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Properties

Label 32T6
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4.C_2^3$

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Show commands: Magma

magma: G := TransitiveGroup(32, 6);
 

Group invariants

Abstract group:  $C_4.C_2^3$
magma: IdentifyGroup(G);
 
Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $6$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $32$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10,4,12)(2,9,3,11)(5,14,8,15)(6,13,7,16)(17,27,19,26)(18,28,20,25)(21,32,24,29)(22,31,23,30)$, $(1,17,4,19)(2,18,3,20)(5,30,8,31)(6,29,7,32)(9,28,11,25)(10,27,12,26)(13,21,16,24)(14,22,15,23)$, $(1,20)(2,19)(3,17)(4,18)(5,32)(6,31)(7,30)(8,29)(9,26)(10,25)(11,27)(12,28)(13,23)(14,24)(15,21)(16,22)$, $(1,6,4,7)(2,5,3,8)(9,15,11,14)(10,16,12,13)(17,32,19,29)(18,31,20,30)(21,26,24,27)(22,25,23,28)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15

Degree 16: $C_2^4$, $(C_2 \times Q_8):C_2$ x 5

Low degree siblings

16T20 x 5

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

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J
Size 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A 2A 2A 2A 2A
Type
32.50.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.50.1b R 1 1 −1 −1 −1 −1 1 1 −1 1 1 1 −1 1 −1 −1 1
32.50.1c R 1 1 −1 −1 −1 1 −1 −1 −1 1 −1 1 1 1 1 1 −1
32.50.1d R 1 1 −1 −1 1 −1 −1 1 1 1 1 −1 1 −1 1 −1 −1
32.50.1e R 1 1 −1 −1 1 1 1 −1 1 1 −1 −1 −1 −1 −1 1 1
32.50.1f R 1 1 −1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 −1 1 1
32.50.1g R 1 1 −1 1 −1 1 1 −1 1 −1 1 −1 −1 1 1 −1 −1
32.50.1h R 1 1 −1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 1 1 −1
32.50.1i R 1 1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 −1 1
32.50.1j R 1 1 1 −1 −1 −1 −1 −1 1 −1 1 1 −1 −1 1 1 1
32.50.1k R 1 1 1 −1 −1 1 1 1 1 −1 −1 1 1 −1 −1 −1 −1
32.50.1l R 1 1 1 −1 1 −1 1 −1 −1 −1 1 −1 1 1 −1 1 −1
32.50.1m R 1 1 1 −1 1 1 −1 1 −1 −1 −1 −1 −1 1 1 −1 1
32.50.1n R 1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 1 −1 1 −1 1
32.50.1o R 1 1 1 1 −1 1 −1 1 −1 1 1 −1 −1 −1 −1 1 −1
32.50.1p R 1 1 1 1 1 −1 −1 −1 1 1 −1 1 −1 1 −1 −1 −1
32.50.4a S 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed