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Properties

Label 16T20
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2 \times Q_8):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 20);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_2 \times Q_8):C_2$
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:  (1,7,2,8)(3,6,4,5)(9,12,10,11)(13,15,14,16), (1,2)(3,4)(5,6)(7,8), (1,4,2,3)(5,7,6,8)(9,16,10,15)(11,13,12,14), (1,16)(2,15)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$

Low degree siblings

16T20 x 4, 32T6

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{16}$ $1$ $1$ $()$
$2^{4},1^{8}$ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$2^{8}$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$4^{4}$ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,15,10,16)(11,14,12,13)$
$4^{4}$ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,16,10,15)(11,13,12,14)$
$4^{4}$ $2$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
$4^{4}$ $2$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
$4^{4}$ $2$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,11,10,12)(13,16,14,15)$
$4^{4}$ $2$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,12,10,11)(13,15,14,16)$
$2^{8}$ $2$ $2$ $( 1, 9)( 2,10)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)$
$4^{4}$ $2$ $4$ $( 1, 9, 2,10)( 3,15, 4,16)( 5,14, 6,13)( 7,11, 8,12)$
$4^{4}$ $2$ $4$ $( 1,11, 2,12)( 3,13, 4,14)( 5,15, 6,16)( 7,10, 8, 9)$
$2^{8}$ $2$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,10)( 8, 9)$
$2^{8}$ $2$ $2$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7,15)( 8,16)$
$4^{4}$ $2$ $4$ $( 1,13, 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7,15, 8,16)$
$4^{4}$ $2$ $4$ $( 1,15, 2,16)( 3,10, 4, 9)( 5,12, 6,11)( 7,14, 8,13)$
$2^{8}$ $2$ $2$ $( 1,15)( 2,16)( 3,10)( 4, 9)( 5,12)( 6,11)( 7,14)( 8,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  32.50
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J
Size 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A
Type
32.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.50.1b R 1 1 −1 −1 −1 −1 1 −1 −1 1 −1 1 1 −1 1 1 1
32.50.1c R 1 1 −1 −1 −1 1 −1 1 1 1 −1 1 −1 1 −1 −1 1
32.50.1d R 1 1 −1 −1 1 −1 −1 1 1 −1 1 −1 −1 −1 1 1 1
32.50.1e R 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 1 1 −1 −1 1
32.50.1f R 1 1 −1 1 −1 −1 −1 1 −1 1 1 −1 1 1 1 −1 −1
32.50.1g R 1 1 −1 1 −1 1 1 −1 1 1 1 −1 −1 −1 −1 1 −1
32.50.1h R 1 1 −1 1 1 −1 1 −1 1 −1 −1 1 −1 1 1 −1 −1
32.50.1i R 1 1 −1 1 1 1 −1 1 −1 −1 −1 1 1 −1 −1 1 −1
32.50.1j R 1 1 1 −1 −1 −1 −1 −1 1 −1 1 1 1 1 −1 1 −1
32.50.1k R 1 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1
32.50.1l R 1 1 1 −1 1 −1 1 1 −1 1 −1 −1 −1 1 −1 1 −1
32.50.1m R 1 1 1 −1 1 1 −1 −1 1 1 −1 −1 1 −1 1 −1 −1
32.50.1n R 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 1 −1 −1 −1 1
32.50.1o R 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
32.50.1p R 1 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 −1 −1 1
32.50.4a S 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);