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Properties

Label 20T28
Degree $20$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5^2$

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Show commands: Magma

magma: G := TransitiveGroup(20, 28);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5^2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,5)(2,6)(3,20)(4,19)(7,16)(8,15)(9,17)(10,18)(11,12)(13,14), (1,12,14,3,6,16,17,8,10,19)(2,11,13,4,5,15,18,7,9,20)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $D_5^2$

Low degree siblings

10T9 x 2, 20T28, 25T12

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3, 8,12,16,19)( 4, 7,11,15,20)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $5$ $( 3,12,19, 8,16)( 4,11,20, 7,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $25$ $2$ $( 1, 2)( 3, 4)( 5,17)( 6,18)( 7,19)( 8,20)( 9,14)(10,13)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)$
$ 10, 10 $ $10$ $10$ $( 1, 3, 6, 8,10,12,14,16,17,19)( 2, 4, 5, 7, 9,11,13,15,18,20)$
$ 10, 10 $ $10$ $10$ $( 1, 3,10,12,17,19, 6, 8,14,16)( 2, 4, 9,11,18,20, 5, 7,13,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $5$ $2$ $( 1, 4)( 2, 3)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$
$ 10, 10 $ $10$ $10$ $( 1, 4, 6,20,10,15,14,11,17, 7)( 2, 3, 5,19, 9,16,13,12,18, 8)$
$ 10, 10 $ $10$ $10$ $( 1, 4,10,15,17, 7, 6,20,14,11)( 2, 3, 9,16,18, 8, 5,19,13,12)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3, 8,12,16,19)( 4, 7,11,15,20)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,12,19, 8,16)( 4,11,20, 7,15)$
$ 5, 5, 5, 5 $ $4$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,16, 8,19,12)( 4,15, 7,20,11)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1, 6,10,14,17)( 2, 5, 9,13,18)( 3,19,16,12, 8)( 4,20,15,11, 7)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)( 3,12,19, 8,16)( 4,11,20, 7,15)$
$ 5, 5, 5, 5 $ $2$ $5$ $( 1,10,17, 6,14)( 2, 9,18, 5,13)( 3,16, 8,19,12)( 4,15, 7,20,11)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $100=2^{2} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  100.13
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 5A1 5A2 5B1 5B2 5C1 5C2 5D1 5D2 10A1 10A3 10B1 10B3
Size 1 5 5 25 2 2 2 2 4 4 4 4 10 10 10 10
2 P 1A 1A 1A 1A 5A2 5A1 5B2 5B1 5D2 5D1 5C2 5C1 5A2 5B1 5A1 5B2
5 P 1A 2A 2B 2C 1A 1A 1A 1A 1A 1A 1A 1A 2A 2B 2A 2B
Type
100.13.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
100.13.1b R 1 −1 −1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
100.13.1c R 1 −1 1 −1 1 1 1 1 1 1 1 1 −1 −1 1 1
100.13.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 −1 −1
100.13.2a1 R 2 0 2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52
100.13.2a2 R 2 0 2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5
100.13.2b1 R 2 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0
100.13.2b2 R 2 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0
100.13.2c1 R 2 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0
100.13.2c2 R 2 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0
100.13.2d1 R 2 0 −2 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52
100.13.2d2 R 2 0 −2 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5
100.13.4a1 R 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 0 0 0 0
100.13.4a2 R 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 0 0 0 0
100.13.4b1 R 4 0 0 0 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 −1 0 0 0 0
100.13.4b2 R 4 0 0 0 2ζ5−1+2ζ5 2ζ5−2+2ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 −1 0 0 0 0

magma: CharacterTable(G);