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Properties

Label 32T37
Degree $32$
Order $32$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2\times C_8$

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Show commands: Magma

magma: G := TransitiveGroup(32, 37);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2\times C_8$
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:  (1,13,12,23,2,14,11,24)(3,16,9,22,4,15,10,21)(5,20,29,28,6,19,30,27)(7,18,31,26,8,17,32,25), (1,4)(2,3)(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), (1,19,2,20)(3,18,4,17)(5,24,6,23)(7,22,8,21)(9,26,10,25)(11,28,12,27)(13,30,14,29)(15,32,16,31)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $C_8$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$, $C_8\times C_2$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_8$ x 4, $C_4\times C_2$ x 6, $C_2^3$

Degree 16: $C_4\times C_2^2$, $C_8\times C_2$ x 6

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

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:  32.36
magma: IdentifyGroup(G);
 
Character table:    32 x 32 character table

magma: CharacterTable(G);