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Properties

Label 42T37
Degree $42$
Order $168$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PSL(2,7)$

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Show commands: Magma

magma: G := TransitiveGroup(42, 37);
 

Group action invariants

Degree $n$:  $42$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $37$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,7)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (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), (1,28,33)(2,27,34)(3,30,35)(4,29,36)(5,25,31)(6,26,32)(7,10,11)(8,9,12)(13,41,21)(14,42,22)(15,40,19)(16,39,20)(17,38,24)(18,37,23)
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: None

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

Low degree siblings

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

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

Malle's constant $a(G)$:     $1/20$

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);