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Properties

Label 20T37
Degree $20$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_5\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(20, 37);
 

Group action invariants

Degree $n$:  $20$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_5\times A_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,16,2,13,4,14)(3,15)(5,12,6,9,8,10)(7,11)(17,20,18), (1,8,9,16,17,4,5,12,13,20)(2,7,10,15,18,3,6,11,14,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$30$:  $D_5\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 5: $D_{5}$

Degree 10: None

Low degree siblings

30T20, 30T28, 40T65

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  120.39
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A1 3A-1 5A1 5A2 6A1 6A-1 10A1 10A3 15A1 15A-1 15A2 15A-2
Size 1 3 5 15 4 4 2 2 20 20 6 6 8 8 8 8
2 P 1A 1A 1A 1A 3A-1 3A1 5A2 5A1 3A1 3A-1 5A2 5A1 15A-2 15A2 15A-1 15A1
3 P 1A 2A 2B 2C 1A 1A 5A2 5A1 2B 2B 10A3 10A1 5A1 5A1 5A2 5A2
5 P 1A 2A 2B 2C 3A-1 3A1 1A 1A 6A-1 6A1 2A 2A 3A-1 3A1 3A1 3A-1
Type
120.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
120.39.1b R 1 1 −1 −1 1 1 1 1 −1 −1 1 1 1 1 1 1
120.39.1c1 C 1 1 1 1 ζ3−1 ζ3 1 1 ζ3 ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
120.39.1c2 C 1 1 1 1 ζ3 ζ3−1 1 1 ζ3−1 ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
120.39.1d1 C 1 1 −1 −1 ζ3−1 ζ3 1 1 −ζ3 −ζ3−1 1 1 ζ3 ζ3−1 ζ3−1 ζ3
120.39.1d2 C 1 1 −1 −1 ζ3 ζ3−1 1 1 −ζ3−1 −ζ3 1 1 ζ3−1 ζ3 ζ3 ζ3−1
120.39.2a1 R 2 2 0 0 2 2 ζ5−2+ζ52 ζ5−1+ζ5 0 0 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52
120.39.2a2 R 2 2 0 0 2 2 ζ5−1+ζ5 ζ5−2+ζ52 0 0 ζ5−1+ζ5 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−2+ζ52 ζ5−1+ζ5 ζ5−1+ζ5
120.39.2b1 C 2 2 0 0 2ζ15−5 2ζ155 ζ15−6+ζ156 ζ15−3+ζ153 0 0 ζ15−6+ζ156 ζ15−3+ζ153 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157
120.39.2b2 C 2 2 0 0 2ζ155 2ζ15−5 ζ15−6+ζ156 ζ15−3+ζ153 0 0 ζ15−6+ζ156 ζ15−3+ζ153 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154
120.39.2b3 C 2 2 0 0 2ζ15−5 2ζ155 ζ15−3+ζ153 ζ15−6+ζ156 0 0 ζ15−3+ζ153 ζ15−6+ζ156 1−ζ15−ζ152+ζ153−ζ154−ζ157 ζ15+ζ154 1−ζ15−ζ154+ζ155 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157
120.39.2b4 C 2 2 0 0 2ζ155 2ζ15−5 ζ15−3+ζ153 ζ15−6+ζ156 0 0 ζ15−3+ζ153 ζ15−6+ζ156 ζ15+ζ154 1−ζ15−ζ152+ζ153−ζ154−ζ157 −1+ζ15+ζ152−ζ153+ζ154−ζ155+ζ157 1−ζ15−ζ154+ζ155
120.39.3a R 3 −1 3 −1 0 0 3 3 0 0 −1 −1 0 0 0 0
120.39.3b R 3 −1 −3 1 0 0 3 3 0 0 −1 −1 0 0 0 0
120.39.6a1 R 6 −2 0 0 0 0 3ζ5−2+3ζ52 3ζ5−1+3ζ5 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 0 0 0 0
120.39.6a2 R 6 −2 0 0 0 0 3ζ5−1+3ζ5 3ζ5−2+3ζ52 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 0 0 0 0

magma: CharacterTable(G);