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Properties

Label 32T18
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_{16}:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(32, 18);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_{16}:C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $32$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,15,30,17,3,13,31,20)(2,16,29,18,4,14,32,19)(5,25,9,21,8,27,11,24)(6,26,10,22,7,28,12,23), (1,6,3,7)(2,5,4,8)(9,29,11,32)(10,30,12,31)(13,22,15,23)(14,21,16,24)(17,28,20,26)(18,27,19,25), (1,25,30,21,3,27,31,24)(2,26,29,22,4,28,32,23)(5,13,9,20,8,15,11,17)(6,14,10,19,7,16,12,18)
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, $C_2^3$
$16$:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, 16T32, 16T50

Low degree siblings

16T32, 16T50

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,19)(14,20)(15,18)(16,17)(21,26) (22,25)(23,27)(24,28)(29,31)(30,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,31,11,30)(10,32,12,29)(13,25,15,27)(14,26,16,28) (17,24,20,21)(18,23,19,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 3, 7)( 2, 5, 4, 8)( 9,29,11,32)(10,30,12,31)(13,22,15,23)(14,21,16,24) (17,28,20,26)(18,27,19,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,31)( 6,32)( 7,29)( 8,30)(13,21)(14,22)(15,24) (16,23)(17,25)(18,26)(19,28)(20,27)$
$ 8, 8, 8, 8 $ $4$ $8$ $( 1,13,30,20, 3,15,31,17)( 2,14,29,19, 4,16,32,18)( 5,27, 9,24, 8,25,11,21) ( 6,28,10,23, 7,26,12,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,14, 3,16)( 2,13, 4,15)( 5,28, 8,26)( 6,27, 7,25)( 9,22,11,23)(10,21,12,24) (17,32,20,29)(18,31,19,30)$
$ 8, 8, 8, 8 $ $4$ $8$ $( 1,21,31,25, 3,24,30,27)( 2,22,32,26, 4,23,29,28)( 5,20,11,13, 8,17, 9,15) ( 6,19,12,14, 7,18,10,16)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,22, 3,23)( 2,21, 4,24)( 5,19, 8,18)( 6,20, 7,17)( 9,14,11,16)(10,13,12,15) (25,32,27,29)(26,31,28,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,30, 3,31)( 2,29, 4,32)( 5, 9, 8,11)( 6,10, 7,12)(13,20,15,17)(14,19,16,18) (21,27,24,25)(22,28,23,26)$

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:  $3$
Label:  32.44
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 4B 4C 4D 4E 8A 8B
Size 1 1 2 4 2 2 4 4 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2A 2A 2A 4A 4A
Type
32.44.1a R 1 1 1 1 1 1 1 1 1 1 1
32.44.1b R 1 1 −1 −1 1 −1 1 −1 1 1 −1
32.44.1c R 1 1 −1 −1 1 −1 1 1 −1 −1 1
32.44.1d R 1 1 −1 1 1 −1 −1 −1 1 −1 1
32.44.1e R 1 1 −1 1 1 −1 −1 1 −1 1 −1
32.44.1f R 1 1 1 −1 1 1 −1 −1 −1 1 1
32.44.1g R 1 1 1 −1 1 1 −1 1 1 −1 −1
32.44.1h R 1 1 1 1 1 1 1 −1 −1 −1 −1
32.44.2a R 2 2 −2 0 −2 2 0 0 0 0 0
32.44.2b R 2 2 2 0 −2 −2 0 0 0 0 0
32.44.4a S 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);