↑

Properties

Label 20T471
20T471 1 2 1->2 5 1->5 3 2->3 7 2->7 3->1 8 3->8 4 6 4->6 13 5->13 18 5->18 15 6->15 19 6->19 16 7->16 17 7->17 14 8->14 20 8->20 9 9->3 9->16 10 10->1 10->13 11 11->2 11->15 12 12->4 12->14 13->10 14->11 14->12 15->9 15->11 16->9 16->12 17->20 18->6 19->8 19->17 20->5 20->19
Degree $20$
Order $15360$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^8.(C_3\times F_5)$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(20, 471);
 

Group invariants

Abstract group:  $C_2^8.(C_3\times F_5)$
Copy content magma:IdentifyGroup(G);
 
Order:  $15360=2^{10} \cdot 3 \cdot 5$
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 $t$:  $471$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,3)(5,18,6,19,8,20)(7,17)(9,16,12,14,11,15)(10,13)$, $(1,5,13,10)(2,7,16,9,3,8,14,12,4,6,15,11)(17,20,19)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$4$:  $C_4$
$6$:  $C_6$
$12$:  $C_{12}$
$20$:  $F_5$
$60$:  $F_5\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $F_5$

Degree 10: None

Low degree siblings

30T897, 30T901, 32T723616, 40T10657, 40T10666, 40T10667

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 $2^{8},1^{4}$ $15$ $2$ $8$ $( 1, 2)( 3, 4)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{4},1^{12}$ $30$ $2$ $4$ $( 5, 7)( 6, 8)( 9,11)(10,12)$
2C $2^{8},1^{4}$ $30$ $2$ $8$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)(13,16)(14,15)(17,19)(18,20)$
2D $2^{6},1^{8}$ $60$ $2$ $6$ $( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)$
2E $2^{8},1^{4}$ $60$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,19)(18,20)$
2F $2^{10}$ $60$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)$
2G $2^{8},1^{4}$ $80$ $2$ $8$ $( 5,20)( 6,19)( 7,18)( 8,17)( 9,14)(10,13)(11,16)(12,15)$
3A1 $3^{5},1^{5}$ $256$ $3$ $10$ $( 1, 4, 2)( 5, 6, 7)(10,12,11)(13,15,16)(18,20,19)$
3A-1 $3^{5},1^{5}$ $256$ $3$ $10$ $( 1, 2, 4)( 5, 7, 6)(10,11,12)(13,16,15)(18,19,20)$
4A $4^{4},1^{4}$ $240$ $4$ $12$ $( 1,10, 2, 9)( 3,12, 4,11)(13,20,14,19)(15,18,16,17)$
4B1 $4^{4},1^{4}$ $320$ $4$ $12$ $( 5,15,20,12)( 6,16,19,11)( 7,13,18,10)( 8,14,17, 9)$
4B-1 $4^{4},1^{4}$ $320$ $4$ $12$ $( 5,12,20,15)( 6,11,19,16)( 7,10,18,13)( 8, 9,17,14)$
4C $4^{2},2^{6}$ $480$ $4$ $12$ $( 1,16)( 2,15)( 3,14)( 4,13)( 5,11, 7, 9)( 6,12, 8,10)(17,19)(18,20)$
4D $4^{4},2^{2}$ $480$ $4$ $14$ $( 1,20, 3,18)( 2,19, 4,17)( 5,14, 8,15)( 6,13, 7,16)( 9,10)(11,12)$
5A $5^{4}$ $1024$ $5$ $16$ $( 1,10,18, 7,14)( 2, 9,17, 8,13)( 3,12,20, 5,16)( 4,11,19, 6,15)$
6A1 $6^{2},3,2^{2},1$ $1280$ $6$ $14$ $( 1, 2, 4)( 5,18, 6,20, 7,19)( 8,17)( 9,14)(10,16,12,13,11,15)$
6A-1 $6^{2},3,2^{2},1$ $1280$ $6$ $14$ $( 1, 4, 2)( 5,19, 7,20, 6,18)( 8,17)( 9,14)(10,15,11,13,12,16)$
8A1 $8^{2},2^{2}$ $960$ $8$ $16$ $( 1,13,10,20, 2,14, 9,19)( 3,15,12,18, 4,16,11,17)( 5, 6)( 7, 8)$
8A-1 $8^{2},2^{2}$ $960$ $8$ $16$ $( 1,19, 9,14, 2,20,10,13)( 3,17,11,16, 4,18,12,15)( 5, 6)( 7, 8)$
12A1 $12,4,3,1$ $1280$ $12$ $16$ $( 1, 4, 2)( 5,11,18,15, 6,10,20,16, 7,12,19,13)( 8, 9,17,14)$
12A-1 $12,4,3,1$ $1280$ $12$ $16$ $( 1, 2, 4)( 5,13,19,12, 7,16,20,10, 6,15,18,11)( 8,14,17, 9)$
12A5 $12,4,3,1$ $1280$ $12$ $16$ $( 1, 2, 4)( 5,10,19,15, 7,11,20,13, 6,12,18,16)( 8, 9,17,14)$
12A-5 $12,4,3,1$ $1280$ $12$ $16$ $( 1, 4, 2)( 5,16,18,12, 6,13,20,11, 7,15,19,10)( 8,14,17, 9)$
