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Properties

Label 16T252
16T252 1 2 1->2 13 1->13 14 2->14 3 7 3->7 8 3->8 12 3->12 4 4->7 4->8 11 4->11 5 6 5->6 10 5->10 9 6->9 16 7->16 15 8->15 9->3 9->13 9->13 10->4 10->14 10->14 11->1 12->2 13->8 14->7 15->5 16->6
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3:\OD_{16}$

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Copy content magma:G := TransitiveGroup(16, 252);
 

Group invariants

Abstract group:  $C_2^3:\OD_{16}$
Copy content magma:IdentifyGroup(G);
 
Order:  $128=2^{7}$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $252$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,13,8,15,5,10,4,11)(2,14,7,16,6,9,3,12)$, $(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)$, $(3,8)(4,7)(9,13)(10,14)$
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 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_8:C_2$ x 4, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
$32$:  $(C_8:C_2):C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_8:C_2)$ x 2, $C_2 \times (C_2^2:C_4)$
$64$:  16T72, 16T76, 16T95

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8:C_2$, $(C_8:C_2):C_2$, $C_2^3: C_4$

Low degree siblings

16T252 x 15, 32T574 x 8, 32T575 x 8, 32T576 x 4, 32T577 x 4, 32T578 x 8

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 $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2D $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,10)(11,12)(13,14)(15,16)$
2E $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,14)(10,13)(11,15)(12,16)$
2F $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,13)(10,14)(11,16)(12,15)$
2G $2^{8}$ $2$ $2$ $8$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,10)(11,12)(13,14)(15,16)$
2H $2^{8}$ $2$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
2I $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2J $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 6)( 2, 5)( 9,14)(10,13)(11,12)(15,16)$
2K $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)(11,16)(12,15)$
2L $2^{4},1^{8}$ $4$ $2$ $4$ $( 1, 6)( 2, 5)(11,16)(12,15)$
2M $2^{8}$ $4$ $2$ $8$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,14)(10,13)(11,12)(15,16)$
4A1 $4^{4}$ $2$ $4$ $12$ $( 1, 8, 5, 4)( 2, 7, 6, 3)( 9,15,14,11)(10,16,13,12)$
4A-1 $4^{4}$ $2$ $4$ $12$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,11,14,15)(10,12,13,16)$
4B1 $4^{4}$ $2$ $4$ $12$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,16,14,12)(10,15,13,11)$
4B-1 $4^{4}$ $2$ $4$ $12$ $( 1, 8, 5, 4)( 2, 7, 6, 3)( 9,12,14,16)(10,11,13,15)$
4C $4^{4}$ $4$ $4$ $12$ $( 1, 8, 5, 4)( 2, 7, 6, 3)( 9,11,14,15)(10,12,13,16)$
4D $4^{4}$ $4$ $4$ $12$ $( 1, 4, 5, 8)( 2, 3, 6, 7)( 9,12,14,16)(10,11,13,15)$
4E1 $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
4E-1 $4^{4}$ $4$ $4$ $12$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,15,10,16)(11,13,12,14)$
4F1 $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,13,12,14)$
4F-1 $4^{4}$ $4$ $4$ $12$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,11,10,12)(13,16,14,15)$
8A1 $8^{2}$ $8$ $8$ $14$ $( 1,15, 8,14, 5,11, 4, 9)( 2,16, 7,13, 6,12, 3,10)$
8A-1 $8^{2}$ $8$ $8$ $14$ $( 1,14, 4,15, 5, 9, 8,11)( 2,13, 3,16, 6,10, 7,12)$
8B1 $8^{2}$ $8$ $8$ $14$ $( 1,15, 4, 9, 5,11, 8,14)( 2,16, 3,10, 6,12, 7,13)$
8B-1 $8^{2}$ $8$ $8$ $14$ $( 1,14, 8,11, 5, 9, 4,15)( 2,13, 7,12, 6,10, 3,16)$
8C1 $8^{2}$ $8$ $8$ $14$ $( 1,12, 4, 9, 5,16, 8,14)( 2,11, 3,10, 6,15, 7,13)$
8C-1 $8^{2}$ $8$ $8$ $14$ $( 1,10, 3,16, 5,13, 7,12)( 2, 9, 4,15, 6,14, 8,11)$
8D1 $8^{2}$ $8$ $8$ $14$ $( 1,12, 8,14, 5,16, 4, 9)( 2,11, 7,13, 6,15, 3,10)$
8D-1 $8^{2}$ $8$ $8$ $14$ $( 1,10, 7,12, 5,13, 3,16)( 2, 9, 8,11, 6,14, 4,15)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

32 x 32 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed