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Properties

Label 32T27
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\times \SD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(32, 27);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $27$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times \SD_{16}$
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:  (1,7,17,21)(2,8,18,22)(3,5,19,23)(4,6,20,24)(9,32,26,14)(10,31,25,13)(11,29,28,16)(12,30,27,15), (1,11)(2,12)(3,9)(4,10)(5,24)(6,23)(7,22)(8,21)(13,14)(15,16)(17,28)(18,27)(19,26)(20,25)(29,30)(31,32), (1,3)(2,4)(5,15)(6,16)(7,13)(8,14)(9,28)(10,27)(11,26)(12,25)(17,19)(18,20)(21,31)(22,32)(23,30)(24,29)
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, $C_2^3$
$16$:  $QD_{16}$ x 2, $D_4\times C_2$

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, $QD_{16}$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, $QD_{16}$ x 2, 16T48 x 2

Low degree siblings

16T48 x 2

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

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:  32.40
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C 4D 8A1 8A-1 8B1 8B-1
Size 1 1 1 1 4 4 2 2 4 4 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 4A 4A 4A 4A
Type
32.40.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.40.1b R 1 1 −1 −1 −1 1 1 −1 −1 1 −1 −1 1 1
32.40.1c R 1 1 −1 −1 −1 1 1 −1 1 −1 1 1 −1 −1
32.40.1d R 1 1 −1 −1 1 −1 1 −1 −1 1 1 1 −1 −1
32.40.1e R 1 1 −1 −1 1 −1 1 −1 1 −1 −1 −1 1 1
32.40.1f R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1
32.40.1g R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.40.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
32.40.2a R 2 2 −2 −2 0 0 −2 2 0 0 0 0 0 0
32.40.2b R 2 2 2 2 0 0 −2 −2 0 0 0 0 0 0
32.40.2c1 C 2 −2 −2 2 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83
32.40.2c2 C 2 −2 −2 2 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 ζ8+ζ83 −ζ8−ζ83
32.40.2d1 C 2 −2 2 −2 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83
32.40.2d2 C 2 −2 2 −2 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83

magma: CharacterTable(G);