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Properties

Label 20T472
20T472 1 9 1->9 19 1->19 2 12 2->12 17 2->17 3 10 3->10 20 3->20 4 11 4->11 18 4->18 5 5->12 16 5->16 6 6->10 13 6->13 7 7->11 15 7->15 8 8->9 14 8->14 9->1 9->17 10->3 10->20 11->2 11->18 12->4 12->19 13->1 13->15 14->4 14->13 15->2 15->16 16->3 16->14 17->8 17->8 18->5 18->6 19->7 19->7 20->5 20->6
Degree $20$
Order $15360$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^4.F_{16}:C_4$

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Copy content magma:G := TransitiveGroup(20, 472);
 

Group invariants

Abstract group:  $C_2^4.F_{16}:C_4$
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 $t$:  $472$
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:  $(1,9,17,8,14,4,11,18,5,16,3,10,20,6,13)(2,12,19,7,15)$, $(1,19,7,11,2,17,8,9)(3,20,5,12,4,18,6,10)(13,15,16,14)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$
$6$:  $S_3$
$12$:  $C_3 : C_4$
$20$:  $F_5$
$60$:  $C_{15} : C_4$
$960$:  16T1079 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 5: $F_5$

Degree 10: None

Low degree siblings

30T886, 30T900, 40T10658, 40T10665

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 $2^{8},1^{4}$ $15$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B $2^{8},1^{4}$ $15$ $2$ $8$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
2C $2^{8},1^{4}$ $15$ $2$ $8$ $( 1, 4)( 2, 3)( 9,12)(10,11)(13,14)(15,16)(17,18)(19,20)$
2D $2^{4},1^{12}$ $30$ $2$ $4$ $(1,3)(2,4)(5,7)(6,8)$
2E $2^{6},1^{8}$ $60$ $2$ $6$ $( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)$
2F $2^{8},1^{4}$ $60$ $2$ $8$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,19)(18,20)$
2G $2^{10}$ $60$ $2$ $10$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,15)(14,16)(17,20)(18,19)$
2H $2^{8},1^{4}$ $80$ $2$ $8$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,15)( 6,16)( 7,13)( 8,14)$
3A $3^{5},1^{5}$ $512$ $3$ $10$ $( 1, 3, 2)( 6, 7, 8)( 9,10,12)(13,15,14)(17,18,20)$
4A $4^{4},1^{4}$ $240$ $4$ $12$ $( 5,19, 6,20)( 7,17, 8,18)( 9,15,10,16)(11,13,12,14)$
4B $4^{4},2^{2}$ $240$ $4$ $14$ $( 1,15, 4,14)( 2,16, 3,13)( 5, 9, 7,11)( 6,10, 8,12)(17,18)(19,20)$
4C $4^{4},2^{2}$ $240$ $4$ $14$ $( 1,10, 4,11)( 2, 9, 3,12)( 5, 7)( 6, 8)(13,20,14,19)(15,18,16,17)$
4D $4^{2},2^{6}$ $480$ $4$ $12$ $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,17)(10,18)(11,19)(12,20)(13,15)(14,16)$
4E1 $4^{4},2,1^{2}$ $960$ $4$ $13$ $( 1, 7,19,13)( 2, 8,20,14)( 3, 6,17,16)( 4, 5,18,15)(11,12)$
4E-1 $4^{4},2,1^{2}$ $960$ $4$ $13$ $( 1,13,19, 7)( 2,14,20, 8)( 3,16,17, 6)( 4,15,18, 5)(11,12)$
5A $5^{4}$ $1024$ $5$ $16$ $( 1,10,18, 8,13)( 2, 9,17, 7,14)( 3,12,20, 6,15)( 4,11,19, 5,16)$
6A $6^{2},3,2^{2},1$ $2560$ $6$ $14$ $( 1, 4, 2)( 5,18, 7,17, 6,19)( 8,20)( 9,15)(10,14,11,16,12,13)$
8A1 $8^{2},2,1^{2}$ $960$ $8$ $15$ $( 1, 2)( 5,12,19,14, 6,11,20,13)( 7, 9,17,15, 8,10,18,16)$
8A-1 $8^{2},2,1^{2}$ $960$ $8$ $15$ $( 1, 2)( 5,13,20,11, 6,14,19,12)( 7,16,18,10, 8,15,17, 9)$
8B1 $8^{2},4$ $960$ $8$ $17$ $( 1, 8,15,12, 4, 6,14,10)( 2, 7,16,11, 3, 5,13, 9)(17,19,18,20)$
8B-1 $8^{2},4$ $960$ $8$ $17$ $( 1,10,14, 6, 4,12,15, 8)( 2, 9,13, 5, 3,11,16, 7)(17,20,18,19)$