15A1 $15,5$ $1024$ $15$ $18$ $( 1,17,16,10, 8, 3,18,13,12, 7, 2,20,14, 9, 5)( 4,19,15,11, 6)$
15A-1 $15,5$ $1024$ $15$ $18$ $( 1, 5, 9,14,20, 2, 7,12,13,18, 3, 8,10,16,17)( 4, 6,11,15,19)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 3A1 3A-1 4A 4B1 4B-1 4C 4D 5A 6A1 6A-1 8A1 8A-1 12A1 12A-1 12A5 12A-5 15A1 15A-1
Size 1 15 30 30 60 60 60 80 256 256 240 320 320 480 480 1024 1280 1280 960 960 1280 1280 1280 1280 1024 1024
2 P 1A 1A 1A 1A 1A 1A 1A 1A 3A-1 3A1 2A 2G 2G 2B 2C 5A 3A1 3A-1 4A 4A 6A1 6A-1 6A-1 6A1 15A-1 15A1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 1A 1A 4A 4B-1 4B1 4C 4D 5A 2G 2G 8A-1 8A1 4B1 4B-1 4B1 4B-1 5A 5A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 3A-1 3A1 4A 4B1 4B-1 4C 4D 1A 6A-1 6A1 8A1 8A-1 12A5 12A-5 12A1 12A-1 3A-1 3A1
Type
15360.j.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 1
15360.j.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 1
15360.j.1c1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 1 1 1 1 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3 ζ3−1 ζ3−1 ζ3−1 ζ3
15360.j.1c2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 1 1 1 1 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3−1 ζ3 ζ3 ζ3 ζ3−1
15360.j.1d1 C 1 1 1 1 1 1 1 −1 1 1 −1 −i i −1 −1 1 −1 −1 −i i −i i i −i 1 1
15360.j.1d2 C 1 1 1 1 1 1 1 −1 1 1 −1 i −i −1 −1 1 −1 −1 i −i i −i −i i 1 1
15360.j.1e1 C 1 1 1 1 1 1 1 1 ζ3−1 ζ3 1 −1 −1 1 1 1 ζ3 ζ3−1 −1 −1 −ζ3 −ζ3 −ζ3−1 −ζ3−1 ζ3−1 ζ3
15360.j.1e2 C 1 1 1 1 1 1 1 1 ζ3 ζ3−1 1 −1 −1 1 1 1 ζ3−1 ζ3 −1 −1 −ζ3−1 −ζ3−1 −ζ3 −ζ3 ζ3 ζ3−1
15360.j.1f1 C 1 1 1 1 1 1 1 −1 −ζ122 ζ124 −1 −ζ123 ζ123 −1 −1 1 −ζ124 ζ122 −ζ123 ζ123 ζ12 −ζ12 −ζ125 ζ125 −ζ122 ζ124
15360.j.1f2 C 1 1 1 1 1 1 1 −1 ζ124 −ζ122 −1 ζ123 −ζ123 −1 −1 1 ζ122 −ζ124 ζ123 −ζ123 −ζ125 ζ125 ζ12 −ζ12 ζ124 −ζ122
15360.j.1f3 C 1 1 1 1 1 1 1 −1 −ζ122 ζ124 −1 ζ123 −ζ123 −1 −1 1 −ζ124 ζ122 ζ123 −ζ123 −ζ12 ζ12 ζ125 −ζ125 −ζ122 ζ124
15360.j.1f4 C 1 1 1 1 1 1 1 −1 ζ124 −ζ122 −1 −ζ123 ζ123 −1 −1 1 ζ122 −ζ124 −ζ123 ζ123 ζ125 −ζ125 −ζ12 ζ12 ζ124 −ζ122
15360.j.4a R 4 4 4 4 4 4 4 0 4 4 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 −1
15360.j.4b1 C 4 4 4 4 4 4 4 0 4ζ3−1 4ζ3 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −ζ3−1 −ζ3
15360.j.4b2 C 4 4 4 4 4 4 4 0 4ζ3 4ζ3−1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −ζ3 −ζ3−1
15360.j.15a R 15 −1 7 −1 3 −1 −5 3 0 0 3 3 3 −1 −1 0 0 0 −1 −1 0 0 0 0 0 0
15360.j.15b R 15 −1 7 −1 3 −1 −5 3 0 0 3 −3 −3 −1 −1 0 0 0 1 1 0 0 0 0 0 0
15360.j.15c1 C 15 −1 7 −1 3 −1 −5 −3 0 0 −3 −3i 3i 1 1 0 0 0 i −i 0 0 0 0 0 0
15360.j.15c2 C 15 −1 7 −1 3 −1 −5 −3 0 0 −3 3i −3i 1 1 0 0 0 −i i 0 0 0 0 0 0
15360.j.30a R 30 −2 −2 14 −2 −2 −2 6 0 0 −2 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0
15360.j.30b R 30 14 6 −2 −6 −2 2 6 0 0 −2 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0
15360.j.30c R 30 −2 −2 14 −2 −2 −2 −6 0 0 2 0 0 2 −2 0 0 0 0 0 0 0 0 0 0 0
15360.j.30d R 30 14 6 −2 −6 −2 2 −6 0 0 2 0 0 −2 2 0 0 0 0 0 0 0 0 0 0 0
15360.j.60a R 60 −20 4 −4 0 −4 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15360.j.60b R 60 −4 −4 −4 −4 12 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15360.j.60c R 60 12 −12 −4 8 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed