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Properties

Label 16T1300
16T1300 1 3 1->3 16 1->16 2 4 2->4 14 2->14 3->2 13 3->13 4->1 15 4->15 5 5->3 6 6->1 7 7->2 8 7->8 8->4 9 9->5 10 9->10 10->7 11 11->8 12 12->6 13->10 13->16 14->12 14->15 15->11 15->13 16->9 16->14
Degree $16$
Order $1536$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^6:(C_4\times S_3)$

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Copy content magma:G := TransitiveGroup(16, 1300);
 

Group invariants

Abstract group:  $C_2^6:(C_4\times S_3)$
Copy content magma:IdentifyGroup(G);
 
Order:  $1536=2^{9} \cdot 3$
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 $t$:  $1300$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16,9,5,3,13,10,7,2,14,12,6)(4,15,11,8)$, $(1,3,2,4)(7,8)(9,10)(13,16,14,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$6$:  $S_3$
$8$:  $C_4\times C_2$
$12$:  $D_{6}$
$24$:  $S_4$, $S_3 \times C_4$
$48$:  $S_4\times C_2$
$96$:  12T53
$192$:  $V_4^2:(S_3\times C_2)$
$384$:  12T153

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: None

Low degree siblings

16T1300, 24T3348 x 2, 24T3353 x 2, 24T4486 x 2, 24T4491 x 2, 24T4584 x 2, 24T4587 x 2, 24T4767 x 2, 24T4768 x 2, 32T96743, 32T96757, 32T96808, 32T96822 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 $2^{8}$ $3$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
2B $2^{8}$ $4$ $2$ $8$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,15)( 6,16)( 7,13)( 8,14)$
2C $2^{8}$ $6$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,10)(11,12)(13,15)(14,16)$
2D $2^{4},1^{8}$ $6$ $2$ $4$ $( 5, 6)( 7, 8)(13,14)(15,16)$
2E $2^{8}$ $12$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)$
2F $2^{8}$ $12$ $2$ $8$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,16)( 6,15)( 7,14)( 8,13)$
2G $2^{4},1^{8}$ $12$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2H $2^{6},1^{4}$ $24$ $2$ $6$ $( 5, 7)( 6, 8)( 9,10)(11,12)(13,16)(14,15)$
2I $2^{4},1^{8}$ $24$ $2$ $4$ $( 1, 2)( 7, 8)(11,12)(13,14)$
2J $2^{8}$ $24$ $2$ $8$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,16)( 6,15)( 7,13)( 8,14)$
3A $3^{4},1^{4}$ $128$ $3$ $8$ $( 1, 2, 3)( 5, 8, 6)( 9,11,12)(14,16,15)$
4A1 $4^{4}$ $16$ $4$ $12$ $( 1, 6,11,16)( 2, 5,12,15)( 3, 8, 9,14)( 4, 7,10,13)$
4A-1 $4^{4}$ $16$ $4$ $12$ $( 1,16,11, 6)( 2,15,12, 5)( 3,14, 9, 8)( 4,13,10, 7)$
4B $4^{4}$ $24$ $4$ $12$ $( 1, 4, 3, 2)( 5, 6, 7, 8)( 9,10,11,12)(13,16,15,14)$
4C $4^{4}$ $24$ $4$ $12$ $( 1,12, 2,11)( 3,10, 4, 9)( 5,16, 6,15)( 7,14, 8,13)$
4D $4^{4}$ $24$ $4$ $12$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,16, 6,15)( 7,13, 8,14)$
4E $4^{4}$ $24$ $4$ $12$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,14, 6,13)( 7,16, 8,15)$
4F $4^{2},2^{4}$ $48$ $4$ $10$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,14, 6,13)( 7,15, 8,16)$
4G $4^{2},2^{2},1^{4}$ $48$ $4$ $8$ $( 1, 3, 4, 2)( 5, 8)( 9,11,12,10)(13,16)$
4H1 $4^{4}$ $48$ $4$ $12$ $( 1,15,11, 6)( 2,16,12, 5)( 3,14, 9, 7)( 4,13,10, 8)$
4H-1 $4^{4}$ $48$ $4$ $12$ $( 1, 6,11,15)( 2, 5,12,16)( 3, 7, 9,14)( 4, 8,10,13)$
4I1 $4^{4}$ $48$ $4$ $12$ $( 1, 5,12,16)( 2, 8,11,13)( 3, 7,10,14)( 4, 6, 9,15)$
4I-1 $4^{4}$ $48$ $4$ $12$ $( 1,16,12, 5)( 2,13,11, 8)( 3,14,10, 7)( 4,15, 9, 6)$
4J1 $4^{4}$ $48$ $4$ $12$ $( 1,14, 9, 8)( 2,13,10, 7)( 3,16,11, 6)( 4,15,12, 5)$
4J-1 $4^{4}$ $48$ $4$ $12$ $( 1, 8, 9,14)( 2, 7,10,13)( 3, 6,11,16)( 4, 5,12,15)$
4K $4^{4}$ $96$ $4$ $12$ $( 1,12, 2, 9)( 3,10, 4,11)( 5,14, 6,15)( 7,16, 8,13)$
4L $4^{2},2^{2},1^{4}$ $96$ $4$ $8$ $( 3, 4)( 7, 8)( 9,12,10,11)(13,16,14,15)$
6A $6^{2},2^{2}$ $128$ $6$ $12$ $( 1, 9, 2,11, 3,12)( 4,10)( 5,16, 8,15, 6,14)( 7,13)$
8A1 $8^{2}$ $96$ $8$ $14$ $( 1, 7,11,16, 2, 8,12,15)( 3, 6, 9,13, 4, 5,10,14)$
8A-1 $8^{2}$ $96$ $8$ $14$ $( 1,15,12, 8, 2,16,11, 7)( 3,14,10, 5, 4,13, 9, 6)$
12A1 $12,4$ $128$ $12$ $14$ $( 1,15, 9, 6, 2,14,11, 5, 3,16,12, 8)( 4,13,10, 7)$
12A-1 $12,4$ $128$ $12$ $14$ $( 1, 8,12,16, 3, 5,11,14, 2, 6, 9,15)( 4, 7,10,13)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

33 x 33 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed