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Properties

Label 32T16
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4^2:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(32, 16);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $16$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4^2:C_2$
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:  (1,22,31,25)(2,21,32,26)(3,23,30,28)(4,24,29,27)(5,17,10,14)(6,18,9,13)(7,19,12,16)(8,20,11,15), (1,8)(2,7)(3,5)(4,6)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28), (1,16,30,18)(2,15,29,17)(3,13,31,19)(4,14,32,20)(5,26,11,24)(6,25,12,23)(7,28,9,22)(8,27,10,21)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$, $Q_8:C_2$ x 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, $Q_8:C_2$ x 6

Degree 16: $D_4\times C_2$, $Q_8 : C_2$ x 2, 16T30 x 2

Low degree siblings

16T30 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2B $2^{16}$ $1$ $2$ $16$ $( 1,30)( 2,29)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,19)(14,20)(15,17)(16,18)(21,27)(22,28)(23,25)(24,26)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,31)( 2,32)( 3,30)( 4,29)( 5,10)( 6, 9)( 7,12)( 8,11)(13,18)(14,17)(15,20)(16,19)(21,26)(22,25)(23,28)(24,27)$
2D $2^{16}$ $4$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 6)( 7, 8)( 9,10)(11,12)(13,20)(14,19)(15,18)(16,17)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32)$
2E $2^{16}$ $4$ $2$ $16$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)$
4A $4^{8}$ $2$ $4$ $24$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,29, 8,32)( 6,30, 7,31)(13,23,16,22)(14,24,15,21)(17,27,20,26)(18,28,19,25)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,18,30,16)( 2,17,29,15)( 3,19,31,13)( 4,20,32,14)( 5,24,11,26)( 6,23,12,25)( 7,22, 9,28)( 8,21,10,27)$
4C1 $4^{8}$ $2$ $4$ $24$ $( 1,22,31,25)( 2,21,32,26)( 3,23,30,28)( 4,24,29,27)( 5,17,10,14)( 6,18, 9,13)( 7,19,12,16)( 8,20,11,15)$
4C-1 $4^{8}$ $2$ $4$ $24$ $( 1, 7, 3, 6)( 2, 8, 4, 5)( 9,31,12,30)(10,32,11,29)(13,25,16,28)(14,26,15,27)(17,21,20,24)(18,22,19,23)$
4D1 $4^{8}$ $2$ $4$ $24$ $( 1,16,30,18)( 2,15,29,17)( 3,13,31,19)( 4,14,32,20)( 5,26,11,24)( 6,25,12,23)( 7,28, 9,22)( 8,27,10,21)$
4D-1 $4^{8}$ $2$ $4$ $24$ $( 1,25,31,22)( 2,26,32,21)( 3,28,30,23)( 4,27,29,24)( 5,14,10,17)( 6,13, 9,18)( 7,16,12,19)( 8,15,11,20)$
4E $4^{8}$ $4$ $4$ $24$ $( 1,27, 3,26)( 2,28, 4,25)( 5,13, 8,16)( 6,14, 7,15)( 9,17,12,20)(10,18,11,19)(21,31,24,30)(22,32,23,29)$
4F $4^{8}$ $4$ $4$ $24$ $( 1,20, 3,17)( 2,19, 4,18)( 5,23, 8,22)( 6,24, 7,21)( 9,27,12,26)(10,28,11,25)(13,32,16,29)(14,31,15,30)$

Malle's constant $a(G)$:     $1/16$

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:  $2$
Label:  32.31
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A 4B 4C1 4C-1 4D1 4D-1 4E 4F
Size 1 1 1 1 4 4 2 2 2 2 2 2 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2B 2C 2A 2B 2C 2A 2A
Type
32.31.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.31.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
32.31.1c R 1 1 1 1 −1 −1 1 1 1 1 1 1 −1 −1
32.31.1d R 1 1 1 1 −1 1 −1 −1 −1 −1 1 1 −1 1
32.31.1e R 1 1 1 1 −1 1 −1 −1 1 1 −1 −1 1 −1
32.31.1f R 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 −1
32.31.1g R 1 1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 1
32.31.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
32.31.2a R 2 −2 −2 2 0 0 −2 2 0 0 0 0 0 0
32.31.2b R 2 −2 −2 2 0 0 2 −2 0 0 0 0 0 0
32.31.2c1 C 2 −2 2 −2 0 0 0 0 −2i 2i 0 0 0 0
32.31.2c2 C 2 −2 2 −2 0 0 0 0 2i −2i 0 0 0 0
32.31.2d1 C 2 2 −2 −2 0 0 0 0 0 0 −2i 2i 0 0
32.31.2d2 C 2 2 −2 −2 0 0 0 0 0 0 2i −2i 0 0

magma: CharacterTable(G);