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Properties

Label 16T23
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_8 : C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 23);
 

Group invariants

Abstract group:  $Q_8 : C_2^2$
magma: IdentifyGroup(G);
 
Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $23$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13)$, $(1,5)(2,6)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13)$, $(3,6)(4,5)(11,14)(12,13)$, $(1,2)(3,5)(4,6)(7,9)(8,10)(11,12)(13,14)(15,16)$, $(1,10)(2,9)(3,11)(4,12)(5,13)(6,14)(7,16)(8,15)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$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: $C_2^3$, $Q_8:C_2^2$ x 2

Low degree siblings

8T22 x 6, 16T23 x 8, 32T9

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

Malle's constant $a(G)$:     $1/4$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C 4D 4E 4F
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 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
32.49.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.49.1b R 1 1 −1 −1 −1 −1 −1 1 1 1 −1 −1 1 −1 1 1 1
32.49.1c R 1 1 −1 −1 −1 1 1 1 1 −1 1 −1 −1 1 −1 −1 1
32.49.1d R 1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 1 1 −1 1 −1
32.49.1e R 1 1 −1 −1 1 1 −1 −1 1 1 1 1 −1 −1 1 −1 −1
32.49.1f R 1 1 −1 1 −1 −1 −1 1 −1 −1 1 1 −1 1 1 1 −1
32.49.1g R 1 1 −1 1 −1 1 1 1 −1 1 −1 1 1 −1 −1 −1 −1
32.49.1h R 1 1 −1 1 1 −1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.49.1i R 1 1 −1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 1 −1 1
32.49.1j R 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 −1 1 −1 1
32.49.1k R 1 1 1 −1 −1 1 −1 −1 −1 1 −1 1 −1 1 −1 1 1
32.49.1l R 1 1 1 −1 1 −1 −1 1 −1 1 1 −1 1 1 −1 −1 −1
32.49.1m R 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1 −1
32.49.1n R 1 1 1 1 −1 −1 1 −1 1 1 −1 −1 −1 1 1 −1 −1
32.49.1o R 1 1 1 1 −1 1 −1 −1 1 −1 1 −1 1 −1 −1 1 −1
32.49.1p R 1 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 −1 1
32.49.4a R 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);