↑

Properties

Label 42T38
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PSL(2,7)$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(42, 38);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $38$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,7)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $6$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,24,40,9,25,35,14)(2,23,39,10,26,36,13)(3,19,42,11,28,31,18)(4,20,41,12,27,32,17)(5,21,38,8,30,34,15)(6,22,37,7,29,33,16), (1,32,39,24)(2,31,40,23)(3,35,37,21)(4,36,38,22)(5,33,42,19)(6,34,41,20)(7,25,11,29)(8,26,12,30)(9,28)(10,27)(13,17,14,18)(15,16)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 6: None

Degree 7: $\GL(3,2)$ x 2

Degree 14: $\PSL(2,7)$

Degree 21: $\PSL(2,7)$

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{42}$ $1$ $1$ $0$ $()$
2A $2^{18},1^{6}$ $21$ $2$ $18$ $( 1, 8)( 2, 7)( 3,10)( 4, 9)( 5,12)( 6,11)(13,15)(14,16)(19,20)(23,24)(25,39)(26,40)(27,37)(28,38)(29,42)(30,41)(31,33)(32,34)$
3A $3^{14}$ $56$ $3$ $28$ $( 1,32,22)( 2,31,21)( 3,34,23)( 4,33,24)( 5,36,19)( 6,35,20)( 7,25,17)( 8,26,18)( 9,27,16)(10,28,15)(11,30,13)(12,29,14)(37,40,41)(38,39,42)$
4A $4^{9},2^{3}$ $42$ $4$ $30$ $( 1,42, 8,29)( 2,41, 7,30)( 3,39,10,25)( 4,40, 9,26)( 5,37,12,27)( 6,38,11,28)(13,34,15,32)(14,33,16,31)(17,36)(18,35)(19,24,20,23)(21,22)$
7A1 $7^{6}$ $24$ $7$ $36$ $( 1,22,31,30,40,10,14)( 2,21,32,29,39, 9,13)( 3,19,36,25,41,11,16)( 4,20,35,26,42,12,15)( 5,23,33,28,37, 7,17)( 6,24,34,27,38, 8,18)$
7A-1 $7^{6}$ $24$ $7$ $36$ $( 1,10,30,22,14,40,31)( 2, 9,29,21,13,39,32)( 3,11,25,19,16,41,36)( 4,12,26,20,15,42,35)( 5, 7,28,23,17,37,33)( 6, 8,27,24,18,38,34)$

Malle's constant $a(G)$:     $1/18$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  168.42
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 4A 7A1 7A-1
Size 1 21 56 42 24 24
2 P 1A 1A 3A 2A 7A1 7A-1
3 P 1A 2A 1A 4A 7A-1 7A1
7 P 1A 2A 3A 4A 1A 1A
Type
168.42.1a R 1 1 1 1 1 1
168.42.3a1 C 3 −1 0 1 −ζ7−3−1−ζ7−ζ72 ζ7−3+ζ7+ζ72
168.42.3a2 C 3 −1 0 1 ζ7−3+ζ7+ζ72 −ζ7−3−1−ζ7−ζ72
168.42.6a R 6 2 0 0 −1 −1
168.42.7a R 7 −1 1 −1 0 0
168.42.8a R 8 0 −1 0 1 1

magma: CharacterTable(G);