},2^{3},1^{2}$ $32$ $4$ $9$ $( 3, 4)( 5,14, 6,13)( 7,15, 8,16)( 9,11)(10,12)$
8A $8^{2}$ $8$ $8$ $14$ $( 1,11, 4, 9, 2,12, 3,10)( 5,14, 8,15, 6,13, 7,16)$
8B $8,1^{8}$ $8$ $8$ $7$ $( 1,10, 3,12, 2, 9, 4,11)$
8C $8^{2}$ $8$ $8$ $14$ $( 1,10, 3,12, 2, 9, 4,11)( 5,14, 8,15, 6,13, 7,16)$
8D $8,2^{4}$ $8$ $8$ $11$ $( 1, 9, 3,11, 2,10, 4,12)( 5, 6)( 7, 8)(13,14)(15,16)$
8E $8,4^{2}$ $8$ $8$ $13$ $( 1,11, 4, 9, 2,12, 3,10)( 5, 8, 6, 7)(13,16,14,15)$
8F $8,4^{2}$ $8$ $8$ $13$ $( 1,12, 4,10, 2,11, 3, 9)( 5, 7, 6, 8)(13,15,14,16)$
8G $8^{2}$ $8$ $8$ $14$ $( 1,10, 4,11, 2, 9, 3,12)( 5,14, 7,16, 6,13, 8,15)$
8H $8^{2}$ $8$ $8$ $14$ $( 1, 9, 4,12, 2,10, 3,11)( 5,16, 8,14, 6,15, 7,13)$
8I $8,4^{2}$ $16$ $8$ $13$ $( 1, 3, 2, 4)( 5,13, 7,15, 6,14, 8,16)( 9,12,10,11)$
8J $8,2^{2},1^{4}$ $16$ $8$ $9$ $( 5,16, 8,14, 6,15, 7,13)( 9,10)(11,12)$
8K1 $8^{2}$ $16$ $8$ $14$ $( 1, 6, 3, 7, 2, 5, 4, 8)( 9,13,12,15,10,14,11,16)$
8K-1 $8^{2}$ $16$ $8$ $14$ $( 1,14, 3,15, 2,13, 4,16)( 5, 9, 7,12, 6,10, 8,11)$
8L1 $8^{2}$ $16$ $8$ $14$ $( 1,13, 4,16, 2,14, 3,15)( 5,12, 7, 9, 6,11, 8,10)$
8L-1 $8^{2}$ $16$ $8$ $14$ $( 1, 5, 4, 8, 2, 6, 3, 7)( 9,15,11,14,10,16,12,13)$
16A1 $16$ $32$ $16$ $15$ $( 1, 7,10,16, 4, 6,11,13, 2, 8, 9,15, 3, 5,12,14)$
16A-1 $16$ $32$ $16$ $15$ $( 1,14,12, 5, 3,15, 9, 8, 2,13,11, 6, 4,16,10, 7)$
16B1 $16$ $32$ $16$ $15$ $( 1,13, 9, 5, 4,16,12, 8, 2,14,10, 6, 3,15,11, 7)$
16B-1 $16$ $32$ $16$ $15$ $( 1, 7,11,15, 3, 6,10,14, 2, 8,12,16, 4, 5, 9,13)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

38 x 38 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed