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Properties

Label 24T284
Degree $24$
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(24, 284);
 

Group action invariants

Degree $n$:  $24$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $284$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,7)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $3$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,8,24,21,17,10,13)(2,9,22,19,18,11,14)(3,7,23,20,16,12,15), (1,7,12,13,22,5,18)(2,8,10,14,23,6,16)(3,9,11,15,24,4,17)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 8: $\PSL(2,7)$

Degree 12: None

Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 28T32, 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 TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 7, 7, 7, 1, 1, 1 $ $24$ $7$ $( 4, 8,16,11,22,20,13)( 5, 9,17,12,23,21,14)( 6, 7,18,10,24,19,15)$
$ 7, 7, 7, 1, 1, 1 $ $24$ $7$ $( 4,11,13,16,20, 8,22)( 5,12,14,17,21, 9,23)( 6,10,15,18,19, 7,24)$
$ 3, 3, 3, 3, 3, 3, 3, 3 $ $56$ $3$ $( 1, 2, 3)( 4, 6, 5)( 7,17,22)( 8,18,23)( 9,16,24)(10,14,20)(11,15,21) (12,13,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $21$ $2$ $( 1, 4)( 2, 5)( 3, 6)( 7,15)( 8,13)( 9,14)(10,16)(11,17)(12,18)(19,23)(20,24) (21,22)$
$ 4, 4, 4, 4, 4, 4 $ $42$ $4$ $( 1, 4,17,22)( 2, 5,18,23)( 3, 6,16,24)( 7,20,13,10)( 8,21,14,11)( 9,19,15,12)$

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);