↑

Properties

Label 32T2134
Degree $32$
Order $160$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2.C_2^4:C_5$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(32, 2134);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2134$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2.C_2^4:C_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,12,6,30,28,2,11,5,29,27)(3,31,18,7,20,4,32,17,8,19)(9,10)(13,23,16,21,25,14,24,15,22,26), (1,18,13,20,5)(2,17,14,19,6)(3,26,24,31,9)(4,25,23,32,10)(7,28,16,22,30)(8,27,15,21,29)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$5$:  $C_5$
$80$:  $C_2^4 : C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 8: None

Degree 16: 16T178

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 TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B $2^{16}$ $10$ $2$ $16$ $( 1,13)( 2,14)( 3, 7)( 4, 8)( 5, 9)( 6,10)(11,31)(12,32)(15,28)(16,27)(17,30)(18,29)(19,24)(20,23)(21,26)(22,25)$
4A $4^{8}$ $10$ $4$ $24$ $( 1,23, 2,24)( 3,30, 4,29)( 5,28, 6,27)( 7,18, 8,17)( 9,16,10,15)(11,21,12,22)(13,19,14,20)(25,32,26,31)$
4B $4^{8}$ $10$ $4$ $24$ $( 1,12, 2,11)( 3, 9, 4,10)( 5, 8, 6, 7)(13,32,14,31)(15,30,16,29)(17,27,18,28)(19,26,20,25)(21,23,22,24)$
5A1 $5^{6},1^{2}$ $16$ $5$ $24$ $( 1,18,13,20, 5)( 2,17,14,19, 6)( 3,26,24,31, 9)( 4,25,23,32,10)( 7,28,16,22,30)( 8,27,15,21,29)$
5A-1 $5^{6},1^{2}$ $16$ $5$ $24$ $( 1, 5,20,13,18)( 2, 6,19,14,17)( 3, 9,31,24,26)( 4,10,32,23,25)( 7,30,22,16,28)( 8,29,21,15,27)$
5A2 $5^{6},1^{2}$ $16$ $5$ $24$ $( 1,20,18, 5,13)( 2,19,17, 6,14)( 3,31,26, 9,24)( 4,32,25,10,23)( 7,22,28,30,16)( 8,21,27,29,15)$
5A-2 $5^{6},1^{2}$ $16$ $5$ $24$ $( 1,13, 5,18,20)( 2,14, 6,17,19)( 3,24, 9,26,31)( 4,23,10,25,32)( 7,16,30,28,22)( 8,15,29,27,21)$
10A1 $10^{3},2$ $16$ $10$ $28$ $( 1,14, 5,17,20, 2,13, 6,18,19)( 3,23, 9,25,31, 4,24,10,26,32)( 7,15,30,27,22, 8,16,29,28,21)(11,12)$
10A-1 $10^{3},2$ $16$ $10$ $28$ $( 1,19,18, 6,13, 2,20,17, 5,14)( 3,32,26,10,24, 4,31,25, 9,23)( 7,21,28,29,16, 8,22,27,30,15)(11,12)$
10A3 $10^{3},2$ $16$ $10$ $28$ $( 1, 6,20,14,18, 2, 5,19,13,17)( 3,10,31,23,26, 4, 9,32,24,25)( 7,29,22,15,28, 8,30,21,16,27)(11,12)$
10A-3 $10^{3},2$ $16$ $10$ $28$ $( 1,17,13,19, 5, 2,18,14,20, 6)( 3,25,24,32, 9, 4,26,23,31,10)( 7,27,16,21,30, 8,28,15,22,29)(11,12)$

Malle's constant $a(G)$:     $1/16$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $160=2^{5} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  160.199
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 4A 4B 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3
Size 1 1 10 10 10 16 16 16 16 16 16 16 16
2 P 1A 1A 1A 2A 2A 5A-1 5A1 5A2 5A-2 5A-2 5A2 5A1 5A-1
5 P 1A 2A 2B 4A 4B 1A 1A 1A 1A 2A 2A 2A 2A
Type
160.199.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
160.199.1b1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2
160.199.1b2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52
160.199.1b3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1
160.199.1b4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5
160.199.4a S 4 −4 0 0 0 −1 −1 −1 −1 1 1 1 1
160.199.4b1 C 4 −4 0 0 0 −ζ5 −ζ5−1 −ζ52 −ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5
160.199.4b2 C 4 −4 0 0 0 −ζ5−1 −ζ5 −ζ5−2 −ζ52 ζ52 ζ5−2 ζ5 ζ5−1
160.199.4b3 C 4 −4 0 0 0 −ζ52 −ζ5−2 −ζ5−1 −ζ5 ζ5 ζ5−1 ζ5−2 ζ52
160.199.4b4 C 4 −4 0 0 0 −ζ5−2 −ζ52 −ζ5 −ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2
160.199.5a R 5 5 −3 1 1 0 0 0 0 0 0 0 0
160.199.5b R 5 5 1 −3 1 0 0 0 0 0 0 0 0
160.199.5c R 5 5 1 1 −3 0 0 0 0 0 0 0 0

magma: CharacterTable(G);