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Properties

Label 32T32
Degree $32$
Order $32$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\times C_{16}$

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Show commands: Magma

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

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_8$ x 2, $C_4\times C_2$
$16$:  $C_{16}$ x 2, $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_8$ x 2, $C_4\times C_2$

Degree 16: $C_{16}$ x 2, $C_8\times C_2$

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  32.16
magma: IdentifyGroup(G);
 
Character table:    32 x 32 character table

magma: CharacterTable(G);