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Properties

Label 20T73
Degree $20$
Order $320$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\wr D_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 73);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $73$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\wr D_5$
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:  (1,6)(2,5)(3,4)(7,19,18,10)(8,20,17,9)(11,16)(12,15)(13,14), (1,14,12,3)(2,13,11,4)(5,9,16,20)(6,10,15,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$10$:  $D_{5}$
$20$:  $D_{10}$
$160$:  $(C_2^4 : C_5) : C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: None

Degree 5: $D_{5}$

Degree 10: $D_{10}$, $(C_2^4 : C_5) : C_2$, $C_2\times (C_2^4 : D_5)$

Low degree siblings

10T23 x 6, 20T71 x 6, 20T73 x 5, 20T76 x 6, 20T81 x 3, 20T85 x 6, 20T87 x 6, 32T9313 x 2, 40T204 x 3, 40T270 x 12, 40T271 x 12, 40T272 x 3, 40T273 x 2, 40T284 x 6, 40T286 x 6, 40T288 x 3, 40T293 x 3, 40T295 x 6

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{20}$ $1$ $1$ $()$
$2^{4},1^{12}$ $5$ $2$ $( 7,18)( 8,17)( 9,20)(10,19)$
$2^{4},1^{12}$ $5$ $2$ $( 5,16)( 6,15)( 9,20)(10,19)$
$2^{8},1^{4}$ $20$ $2$ $( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,19)(14,20)(15,17)(16,18)$
$4^{4},1^{4}$ $20$ $4$ $( 3, 9,14,20)( 4,10,13,19)( 5, 7,16,18)( 6, 8,15,17)$
$2^{8},1^{4}$ $5$ $2$ $( 3,14)( 4,13)( 5,16)( 6,15)( 7,18)( 8,17)( 9,20)(10,19)$
$2^{10}$ $5$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,19)(10,20)(11,12)(13,14)(15,16)(17,18)$
$2^{10}$ $5$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)(11,12)(13,14)$
$4^{2},2^{6}$ $20$ $4$ $( 1, 2)( 3,10,14,19)( 4, 9,13,20)( 5, 8)( 6, 7)(11,12)(15,18)(16,17)$
$4^{2},2^{6}$ $20$ $4$ $( 1, 2)( 3,10)( 4, 9)( 5, 8,16,17)( 6, 7,15,18)(11,12)(13,20)(14,19)$
$2^{10}$ $5$ $2$ $( 1, 2)( 3,13)( 4,14)( 5, 6)( 7,17)( 8,18)( 9,19)(10,20)(11,12)(15,16)$
$4^{2},2^{6}$ $20$ $4$ $( 1, 3)( 2, 4)( 5, 9,16,20)( 6,10,15,19)( 7,18)( 8,17)(11,13)(12,14)$
$5^{4}$ $32$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)(11,13,15,17,19)(12,14,16,18,20)$
$4^{2},2^{6}$ $20$ $4$ $( 1, 3,12,14)( 2, 4,11,13)( 5, 9)( 6,10)( 7,18)( 8,17)(15,19)(16,20)$
$2^{10}$ $20$ $2$ $( 1, 4)( 2, 3)( 5,10)( 6, 9)( 7,17)( 8,18)(11,14)(12,13)(15,20)(16,19)$
$10^{2}$ $32$ $10$ $( 1, 4, 5, 8, 9,11,14,15,18,19)( 2, 3, 6, 7,10,12,13,16,17,20)$
$4^{4},2^{2}$ $20$ $4$ $( 1, 4,12,13)( 2, 3,11,14)( 5,10,16,19)( 6, 9,15,20)( 7,17)( 8,18)$
$5^{4}$ $32$ $5$ $( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)(11,15,19,13,17)(12,16,20,14,18)$
$10^{2}$ $32$ $10$ $( 1, 6, 9,13,18,11,16,19, 3, 8)( 2, 5,10,14,17,12,15,20, 4, 7)$
$2^{10}$ $1$ $2$ $( 1,11)( 2,12)( 3,13)( 4,14)( 5,15)( 6,16)( 7,17)( 8,18)( 9,19)(10,20)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $320=2^{6} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  320.1636
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 5A1 5A2 10A1 10A3
Size 1 1 5 5 5 5 5 5 20 20 20 20 20 20 20 20 32 32 32 32
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2B 2B 2C 2D 2D 2C 5A2 5A1 5A1 5A2
5 P 1A 2A 2F 2C 2D 2B 2E 2G 2H 2I 4A 4B 4C 4D 4E 4F 1A 1A 2A 2A
Type
320.1636.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
320.1636.1b R 1 −1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1
320.1636.1c R 1 −1 1 1 1 −1 −1 −1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
320.1636.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 1 1
320.1636.2a1 R 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52
320.1636.2a2 R 2 2 2 2 2 2 2 2 0 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5
320.1636.2b1 R 2 −2 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52
320.1636.2b2 R 2 −2 2 2 2 −2 −2 −2 0 0 0 0 0 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5
320.1636.5a R 5 5 −3 1 1 −3 1 1 1 1 1 1 −1 −1 −1 −1 0 0 0 0
320.1636.5b R 5 5 1 −3 1 1 1 −3 1 1 −1 −1 1 −1 −1 1 0 0 0 0
320.1636.5c R 5 5 1 1 −3 1 −3 1 1 1 −1 −1 −1 1 1 −1 0 0 0 0
320.1636.5d R 5 5 −3 1 1 −3 1 1 −1 −1 −1 −1 1 1 1 1 0 0 0 0
320.1636.5e R 5 5 1 −3 1 1 1 −3 −1 −1 1 1 −1 1 1 −1 0 0 0 0
320.1636.5f R 5 5 1 1 −3 1 −3 1 −1 −1 1 1 1 −1 −1 1 0 0 0 0
320.1636.5g R 5 −5 −3 1 1 3 −1 −1 −1 1 1 −1 1 1 −1 −1 0 0 0 0
320.1636.5h R 5 −5 −3 1 1 3 −1 −1 1 −1 −1 1 −1 −1 1 1 0 0 0 0
320.1636.5i R 5 −5 1 −3 1 −1 −1 3 −1 1 −1 1 −1 1 −1 1 0 0 0 0
320.1636.5j R 5 −5 1 −3 1 −1 −1 3 1 −1 1 −1 1 −1 1 −1 0 0 0 0
320.1636.5k R 5 −5 1 1 −3 −1 3 −1 −1 1 −1 1 1 −1 1 −1 0 0 0 0
320.1636.5l R 5 −5 1 1 −3 −1 3 −1 1 −1 1 −1 −1 1 −1 1 0 0 0 0

magma: CharacterTable(G);