8C1 $8^{2},4$ $960$ $8$ $17$ $( 1,17,10,15, 4,18,11,16)( 2,20, 9,14, 3,19,12,13)( 5, 6, 7, 8)$
8C-1 $8^{2},4$ $960$ $8$ $17$ $( 1,16,11,18, 4,15,10,17)( 2,13,12,19, 3,14, 9,20)( 5, 8, 7, 6)$
15A1 $15,5$ $1024$ $15$ $18$ $( 1,17,15,10, 7, 3,18,14,12, 8, 2,20,13, 9, 6)( 4,19,16,11, 5)$
15A-1 $15,5$ $1024$ $15$ $18$ $( 1, 6, 9,13,20, 2, 8,12,14,18, 3, 7,10,15,17)( 4, 5,11,16,19)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 2H 3A 4A 4B 4C 4D 4E1 4E-1 5A 6A 8A1 8A-1 8B1 8B-1 8C1 8C-1 15A1 15A-1
Size 1 15 15 15 30 60 60 60 80 512 240 240 240 480 960 960 1024 2560 960 960 960 960 960 960 1024 1024
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 3A 2A 2B 2C 2D 2H 2H 5A 3A 4A 4A 4B 4B 4C 4C 15A1 15A-1
3 P 1A 2A 2B 2C 2D 2E 2F 2G 2H 1A 4A 4B 4C 4D 4E-1 4E1 5A 2H 8A-1 8A1 8B-1 8B1 8C-1 8C1 5A 5A
5 P 1A 2A 2B 2C 2D 2E 2F 2G 2H 3A 4A 4B 4C 4D 4E1 4E-1 1A 6A 8A1 8A-1 8B1 8B-1 8C1 8C-1 3A 3A
Type
15360.k.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.k.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.k.1c1 C 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −i i 1 −1 −i −i i i i −i 1 1
15360.k.1c2 C 1 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 i −i 1 −1 i i −i −i −i i 1 1
15360.k.2a R 2 2 2 2 2 2 2 2 2 −1 2 2 2 2 0 0 2 −1 0 0 0 0 0 0 −1 −1
15360.k.2b S 2 2 2 2 2 2 2 2 −2 −1 −2 −2 −2 −2 0 0 2 1 0 0 0 0 0 0 −1 −1
15360.k.4a R 4 4 4 4 4 4 4 4 0 4 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 −1
15360.k.4b1 C 4 4 4 4 4 4 4 4 0 −2 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157
15360.k.4b2 C 4 4 4 4 4 4 4 4 0 −2 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1+2ζ15+ζ152−ζ153+2ζ154−ζ155+ζ157 2−2ζ15−ζ152+ζ153−2ζ154+ζ155−ζ157
15360.k.15a R 15 −1 15 −1 −1 −1 −1 −1 3 0 −1 −1 3 −1 1 1 0 0 −1 −1 1 −1 −1 1 0 0
15360.k.15b R 15 15 −1 −1 −1 −1 −1 −1 3 0 −1 3 −1 −1 1 1 0 0 1 −1 −1 −1 1 −1 0 0
15360.k.15c R 15 −1 −1 −1 7 3 −1 −5 3 0 3 −1 −1 −1 1 1 0 0 −1 1 −1 1 −1 −1 0 0
15360.k.15d R 15 −1 15 −1 −1 −1 −1 −1 3 0 −1 −1 3 −1 −1 −1 0 0 1 1 −1 1 1 −1 0 0
15360.k.15e R 15 15 −1 −1 −1 −1 −1 −1 3 0 −1 3 −1 −1 −1 −1 0 0 −1 1 1 1 −1 1 0 0
15360.k.15f R 15 −1 −1 −1 7 3 −1 −5 3 0 3 −1 −1 −1 −1 −1 0 0 1 −1 1 −1 1 1 0 0
15360.k.15g1 C 15 −1 −1 −1 7 3 −1 −5 −3 0 −3 1 1 1 −i i 0 0 i −i −i i −i i 0 0
15360.k.15g2 C 15 −1 −1 −1 7 3 −1 −5 −3 0 −3 1 1 1 i −i 0 0 −i i i −i i −i 0 0
15360.k.15h1 C 15 −1 15 −1 −1 −1 −1 −1 −3 0 1 1 −3 1 −i i 0 0 i i i −i −i −i 0 0
15360.k.15h2 C 15 −1 15 −1 −1 −1 −1 −1 −3 0 1 1 −3 1 i −i 0 0 −i −i −i i i i 0 0
15360.k.15i1 C 15 15 −1 −1 −1 −1 −1 −1 −3 0 1 −3 1 1 −i i 0 0 −i i −i −i i i 0 0
15360.k.15i2 C 15 15 −1 −1 −1 −1 −1 −1 −3 0 1 −3 1 1 i −i 0 0 i −i i i −i −i 0 0
15360.k.30a R 30 −2 −2 14 6 −6 −2 2 6 0 −2 −2 −2 2 0 0 0 0 0 0 0 0 0 0 0 0
15360.k.30b R 30 −2 −2 14 6 −6 −2 2 −6 0 2 2 2 −2 0 0 0 0 0 0 0 0 0 0 0 0
15360.k.60a R 60 −4 −4 −20 4 0 −4 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15360.k.60b R 60 −4 −4 −4 −4 −4 12 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
15360.k.60c R 60 −4 −4 12 −12 8 −4 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