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Properties

Label 20T223
20T223 1 12 1->12 17 1->17 2 11 2->11 18 2->18 3 5 3->5 6 3->6 4 4->5 4->6 9 5->9 16 5->16 10 6->10 15 6->15 7 7->3 7->4 8 8->3 8->4 9->8 14 9->14 10->7 13 10->13 11->1 11->1 12->2 12->2 13->12 13->17 14->11 14->18 19 15->19 20 15->20 16->19 16->20 17->9 17->16 18->10 18->15 19->8 19->14 20->7 20->13
Degree $20$
Order $1920$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2^4:S_5$

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Copy content magma:G := TransitiveGroup(20, 223);
 

Group invariants

Abstract group:  $C_2^4:S_5$
Copy content magma:IdentifyGroup(G);
 
Order:  $1920=2^{7} \cdot 3 \cdot 5$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  no
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 $t$:  $223$
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:  $(1,12,2,11)(3,5,9,8)(4,6,10,7)(13,17,16,20)(14,18,15,19)$, $(1,17,9,14,11)(2,18,10,13,12)(3,6,15,20,7)(4,5,16,19,8)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: None

Degree 10: $S_5$

Low degree siblings

10T37, 10T38, 16T1328, 20T218, 20T219, 20T222, 20T226, 30T329, 30T332, 30T333, 30T341, 32T97736, 40T1581, 40T1582, 40T1583, 40T1584, 40T1587, 40T1588, 40T1595, 40T1596, 40T1658, 40T1659, 40T1676, 40T1677, 40T1678

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 $2^{4},1^{12}$ $5$ $2$ $4$ $( 1, 2)( 3, 4)( 7, 8)(13,14)$
2B $2^{6},1^{8}$ $10$ $2$ $6$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(17,18)(19,20)$
2C $2^{7},1^{6}$ $20$ $2$ $7$ $( 1,14)( 2,13)( 5,17)( 6,18)( 9,19)(10,20)(15,16)$
2D $2^{10}$ $60$ $2$ $10$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,11)( 6,12)( 7,15)( 8,16)( 9,10)(13,14)$
2E $2^{9},1^{2}$ $60$ $2$ $9$ $( 1,14)( 2,13)( 3, 4)( 5,18)( 6,17)( 9,19)(10,20)(11,12)(15,16)$
3A $3^{6},1^{2}$ $80$ $3$ $12$ $( 1,13,16)( 2,14,15)( 3,17, 5)( 4,18, 6)( 7,20,10)( 8,19, 9)$
4A $4^{3},1^{8}$ $20$ $4$ $9$ $( 3, 8, 4, 7)( 5, 9, 6,10)(17,19,18,20)$
4B $4^{3},2^{2},1^{4}$ $60$ $4$ $11$ $( 1, 7, 2, 8)( 3, 4)( 5,11, 6,12)(13,14)(15,19,16,20)$
4C $4^{2},2^{4},1^{4}$ $60$ $4$ $10$ $( 1, 9, 2,10)( 3,19)( 4,20)( 5,16, 6,15)(11,14)(12,13)$
4D $4^{3},2^{3},1^{2}$ $120$ $4$ $12$ $( 1,11, 2,12)( 3, 4)( 5, 8, 6, 7)(13,18)(14,17)(15,19,16,20)$
4E $4^{5}$ $240$ $4$ $15$ $( 1,15,20, 7)( 2,16,19, 8)( 3, 5,18,11)( 4, 6,17,12)( 9,14,10,13)$
5A $5^{4}$ $384$ $5$ $16$ $( 1,18, 7, 5,19)( 2,17, 8, 6,20)( 3,12,10,16,13)( 4,11, 9,15,14)$
6A $6^{2},3^{2},1^{2}$ $80$ $6$ $14$ $( 1,16,13)( 2,15,14)( 3, 6,17, 4, 5,18)( 7, 9,20, 8,10,19)$
6B $6^{2},3^{2},2$ $160$ $6$ $15$ $( 1,18,10,14, 6,20)( 2,17, 9,13, 5,19)( 3,12, 7)( 4,11, 8)(15,16)$
6C $6,3^{4},2$ $160$ $6$ $14$ $( 1,13, 4, 2,14, 3)( 5,16,18)( 6,15,17)( 7, 8)( 9,19,11)(10,20,12)$
8A $8,4^{2},2^{2}$ $240$ $8$ $15$ $( 1,16, 9, 6, 2,15,10, 5)( 3,13,19,12)( 4,14,20,11)( 7,18)( 8,17)$
12A $12,3^{2},1^{2}$ $160$ $12$ $15$ $( 1,13,16)( 2,14,15)( 3,20, 6, 8,17,10, 4,19, 5, 7,18, 9)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 3A 4A 4B 4C 4D 4E 5A 6A 6B 6C 8A 12A
Size 1 5 10 20 60 60 80 20 60 60 120 240 384 80 160 160 240 160
2 P 1A 1A 1A 1A 1A 1A 3A 2B 2B 2A 2B 2D 5A 3A 3A 3A 4C 6A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 4C 4D 4E 5A 2B 2C 2A 8A 4A
5 P 1A 2A 2B 2C 2D 2E 3A 4A 4B 4C 4D 4E 1A 6A 6B 6C 8A 12A
Type
1920.240996.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1920.240996.1b R 1 1 1 −1 1 −1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1
1920.240996.4a R 4 4 4 2 0 2 1 2 2 0 0 0 −1 1 −1 1 0 −1
1920.240996.4b R 4 4 4 −2 0 −2 1 −2 −2 0 0 0 −1 1 1 1 0 1
1920.240996.5a R 5 5 5 −1 1 −1 −1 −1 −1 1 1 1 0 −1 −1 −1 1 −1
1920.240996.5b R 5 5 5 1 1 1 −1 1 1 1 1 −1 0 −1 1 −1 −1 1
1920.240996.5c R 5 −3 1 3 1 −1 2 −3 1 1 −1 1 0 −2 0 0 −1 0
1920.240996.5d R 5 −3 1 −3 1 1 2 3 −1 1 −1 −1 0 −2 0 0 1 0
1920.240996.6a R 6 6 6 0 −2 0 0 0 0 −2 −2 0 1 0 0 0 0 0
1920.240996.10a R 10 2 −2 4 2 0 1 2 −2 −2 0 0 0 1 1 −1 0 −1
1920.240996.10b R 10 2 −2 −2 −2 2 1 −4 0 2 0 0 0 1 1 −1 0 −1
1920.240996.10c R 10 2 −2 −4 2 0 1 −2 2 −2 0 0 0 1 −1 −1 0 1
1920.240996.10d R 10 2 −2 2 −2 −2 1 4 0 2 0 0 0 1 −1 −1 0 1
1920.240996.10e R 10 −6 2 0 2 0 −2 0 0 2 −2 0 0 2 0 0 0 0
1920.240996.15a R 15 −9 3 −3 −1 1 0 3 −1 −1 1 1 0 0 0 0 −1 0
1920.240996.15b R 15 −9 3 3 −1 −1 0 −3 1 −1 1 −1 0 0 0 0 1 0
1920.240996.20a R 20 4 −4 2 0 2 −1 −2 −2 0 0 0 0 −1 −1 1 0 1
1920.240996.20b R 20 4 −4 −2 0 −2 −1 2 2 0 0 0 0 −1 1 1 0 −1

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed