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Properties

Label 20T22
Degree $20$
Order $80$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{10}:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(20, 22);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{10}:C_4$
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:  (1,14,5,12)(2,13,6,11)(3,17,4,18)(7,15,9,19)(8,16,10,20), (1,5,10,3,7,2,6,9,4,8)(11,15,20,14,18)(12,16,19,13,17)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$20$:  $F_5$
$40$:  $F_{5}\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $F_5$

Degree 10: $F_5$

Low degree siblings

20T19 x 2, 20T22, 40T26, 40T45, 40T55 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrder IndexRepresentative
1A $1^{20}$ $1$ $1$ $0$ $()$
2A $2^{10}$ $1$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{5},1^{10}$ $2$ $2$ $5$ $(11,12)(13,14)(15,16)(17,18)(19,20)$
2C $2^{8},1^{4}$ $5$ $2$ $8$ $( 1, 6)( 2, 5)( 7,10)( 8, 9)(11,14)(12,13)(15,20)(16,19)$
2D $2^{10}$ $5$ $2$ $10$ $( 1, 5)( 2, 6)( 3, 4)( 7, 9)( 8,10)(11,13)(12,14)(15,19)(16,20)(17,18)$
2E $2^{9},1^{2}$ $10$ $2$ $9$ $( 1, 3)( 2, 4)( 5,10)( 6, 9)( 7, 8)(13,19)(14,20)(15,18)(16,17)$
4A1 $4^{5}$ $10$ $4$ $15$ $( 1,12, 5,14)( 2,11, 6,13)( 3,18, 4,17)( 7,19, 9,15)( 8,20,10,16)$
4A-1 $4^{5}$ $10$ $4$ $15$ $( 1,14, 5,12)( 2,13, 6,11)( 3,17, 4,18)( 7,15, 9,19)( 8,16,10,20)$
4B1 $4^{4},2^{2}$ $10$ $4$ $14$ $( 1,18, 7,20)( 2,17, 8,19)( 3,12, 5,16)( 4,11, 6,15)( 9,13)(10,14)$
4B-1 $4^{4},2^{2}$ $10$ $4$ $14$ $( 1,17,10,12)( 2,18, 9,11)( 3,14, 8,15)( 4,13, 7,16)( 5,20)( 6,19)$
5A $5^{4}$ $4$ $5$ $16$ $( 1, 6,10, 4, 7)( 2, 5, 9, 3, 8)(11,15,20,14,18)(12,16,19,13,17)$
10A $10^{2}$ $4$ $10$ $18$ $( 1, 5,10, 3, 7, 2, 6, 9, 4, 8)(11,16,20,13,18,12,15,19,14,17)$
10B1 $10,5^{2}$ $4$ $10$ $17$ $( 1, 4, 6, 7,10)( 2, 3, 5, 8, 9)(11,13,15,17,20,12,14,16,18,19)$
10B3 $10,5^{2}$ $4$ $10$ $17$ $( 1, 7, 4,10, 6)( 2, 8, 3, 9, 5)(11,17,14,19,15,12,18,13,20,16)$

magma: ConjugacyClasses(G);
 

Malle's constant $a(G)$:     $1/5$

Group invariants

Order:  $80=2^{4} \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  80.34
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 4A1 4A-1 4B1 4B-1 5A 10A 10B1 10B3
Size 1 1 2 5 5 10 10 10 10 10 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2D 2D 2C 2C 5A 5A 5A 5A
5 P 1A 2A 2B 2C 2D 2E 4B-1 4B1 4A-1 4A1 1A 2A 2B 2B
Type
80.34.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
80.34.1b R 1 1 −1 1 1 −1 −1 −1 1 1 1 1 −1 −1
80.34.1c R 1 1 −1 1 1 −1 1 1 −1 −1 1 1 −1 −1
80.34.1d R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
80.34.1e1 C 1 1 −1 −1 −1 1 −i i −i i 1 1 −1 −1
80.34.1e2 C 1 1 −1 −1 −1 1 i −i i −i 1 1 −1 −1
80.34.1f1 C 1 1 1 −1 −1 −1 −i i i −i 1 1 1 1
80.34.1f2 C 1 1 1 −1 −1 −1 i −i −i i 1 1 1 1
80.34.2a R 2 −2 0 −2 2 0 0 0 0 0 2 −2 0 0
80.34.2b R 2 −2 0 2 −2 0 0 0 0 0 2 −2 0 0
80.34.4a R 4 4 4 0 0 0 0 0 0 0 −1 −1 −1 −1
80.34.4b R 4 4 −4 0 0 0 0 0 0 0 −1 −1 1 1
80.34.4c1 R 4 −4 0 0 0 0 0 0 0 0 −1 1 2ζ5−2+1+2ζ52 −2ζ5−2−1−2ζ52
80.34.4c2 R 4 −4 0 0 0 0 0 0 0 0 −1 1 −2ζ5−2−1−2ζ52 2ζ5−2+1+2ζ52

magma: CharacterTable(G);