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Properties

Label 20T99
20T99 1 3 1->3 4 1->4 13 1->13 2 2->3 2->4 14 2->14 6 3->6 3->13 15 3->15 5 4->5 4->14 16 4->16 11 5->11 12 5->12 6->11 6->12 7 7->2 7->4 7->13 8 8->1 8->3 8->14 9 17 9->17 19 9->19 20 9->20 10 18 10->18 10->19 10->20 11->1 11->10 12->2 12->9 13->8 13->9 14->7 14->10 15->10 15->17 15->19 16->9 16->18 16->20 17->1 17->16 18->2 18->15 19->5 19->8 19->18 20->6 20->7 20->17
Degree $20$
Order $400$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2\times C_5^2:Q_8$

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Copy content magma:G := TransitiveGroup(20, 99);
 

Group invariants

Abstract group:  $C_2\times C_5^2:Q_8$
Copy content magma:IdentifyGroup(G);
 
Order:  $400=2^{4} \cdot 5^{2}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $20$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $99$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,13,9,17)(2,14,10,18)(3,15,19,8)(4,16,20,7)$, $(1,4,5,11)(2,3,6,12)(7,13)(8,14)(9,20,17,16)(10,19,18,15)$, $(1,3,13,8)(2,4,14,7)(5,12,9,19)(6,11,10,20)(15,17)(16,18)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $C_2^3$, $Q_8$ x 2
$16$:  $Q_8\times C_2$
$200$:  $C_5^2 : Q_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 5: None

Degree 10: $C_5^2 : Q_8$

Low degree siblings

20T99 x 5, 40T321 x 3, 40T325 x 3

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
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 $2^{8},1^{4}$ $25$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)(13,17)(14,18)(15,19)(16,20)$
2C $2^{10}$ $25$ $2$ $10$ $( 1,18)( 2,17)( 3,16)( 4,15)( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(19,20)$
4A $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,16, 9,20)( 2,15,10,19)( 3,14,12,18)( 4,13,11,17)( 5, 7)( 6, 8)$
4B $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,16,17,11)( 2,15,18,12)( 3,10)( 4, 9)( 5,20,13, 7)( 6,19,14, 8)$
4C $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,18, 9,14)( 2,17,10,13)( 3, 7,19,16)( 4, 8,20,15)( 5, 6)(11,12)$
4D $4^{4},1^{4}$ $50$ $4$ $12$ $( 1, 5,13, 9)( 2, 6,14,10)( 3,19, 8,12)( 4,20, 7,11)$
4E $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,15, 9,12)( 2,16,10,11)( 3, 5)( 4, 6)( 7,14,20,18)( 8,13,19,17)$
4F $4^{4},2^{2}$ $50$ $4$ $14$ $( 1,19)( 2,20)( 3, 5,15,17)( 4, 6,16,18)( 7,10,11,14)( 8, 9,12,13)$
5A $5^{2},1^{10}$ $8$ $5$ $8$ $( 1,17,13, 9, 5)( 2,18,14,10, 6)$
5B $5^{4}$ $8$ $5$ $16$ $( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,19,15,12, 8)( 4,20,16,11, 7)$
5C $5^{4}$ $8$ $5$ $16$ $( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3,19,15,12, 8)( 4,20,16,11, 7)$
10A $10,2^{5}$ $8$ $10$ $14$ $( 1,10,17, 6,13, 2, 9,18, 5,14)( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$
10B $10^{2}$ $8$ $10$ $18$ $( 1, 6, 9,14,17, 2, 5,10,13,18)( 3,11,19, 7,15, 4,12,20, 8,16)$
10C $10^{2}$ $8$ $10$ $18$ $( 1,14, 5,18, 9, 2,13, 6,17,10)( 3,11,19, 7,15, 4,12,20, 8,16)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 10A 10B 10C
Size 1 1 25 25 50 50 50 50 50 50 8 8 8 8 8 8
2 P 1A 1A 1A 1A 2B 2B 2B 2B 2B 2B 5A 5B 5C 5A 5B 5C
5 P 1A 2A 2B 2C 4A 4B 4C 4D 4E 4F 1A 1A 1A 2A 2A 2A
Type
400.212.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
400.212.1b R 1 −1 1 −1 −1 −1 −1 1 1 1 1 1 1 −1 −1 −1
400.212.1c R 1 −1 1 −1 −1 1 1 −1 1 −1 1 1 1 −1 −1 −1
400.212.1d R 1 −1 1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1 −1
400.212.1e R 1 −1 1 −1 1 1 −1 1 −1 −1 1 1 1 −1 −1 −1
400.212.1f R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1 1 1
400.212.1g R 1 1 1 1 −1 1 −1 −1 −1 1 1 1 1 1 1 1
400.212.1h R 1 1 1 1 1 −1 −1 −1 1 −1 1 1 1 1 1 1
400.212.2a S 2 −2 −2 2 0 0 0 0 0 0 2 2 2 −2 −2 −2
400.212.2b S 2 2 −2 −2 0 0 0 0 0 0 2 2 2 2 2 2
400.212.8a R 8 8 0 0 0 0 0 0 0 0 −2 −2 3 −2 −2 3
400.212.8b R 8 8 0 0 0 0 0 0 0 0 −2 3 −2 −2 3 −2
400.212.8c R 8 8 0 0 0 0 0 0 0 0 3 −2 −2 3 −2 −2
400.212.8d R 8 −8 0 0 0 0 0 0 0 0 −2 −2 3 2 2 −3
400.212.8e R 8 −8 0 0 0 0 0 0 0 0 −2 3 −2 2 −3 2
400.212.8f R 8 −8 0 0 0 0 0 0 0 0 3 −2 −2 −3 2 2

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed