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Properties

Label 16T397
Degree $16$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2.D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 397);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $397$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2.D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,6,7,2,4,5,8)(9,10)(11,16,12,15)(13,14), (1,10,7,15)(2,9,8,16)(3,12,5,13)(4,11,6,14)
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, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $
$64$:  $((C_8 : C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $C_2^3 : C_4 $

Low degree siblings

16T397, 32T886, 32T1786, 32T1949

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 $2^{8}$ $8$ $2$ $8$ $( 1,10)( 2, 9)( 3,16)( 4,15)( 5,13)( 6,14)( 7,12)( 8,11)$
2D $2^{8}$ $8$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,16)(10,15)(11,14)(12,13)$
4A $4^{2},2^{2},1^{4}$ $4$ $4$ $8$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,10)(13,14)$
4B1 $4^{2},2^{2},1^{4}$ $4$ $4$ $8$ $( 1, 5, 2, 6)( 3, 8, 4, 7)(11,12)(15,16)$
4B-1 $4^{4}$ $4$ $4$ $12$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
4C $4^{4}$ $8$ $4$ $12$ $( 1,10, 2, 9)( 3,16, 4,15)( 5,13, 6,14)( 7,12, 8,11)$
4D $4^{4}$ $8$ $4$ $12$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,16,10,15)(11,13,12,14)$
4E1 $4^{4}$ $16$ $4$ $12$ $( 1,16, 8,10)( 2,15, 7, 9)( 3,14, 6,12)( 4,13, 5,11)$
4E-1 $4^{4}$ $16$ $4$ $12$ $( 1,16, 3,13)( 2,15, 4,14)( 5,11, 8, 9)( 6,12, 7,10)$
8A1 $8,4,2^{2}$ $8$ $8$ $12$ $( 1, 2)( 3, 7, 4, 8)( 5, 6)( 9,16,13,12,10,15,14,11)$
8A-1 $8,4,2^{2}$ $8$ $8$ $12$ $( 1, 5, 2, 6)( 3, 4)( 7, 8)( 9,16,14,11,10,15,13,12)$
8B1 $8,4,1^{4}$ $8$ $8$ $10$ $( 3, 8, 4, 7)( 9,15,13,11,10,16,14,12)$
8B-1 $8,4,1^{4}$ $8$ $8$ $10$ $( 1, 6, 2, 5)( 9,15,14,12,10,16,13,11)$
8C $8^{2}$ $16$ $8$ $14$ $( 1, 9, 5,14, 2,10, 6,13)( 3,16, 7,12, 4,15, 8,11)$

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $128=2^{7}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $5$
Label:  128.136
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A 4B1 4B-1 4C 4D 4E1 4E-1 8A1 8A-1 8B1 8B-1 8C
Size 1 1 2 8 8 4 4 4 8 8 16 16 8 8 8 8 16
2 P 1A 1A 1A 1A 1A 2B 2B 2A 2A 2A 2D 2D 4B1 4B-1 4B1 4B-1 4A
Type
128.136.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.136.1b R 1 1 1 1 −1 1 1 1 1 −1 −1 −1 1 1 1 1 −1
128.136.1c R 1 1 1 1 −1 1 1 1 1 −1 1 1 −1 −1 −1 −1 −1
128.136.1d R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1
128.136.1e1 C 1 1 1 −1 −1 1 −1 −1 1 −1 −i i −i i −i i 1
128.136.1e2 C 1 1 1 −1 −1 1 −1 −1 1 −1 i −i i −i i −i 1
128.136.1f1 C 1 1 1 −1 1 1 −1 −1 1 1 −i i i −i i −i −1
128.136.1f2 C 1 1 1 −1 1 1 −1 −1 1 1 i −i −i i −i i −1
128.136.2a R 2 2 2 −2 0 2 2 2 −2 0 0 0 0 0 0 0 0
128.136.2b R 2 2 2 2 0 2 −2 −2 −2 0 0 0 0 0 0 0 0
128.136.4a R 4 4 4 0 0 −4 0 0 0 0 0 0 0 0 0 0 0
128.136.4b R 4 4 −4 0 −2 0 0 0 0 2 0 0 0 0 0 0 0
128.136.4c R 4 4 −4 0 2 0 0 0 0 −2 0 0 0 0 0 0 0
128.136.4d1 C 4 −4 0 0 0 0 −2i 2i 0 0 0 0 −1−i −1+i 1+i 1−i 0
128.136.4d2 C 4 −4 0 0 0 0 2i −2i 0 0 0 0 −1+i −1−i 1−i 1+i 0
128.136.4e1 C 4 −4 0 0 0 0 −2i 2i 0 0 0 0 1+i 1−i −1−i −1+i 0
128.136.4e2 C 4 −4 0 0 0 0 2i −2i 0 0 0 0 1−i 1+i −1+i −1−i 0

magma: CharacterTable(G);