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Properties

Label 16T1054
16T1054 1 5 1->5 11 1->11 15 1->15 2 6 2->6 12 2->12 16 2->16 3 8 3->8 3->12 13 3->13 4 7 4->7 4->11 14 4->14 5->3 5->13 5->16 6->4 6->14 6->15 7->1 9 7->9 7->9 8->2 10 8->10 8->10 9->8 9->16 10->7 10->15 11->2 11->3 11->13 12->1 12->4 12->14 13->4 13->6 13->10 14->3 14->5 14->9 15->5 15->12 16->6 16->11
Degree $16$
Order $768$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $(C_2^2\times D_4):S_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 1054);
 

Group invariants

Abstract group:  $(C_2^2\times D_4):S_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $768=2^{8} \cdot 3$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1054$
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:  $(1,11,2,12)(3,13,4,14)(5,16,6,15)(7,9,8,10)$, $(1,5,3,8,2,6,4,7)(9,16,11,13,10,15,12,14)$, $(1,15)(2,16)(3,12,4,11)(5,13,6,14)(7,9)(8,10)$
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
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 3
$96$:  12T48
$192$:  $V_4^2:(S_3\times C_2)$
$384$:  12T136

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4\times C_2$

Low degree siblings

16T1048 x 2, 16T1054, 32T34673 x 2, 32T34674, 32T34675, 32T34689, 32T34690, 32T34787, 32T35040

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 $2^{4},1^{8}$ $2$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
2C $2^{4},1^{8}$ $6$ $2$ $4$ $( 1, 2)( 5, 6)( 9,10)(13,14)$
2D $2^{4},1^{8}$ $6$ $2$ $4$ $( 1, 2)( 5, 6)(11,12)(15,16)$
2E $2^{8}$ $8$ $2$ $8$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,11)( 6,12)( 7, 9)( 8,10)$
2F $2^{8}$ $12$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
2G $2^{8}$ $12$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
2H $2^{6},1^{4}$ $24$ $2$ $6$ $( 1, 6)( 2, 5)( 7, 8)( 9,10)(11,15)(12,16)$
2I $2^{6},1^{4}$ $24$ $2$ $6$ $( 1, 6)( 2, 5)( 7, 8)(11,16)(12,15)(13,14)$
2J $2^{8}$ $24$ $2$ $8$ $( 1,12)( 2,11)( 3,13)( 4,14)( 5,15)( 6,16)( 7, 9)( 8,10)$
3A $3^{4},1^{4}$ $32$ $3$ $8$ $( 1, 8, 6)( 2, 7, 5)( 9,11,15)(10,12,16)$
4A $4^{4}$ $8$ $4$ $12$ $( 1,15, 2,16)( 3,14, 4,13)( 5,12, 6,11)( 7, 9, 8,10)$
4B $4^{2},2^{4}$ $12$ $4$ $10$ $( 1, 5, 2, 6)( 3, 4)( 7, 8)( 9,10)(11,15,12,16)(13,14)$
4C $4^{2},1^{8}$ $12$ $4$ $6$ $( 1, 5, 2, 6)(11,15,12,16)$
4D $4^{4}$ $12$ $4$ $12$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
4E $4^{4}$ $12$ $4$ $12$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$
4F $4^{2},2^{2},1^{4}$ $24$ $4$ $8$ $( 1, 5, 2, 6)( 3, 4)( 7, 8)(11,16,12,15)$
4G $4^{2},2^{4}$ $24$ $4$ $10$ $( 1,11, 2,12)( 3,14)( 4,13)( 5,15, 6,16)( 7, 9)( 8,10)$
4H $4^{2},2^{4}$ $24$ $4$ $10$ $( 1,11)( 2,12)( 3,14, 4,13)( 5,15)( 6,16)( 7, 9, 8,10)$
4I $4^{4}$ $24$ $4$ $12$ $( 1,12, 2,11)( 3,13, 4,14)( 5,15, 6,16)( 7, 9, 8,10)$
4J $4^{2},2^{4}$ $24$ $4$ $10$ $( 1,11, 2,12)( 3,10)( 4, 9)( 5,16, 6,15)( 7,13)( 8,14)$
4K $4^{2},2^{4}$ $24$ $4$ $10$ $( 1,11)( 2,12)( 3,10, 4, 9)( 5,16)( 6,15)( 7,13, 8,14)$
4L $4^{4}$ $48$ $4$ $12$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,16, 8,13)( 6,15, 7,14)$
4M $4^{4}$ $48$ $4$ $12$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,16, 7,14)( 6,15, 8,13)$
6A $6^{2},2^{2}$ $32$ $6$ $12$ $( 1, 3, 5, 2, 4, 6)( 7, 8)( 9,10)(11,15,14,12,16,13)$
6B $6,3^{2},2,1^{2}$ $64$ $6$ $10$ $( 1, 5, 8, 2, 6, 7)( 3, 4)( 9,15,11)(10,16,12)$
6C $6^{2},2^{2}$ $64$ $6$ $12$ $( 1,12, 8,16, 6,10)( 2,11, 7,15, 5, 9)( 3,14)( 4,13)$
8A $8^{2}$ $48$ $8$ $14$ $( 1, 8, 5, 3, 2, 7, 6, 4)( 9,11,14,15,10,12,13,16)$
8B $8^{2}$ $48$ $8$ $14$ $( 1, 7, 5, 4, 2, 8, 6, 3)( 9,11,14,15,10,12,13,16)$
12A $12,4$ $64$ $12$ $14$ $( 1,11, 3,15, 5,14, 2,12, 4,16, 6,13)( 7,10, 8, 9)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

31 x 31 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed