↑

Properties

Label 28T32
Degree $28$
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(28, 32);
 

Group action invariants

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

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

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

Degree 14: None

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 42T37, 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 $1^{28}$ $1$ $1$ $0$ $()$
2A $2^{12},1^{4}$ $21$ $2$ $12$ $( 1,27)( 2,26)( 3,28)( 4,25)( 6, 7)( 9,12)(10,11)(13,15)(17,21)(18,24)(19,22)(20,23)$
3A $3^{9},1$ $56$ $3$ $18$ $( 1, 4, 3)( 5,20,25)( 6,17,27)( 7,18,26)( 8,19,28)( 9,24,13)(10,23,15)(11,22,16)(12,21,14)$
4A $4^{6},2^{2}$ $42$ $4$ $20$ $( 1,18,27,24)( 2,19,26,22)( 3,17,28,21)( 4,20,25,23)( 5,14)( 6,13, 7,15)( 8,16)( 9,10,12,11)$
7A1 $7^{4}$ $24$ $7$ $24$ $( 1,17, 6,24, 9,15,26)( 2,20, 8,21,12,16,25)( 3,19, 5,22,10,13,27)( 4,18, 7,23,11,14,28)$
7A-1 $7^{4}$ $24$ $7$ $24$ $( 1,15,24,17,26, 9, 6)( 2,16,21,20,25,12, 8)( 3,13,22,19,27,10, 5)( 4,14,23,18,28,11, 7)$

Malle's constant $a(G)$:     $1/12$

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);