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Properties

Label 20T29
Degree $20$
Order $100$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 29);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $29$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times F_5$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $5$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,17,6,2,12,16,8,3,15,20,10,4,13,19,7,5,11,18,9), (1,15,20,7,3,11,18,6,5,12,16,10,2,13,19,9,4,14,17,8)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$5$:  $C_5$
$10$:  $C_{10}$
$20$:  $F_5$, 20T1

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: None

Low degree siblings

25T7

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

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:  100.9
magma: IdentifyGroup(G);
 
Character table:

1A 2A 4A1 4A-1 5A1 5A-1 5A2 5A-2 5B 5C1 5C-1 5C2 5C-2 10A1 10A-1 10A3 10A-3 20A1 20A-1 20A3 20A-3 20A7 20A-7 20A9 20A-9
Size 1 5 5 5 1 1 1 1 4 4 4 4 4 5 5 5 5 5 5 5 5 5 5 5 5
2 P 1A 1A 2A 2A 5A-2 5A2 5A1 5A-1 5B 5C-1 5C-2 5C1 5C2 5A-2 5A-1 5A2 5A1 10A1 10A1 10A3 10A-3 10A-3 10A-1 10A-1 10A3
5 P 1A 2A 4A1 4A-1 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 4A1 4A-1 4A-1 4A1 4A-1 4A-1 4A1 4A1
Type
100.9.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
100.9.1b R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
100.9.1c1 C 1 −1 −i i 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −i i i −i i −i −i i
100.9.1c2 C 1 −1 i −i 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 i −i −i i −i i i −i
100.9.1d1 C 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ5−2 ζ52
100.9.1d2 C 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ52 ζ5−2
100.9.1d3 C 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5−1 ζ5
100.9.1d4 C 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5 ζ5−1
100.9.1e1 C 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ52 ζ5−2 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ5−1 −ζ5 −ζ5−2 −ζ52
100.9.1e2 C 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5 −ζ5−1 −ζ52 −ζ5−2
100.9.1e3 C 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ52 −ζ5−2 −ζ5−1 −ζ5
100.9.1e4 C 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−2 −ζ52 −ζ5 −ζ5−1
100.9.1f1 C 1 −1 −ζ205 ζ205 −ζ202 ζ208 ζ204 −ζ206 1 −ζ206 ζ204 −ζ202 ζ208 ζ206 −ζ204 −ζ208 ζ202 ζ203 −ζ207 ζ209 −ζ20 ζ20 −ζ209 ζ207 −ζ203
100.9.1f2 C 1 −1 ζ205 −ζ205 ζ208 −ζ202 −ζ206 ζ204 1 ζ204 −ζ206 ζ208 −ζ202 −ζ204 ζ206 ζ202 −ζ208 −ζ207 ζ203 −ζ20 ζ209 −ζ209 ζ20 −ζ203 ζ207
100.9.1f3 C 1 −1 −ζ205 ζ205 ζ208 −ζ202 −ζ206 ζ204 1 ζ204 −ζ206 ζ208 −ζ202 −ζ204 ζ206 ζ202 −ζ208 ζ207 −ζ203 ζ20 −ζ209 ζ209 −ζ20 ζ203 −ζ207
100.9.1f4 C 1 −1 ζ205 −ζ205 −ζ202 ζ208 ζ204 −ζ206 1 −ζ206 ζ204 −ζ202 ζ208 ζ206 −ζ204 −ζ208 ζ202 −ζ203 ζ207 −ζ209 ζ20 −ζ20 ζ209 −ζ207 ζ203
100.9.1f5 C 1 −1 −ζ205 ζ205 −ζ206 ζ204 −ζ202 ζ208 1 ζ208 −ζ202 −ζ206 ζ204 −ζ208 ζ202 −ζ204 ζ206 −ζ209 ζ20 −ζ207 ζ203 −ζ203 ζ207 −ζ20 ζ209
100.9.1f6 C 1 −1 ζ205 −ζ205 ζ204 −ζ206 ζ208 −ζ202 1 −ζ202 ζ208 ζ204 −ζ206 ζ202 −ζ208 ζ206 −ζ204 ζ20 −ζ209 ζ203 −ζ207 ζ207 −ζ203 ζ209 −ζ20
100.9.1f7 C 1 −1 −ζ205 ζ205 ζ204 −ζ206 ζ208 −ζ202 1 −ζ202 ζ208 ζ204 −ζ206 ζ202 −ζ208 ζ206 −ζ204 −ζ20 ζ209 −ζ203 ζ207 −ζ207 ζ203 −ζ209 ζ20
100.9.1f8 C 1 −1 ζ205 −ζ205 −ζ206 ζ204 −ζ202 ζ208 1 ζ208 −ζ202 −ζ206 ζ204 −ζ208 ζ202 −ζ204 ζ206 ζ209 −ζ20 ζ207 −ζ203 ζ203 −ζ207 ζ20 −ζ209
100.9.4a R 4 0 0 0 4 4 4 4 −1 −1 −1 −1 −1 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b1 C 4 0 0 0 4ζ5−2 4ζ52 4ζ5 4ζ5−1 −1 −ζ5−1 −ζ5 −ζ5−2 −ζ52 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b2 C 4 0 0 0 4ζ52 4ζ5−2 4ζ5−1 4ζ5 −1 −ζ5 −ζ5−1 −ζ52 −ζ5−2 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b3 C 4 0 0 0 4ζ5−1 4ζ5 4ζ5−2 4ζ52 −1 −ζ52 −ζ5−2 −ζ5−1 −ζ5 0 0 0 0 0 0 0 0 0 0 0 0
100.9.4b4 C 4 0 0 0 4ζ5 4ζ5−1 4ζ52 4ζ5−2 −1 −ζ5−2 −ζ52 −ζ5 −ζ5−1 0 0 0 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);