Properties

Label 21T4
Degree $21$
Order $42$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_7$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(21, 4);
 

Group invariants

Abstract group:  $F_7$
magma: IdentifyGroup(G);
 
Order:  $42=2 \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $21$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9,5,18,10,14)(2,8,6,16,12,13)(3,7,4,17,11,15)(19,21,20)$, $(1,4,8,12,15,18,19)(2,5,7,11,14,16,20)(3,6,9,10,13,17,21)$
magma: Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 7: $F_7$

Low degree siblings

7T4, 14T4, 42T4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{21}$ $1$ $1$ $0$ $()$
2A $2^{9},1^{3}$ $7$ $2$ $9$ $( 4,19)( 5,20)( 6,21)( 7,16)( 8,18)( 9,17)(10,13)(11,14)(12,15)$
3A1 $3^{7}$ $7$ $3$ $14$ $( 1, 3, 2)( 4,13, 7)( 5,15, 9)( 6,14, 8)(10,16,19)(11,18,21)(12,17,20)$
3A-1 $3^{7}$ $7$ $3$ $14$ $( 1, 2, 3)( 4, 7,13)( 5, 9,15)( 6, 8,14)(10,19,16)(11,21,18)(12,20,17)$
6A1 $6^{3},3$ $7$ $6$ $17$ $( 1, 2, 3)( 4,16,13,19, 7,10)( 5,17,15,20, 9,12)( 6,18,14,21, 8,11)$
6A-1 $6^{3},3$ $7$ $6$ $17$ $( 1, 3, 2)( 4,10, 7,19,13,16)( 5,12, 9,20,15,17)( 6,11, 8,21,14,18)$
7A $7^{3}$ $6$ $7$ $18$ $( 1,15, 4,18, 8,19,12)( 2,14, 5,16, 7,20,11)( 3,13, 6,17, 9,21,10)$

Malle's constant $a(G)$:     $1/9$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 3A1 3A-1 6A1 6A-1 7A
Size 1 7 7 7 7 7 6
2 P 1A 1A 3A-1 3A1 3A1 3A-1 7A
3 P 1A 2A 1A 1A 2A 2A 7A
7 P 1A 2A 3A1 3A-1 6A1 6A-1 1A
Type
42.1.1a R 1 1 1 1 1 1 1
42.1.1b R 1 1 1 1 1 1 1
42.1.1c1 C 1 1 ζ31 ζ3 ζ3 ζ31 1
42.1.1c2 C 1 1 ζ3 ζ31 ζ31 ζ3 1
42.1.1d1 C 1 1 ζ31 ζ3 ζ3 ζ31 1
42.1.1d2 C 1 1 ζ3 ζ31 ζ31 ζ3 1
42.1.6a R 6 0 0 0 0 0 1

magma: CharacterTable(G);
 

Regular extensions

Data not computed