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Properties

Label 20T25
Degree $20$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5:C_{20}$

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Show commands: Magma

magma: G := TransitiveGroup(20, 25);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $25$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5:C_{20}$
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:  (1,12,2,11)(3,14,4,13)(5,15,6,16)(7,18,8,17)(9,19,10,20), (1,9,17,6,13,2,10,18,5,14)(3,19,15,11,7,4,20,16,12,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $D_{5}$, $C_{10}$
$20$:  20T1, 20T2
$50$:  $D_5\times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $D_5\times C_5$

Low degree siblings

20T25

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

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:  100.6
magma: IdentifyGroup(G);
 
Character table:    40 x 40 character table

magma: CharacterTable(G);