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Properties

Label 16T31
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2:Q_8$

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Show commands: Magma

magma: G := TransitiveGroup(16, 31);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2:Q_8$
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:  (1,8,16,10)(2,7,15,9)(3,14,5,12)(4,13,6,11), (1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14), (1,2)(3,4)(5,6)(7,9)(8,10)(11,13)(12,14)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$, $Q_8$ x 2
$16$:  $D_4\times C_2$, $Q_8:C_2$, $Q_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $Q_8$, $D_4\times C_2$, $Q_8:C_2$

Low degree siblings

16T31, 32T17

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  32.29
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C1 4C-1 4D 4E 4F 4G
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2B 2B 2B 2B 2A 2A 2B 2B
Type
32.29.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.29.1b R 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 −1 1
32.29.1c R 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 1 −1
32.29.1d R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.29.1e R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1
32.29.1f R 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 −1
32.29.1g R 1 1 1 1 1 1 −1 −1 −1 −1 1 −1 −1 1
32.29.1h R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.29.2a R 2 −2 2 −2 0 0 −2 2 0 0 0 0 0 0
32.29.2b R 2 −2 2 −2 0 0 2 −2 0 0 0 0 0 0
32.29.2c S 2 −2 −2 2 −2 2 0 0 0 0 0 0 0 0
32.29.2d S 2 −2 −2 2 2 −2 0 0 0 0 0 0 0 0
32.29.2e1 C 2 2 −2 −2 0 0 0 0 −2i 2i 0 0 0 0
32.29.2e2 C 2 2 −2 −2 0 0 0 0 2i −2i 0 0 0 0

magma: CharacterTable(G);