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Properties

Label 20T45
Degree $20$
Order $160$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4:D_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 45);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^4:D_5$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5)(2,6)(3,4)(7,9)(8,10)(11,15)(12,16)(13,14)(17,19)(18,20), (1,19,2,20)(3,17,14,8)(4,18,13,7)(5,16)(6,15)(9,12,10,11)
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$10$:  $D_{5}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $(C_2^4 : C_5) : C_2$, $(C_2^4 : C_5) : C_2$ x 2

Low degree siblings

10T15 x 3, 10T16 x 3, 16T415, 20T38 x 6, 20T39, 20T43 x 3, 20T45 x 2, 32T2132, 40T143 x 3, 40T144 x 3, 40T145 x 6, 40T146

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{8},1^{4}$ $5$ $2$ $8$ $( 1, 2)( 3,14)( 4,13)( 7,18)( 8,17)( 9,10)(11,12)(19,20)$
2B $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,11)( 2,12)( 5,15)( 6,16)( 7,18)( 8,17)( 9,20)(10,19)$
2C $2^{8},1^{4}$ $5$ $2$ $8$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5, 6)( 9,10)(15,16)(19,20)$
2D $2^{10}$ $20$ $2$ $10$ $( 1,10)( 2, 9)( 3, 8)( 4, 7)( 5, 6)(11,20)(12,19)(13,18)(14,17)(15,16)$
4A $4^{4},2^{2}$ $20$ $4$ $14$ $( 1,19, 2,20)( 3,17,14, 8)( 4,18,13, 7)( 5,16)( 6,15)( 9,12,10,11)$
4B $4^{4},1^{4}$ $20$ $4$ $12$ $( 3, 9,13,19)( 4,10,14,20)( 5, 7,16,18)( 6, 8,15,17)$
4C $4^{4},2^{2}$ $20$ $4$ $14$ $( 1, 3,11,13)( 2, 4,12,14)( 5,20, 6,19)( 7,17)( 8,18)( 9,15,10,16)$
5A1 $5^{4}$ $32$ $5$ $16$ $( 1, 5, 9,14,18)( 2, 6,10,13,17)( 3, 7,12,16,20)( 4, 8,11,15,19)$
5A2 $5^{4}$ $32$ $5$ $16$ $( 1, 9,18, 5,14)( 2,10,17, 6,13)( 3,12,20, 7,16)( 4,11,19, 8,15)$

Malle's constant $a(G)$:     $1/8$

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:  160.234
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A 4B 4C 5A1 5A2
Size 1 5 5 5 20 20 20 20 32 32
2 P 1A 1A 1A 1A 1A 2A 2B 2C 5A2 5A1
5 P 1A 2A 2B 2C 2D 4A 4B 4C 1A 1A
Type
160.234.1a R 1 1 1 1 1 1 1 1 1 1
160.234.1b R 1 1 1 1 −1 −1 −1 −1 1 1
160.234.2a1 R 2 2 2 2 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5
160.234.2a2 R 2 2 2 2 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52
160.234.5a R 5 −3 1 1 1 1 −1 −1 0 0
160.234.5b R 5 1 −3 1 1 −1 1 −1 0 0
160.234.5c R 5 1 1 −3 1 −1 −1 1 0 0
160.234.5d R 5 −3 1 1 −1 −1 1 1 0 0
160.234.5e R 5 1 −3 1 −1 1 −1 1 0 0
160.234.5f R 5 1 1 −3 −1 1 1 −1 0 0

magma: CharacterTable(G);