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Properties

Label 16T141
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 141);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $141$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14)(2,13)(3,4)(5,10)(6,9)(7,15)(8,16)(11,12), (1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16), (1,9)(2,10)(3,4)(5,6)(7,15)(8,16)(11,12)(13,14)
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
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4\times C_2$, $(C_4^2 : C_2):C_2$ x 2

Low degree siblings

8T26 x 4, 16T135 x 2, 16T141, 16T142 x 2, 16T152 x 2, 32T147 x 2, 32T148 x 2, 32T155, 32T156

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 $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
2B $2^{4},1^{8}$ $2$ $2$ $4$ $( 3,12)( 4,11)( 7,16)( 8,15)$
2C $2^{8}$ $4$ $2$ $8$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
2D $2^{8}$ $4$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,12)( 8,11)( 9,13)(10,14)$
2E $2^{8}$ $4$ $2$ $8$ $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,14)(12,13)$
2F $2^{8}$ $4$ $2$ $8$ $( 1, 9)( 2,10)( 3, 4)( 5, 6)( 7,15)( 8,16)(11,12)(13,14)$
2G $2^{8}$ $8$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 4)( 7,15)( 8,16)( 9,13)(10,14)(11,12)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,12,15)( 4, 7,11,16)$
4B $4^{4}$ $2$ $4$ $12$ $( 1,13,10, 6)( 2,14, 9, 5)( 3, 8,12,15)( 4, 7,11,16)$
4C $4^{2},1^{8}$ $4$ $4$ $6$ $( 3,15,12, 8)( 4,16,11, 7)$
4D $4^{4}$ $4$ $4$ $12$ $( 1,11,10, 4)( 2,12, 9, 3)( 5, 8,14,15)( 6, 7,13,16)$
4E $4^{2},2^{4}$ $4$ $4$ $10$ $( 1,10)( 2, 9)( 3,15,12, 8)( 4,16,11, 7)( 5,14)( 6,13)$
4F $4^{4}$ $4$ $4$ $12$ $( 1,12,10, 3)( 2,11, 9, 4)( 5,16,14, 7)( 6,15,13, 8)$
8A $8^{2}$ $8$ $8$ $14$ $( 1, 3, 6, 8,10,12,13,15)( 2, 4, 5, 7, 9,11,14,16)$
8B $8^{2}$ $8$ $8$ $14$ $( 1, 7,13,11,10,16, 6, 4)( 2, 8,14,12, 9,15, 5, 3)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $64=2^{6}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
Label:  64.134
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E 4F 8A 8B
Size 1 1 2 4 4 4 4 8 2 2 4 4 4 4 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2B 2A 2B 2A 4A 4B
Type
64.134.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.134.1b R 1 1 1 −1 −1 1 −1 −1 1 1 1 1 1 −1 1 −1
64.134.1c R 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 1 −1 −1 1
64.134.1d R 1 1 1 −1 1 −1 −1 −1 1 1 1 1 −1 1 −1 1
64.134.1e R 1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1 1 1 −1
64.134.1f R 1 1 1 1 −1 −1 1 −1 1 1 −1 −1 −1 −1 1 1
64.134.1g R 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1
64.134.1h R 1 1 1 1 1 1 1 −1 1 1 −1 −1 1 1 −1 −1
64.134.2a R 2 2 −2 0 −2 0 0 0 2 −2 0 0 0 2 0 0
64.134.2b R 2 2 −2 0 0 −2 0 0 −2 2 0 0 2 0 0 0
64.134.2c R 2 2 −2 0 0 2 0 0 −2 2 0 0 −2 0 0 0
64.134.2d R 2 2 −2 0 2 0 0 0 2 −2 0 0 0 −2 0 0
64.134.2e R 2 2 2 −2 0 0 2 0 −2 −2 0 0 0 0 0 0
64.134.2f R 2 2 2 2 0 0 −2 0 −2 −2 0 0 0 0 0 0
64.134.4a R 4 −4 0 0 0 0 0 0 0 0 −2 2 0 0 0 0
64.134.4b R 4 −4 0 0 0 0 0 0 0 0 2 −2 0 0 0 0

magma: CharacterTable(G);