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Properties

Label 20T49
Degree $20$
Order $200$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{10}:F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 49);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $49$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{10}:F_5$
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:  (1,11,18,7)(2,12,17,8)(3,5,16,13)(4,6,15,14)(9,20,10,19), (1,11,6,7)(2,12,5,8)(3,17,16,9)(4,18,15,10)(13,20,14,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $C_4\times C_2$
$20$:  $F_5$ x 2
$40$:  $F_{5}\times C_2$ x 2
$100$:  $C_5^2 : C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 5: None

Degree 10: $C_5^2 : C_4$

Low degree siblings

20T49, 20T52 x 2, 40T157 x 2

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $200=2^{3} \cdot 5^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  200.48
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 5A 5B 5C1 5C2 5D1 5D2 10A 10B 10C1 10C3 10D1 10D3
Size 1 1 25 25 25 25 25 25 4 4 4 4 4 4 4 4 4 4 4 4
2 P 1A 1A 1A 1A 2B 2B 2B 2B 5C1 5D1 5A 5C2 5B 5D2 5A 5D1 5D2 5C1 5C2 5B
5 P 1A 2A 2B 2C 4A1 4A-1 4B1 4B-1 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2A
Type
200.48.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
200.48.1b R 1 −1 1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
200.48.1c R 1 −1 1 −1 1 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
200.48.1d R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 1 1 1 1
200.48.1e1 C 1 −1 −1 1 −i i −i i 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
200.48.1e2 C 1 −1 −1 1 i −i i −i 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
200.48.1f1 C 1 1 −1 −1 −i i i −i 1 1 1 1 1 1 1 1 1 1 1 1
200.48.1f2 C 1 1 −1 −1 i −i −i i 1 1 1 1 1 1 1 1 1 1 1 1
200.48.4a R 4 4 0 0 0 0 0 0 −1 4 −1 −1 −1 −1 −1 4 −1 −1 −1 −1
200.48.4b R 4 4 0 0 0 0 0 0 4 −1 −1 −1 −1 −1 4 −1 −1 −1 −1 −1
200.48.4c R 4 −4 0 0 0 0 0 0 −1 4 −1 −1 −1 −1 1 −4 1 1 1 1
200.48.4d R 4 −4 0 0 0 0 0 0 4 −1 −1 −1 −1 −1 −4 1 1 1 1 1
200.48.4e1 R 4 4 0 0 0 0 0 0 −1 −1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 −1 −1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 ζ5−2+2+ζ52
200.48.4e2 R 4 4 0 0 0 0 0 0 −1 −1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 −1 −1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −ζ5−2+1−ζ52
200.48.4f1 R 4 4 0 0 0 0 0 0 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5
200.48.4f2 R 4 4 0 0 0 0 0 0 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52
200.48.4g1 R 4 −4 0 0 0 0 0 0 −1 −1 2ζ5−2+2ζ52 2ζ5−1+2ζ5 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 1 1 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52 ζ5−2−1+ζ52 −ζ5−2−2−ζ52
200.48.4g2 R 4 −4 0 0 0 0 0 0 −1 −1 2ζ5−1+2ζ5 2ζ5−2+2ζ52 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 1 1 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5 −ζ5−2−2−ζ52 ζ5−2−1+ζ52
200.48.4h1 R 4 −4 0 0 0 0 0 0 −1 −1 ζ5−2+2+ζ52 −ζ5−2+1−ζ52 2ζ5−1+2ζ5 2ζ5−2+2ζ52 1 1 ζ5−2−1+ζ52 −ζ5−2−2−ζ52 −2ζ5−2−2ζ52 −2ζ5−1−2ζ5
200.48.4h2 R 4 −4 0 0 0 0 0 0 −1 −1 −ζ5−2+1−ζ52 ζ5−2+2+ζ52 2ζ5−2+2ζ52 2ζ5−1+2ζ5 1 1 −ζ5−2−2−ζ52 ζ5−2−1+ζ52 −2ζ5−1−2ζ5 −2ζ5−2−2ζ52

magma: CharacterTable(G);