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Properties

Label 32T50
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $Q_8:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 50);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $50$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $Q_8:C_4$
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:  (1,10,2,9)(3,11,4,12)(5,14,6,13)(7,16,8,15)(17,25,18,26)(19,27,20,28)(21,29,22,30)(23,31,24,32), (1,23,11,15,2,24,12,16)(3,22,9,13,4,21,10,14)(5,26,30,20,6,25,29,19)(7,28,31,17,8,27,32,18)
magma: Generators(G);
 

Low degree resolvents

|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, $C_4\times C_2$
$16$:  $QD_{16}$, $C_2^2:C_4$, $Q_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $QD_{16}$, $C_2^2:C_4$ x 2

Degree 16: $C_2^2 : C_4$, $QD_{16}$, $Q_{16}$

Low degree siblings

There are no siblings 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5,22, 6,21)( 7,23, 8,24)( 9,11,10,12)(13,30,14,29)(15,31,16,32) (17,19,18,20)(25,28,26,27)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 5,12,29, 2, 6,11,30)( 3, 8,10,31, 4, 7, 9,32)(13,18,22,27,14,17,21,28) (15,20,23,26,16,19,24,25)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 6,12,30, 2, 5,11,29)( 3, 7,10,32, 4, 8, 9,31)(13,17,22,28,14,18,21,27) (15,19,23,25,16,20,24,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 7,25,14)( 2, 8,26,13)( 3, 5,28,15)( 4, 6,27,16)( 9,29,17,23)(10,30,18,24) (11,32,19,21)(12,31,20,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)(17,26,18,25)(19,28,20,27) (21,30,22,29)(23,32,24,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,30, 6,29)( 7,31, 8,32)(13,21,14,22)(15,24,16,23) (17,27,18,28)(19,26,20,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,13,25, 8)( 2,14,26, 7)( 3,16,28, 6)( 4,15,27, 5)( 9,24,17,30)(10,23,18,29) (11,22,19,31)(12,21,20,32)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,15,12,23, 2,16,11,24)( 3,13,10,22, 4,14, 9,21)( 5,20,29,26, 6,19,30,25) ( 7,17,32,28, 8,18,31,27)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,16,12,24, 2,15,11,23)( 3,14,10,21, 4,13, 9,22)( 5,19,29,25, 6,20,30,26) ( 7,18,32,27, 8,17,31,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,19, 2,20)( 3,17, 4,18)( 5,24, 6,23)( 7,22, 8,21)( 9,27,10,28)(11,26,12,25) (13,32,14,31)(15,30,16,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,15)( 6,16)( 7,14)( 8,13)( 9,17)(10,18)(11,19) (12,20)(21,32)(22,31)(23,29)(24,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,26)( 2,25)( 3,27)( 4,28)( 5,16)( 6,15)( 7,13)( 8,14)( 9,18)(10,17)(11,20) (12,19)(21,31)(22,32)(23,30)(24,29)$

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:  32.10
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 4B 4C 4D 4E1 4E-1 8A1 8A-1 8A3 8A-3
Size 1 1 1 1 2 2 4 4 4 4 2 2 2 2
2 P 1A 1A 1A 1A 2A 2A 2A 2B 2A 2B 4A 4A 4A 4A
Type
32.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.10.1b R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
32.10.1c R 1 1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
32.10.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
32.10.1e1 C 1 1 −1 −1 −1 1 −1 1 −i i i −i −i i
32.10.1e2 C 1 1 −1 −1 −1 1 −1 1 i −i −i i i −i
32.10.1f1 C 1 1 −1 −1 −1 1 1 −1 −i i −i i i −i
32.10.1f2 C 1 1 −1 −1 −1 1 1 −1 i −i i −i −i i
32.10.2a R 2 2 −2 −2 2 −2 0 0 0 0 0 0 0 0
32.10.2b R 2 2 2 2 −2 −2 0 0 0 0 0 0 0 0
32.10.2c1 C 2 −2 2 −2 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83
32.10.2c2 C 2 −2 2 −2 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83
32.10.2d1 S 2 −2 −2 2 0 0 0 0 0 0 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8
32.10.2d2 S 2 −2 −2 2 0 0 0 0 0 0 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8

magma: CharacterTable(G);