↑

Properties

Label 28T46
Degree $28$
Order $336$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PGL(2,7)$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(28, 46);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $46$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PGL(2,7)$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,27,23,7,14,9,21,18)(2,20,15,12,16,10,6,28)(3,5,25,22,17,13,8,26)(4,11,24,19), (1,2,24,21,19,28,14,13)(3,25,16,12)(4,9,6,22,26,20,18,11)(5,15,8,7,23,27,17,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

8T43, 14T16, 16T713, 21T20, 24T707, 28T42, 42T81, 42T82, 42T83

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Cycle 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, 1, 1 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1 $ $56$ $3$ $( 2, 3, 5)( 4, 6, 7)( 8,12,13)( 9,10,11)(14,24,19)(15,26,20)(16,21,17) (18,27,22)(23,28,25)$
$ 6, 6, 6, 6, 3, 1 $ $56$ $6$ $( 2, 4, 5, 7, 3, 6)( 8, 9,13,11,12,10)(14,20,19,26,24,15)(16,25,17,28,21,23) (18,27,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $28$ $2$ $( 2, 7)( 3, 4)( 5, 6)( 8,11)( 9,12)(10,13)(14,26)(15,19)(16,28)(17,23)(20,24) (21,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $21$ $2$ $( 2, 8)( 3,13)( 4,10)( 5,12)( 6, 9)( 7,11)(14,19)(15,26)(16,23)(17,28)(21,25) (22,27)$
$ 7, 7, 7, 7 $ $48$ $7$ $( 1, 2, 5, 4, 3, 6, 7)( 8,22,11,15,28,16,19)( 9,25,14,18,26,21,12) (10,24,20,13,23,27,17)$
$ 4, 4, 4, 4, 4, 4, 2, 2 $ $42$ $4$ $( 1, 2,23,10)( 3, 9)( 4,25,19, 8)( 5,22,26,13)( 6,15,28,12)( 7,24,18,11) (14,20,21,16)(17,27)$
$ 8, 8, 8, 4 $ $42$ $8$ $( 1, 2,24,21,19,28,14,13)( 3,25,16,12)( 4, 9, 6,22,26,20,18,11) ( 5,15, 8, 7,23,27,17,10)$
$ 8, 8, 8, 4 $ $42$ $8$ $( 1, 2,25,27,26,18,17,12)( 3,15,20, 8)( 4,22,11, 6,23,14,21,10) ( 5,24,28,16,19,13, 7, 9)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $336=2^{4} \cdot 3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  336.208
magma: IdentifyGroup(G);
 
Character table:    
     2  4  1  1  2  4  .  3  3  3
     3  1  1  1  1  .  .  .  .  .
     7  1  .  .  .  .  1  .  .  .

       1a 3a 6a 2a 2b 7a 4a 8a 8b
    2P 1a 3a 3a 1a 1a 7a 2b 4a 4a
    3P 1a 1a 2a 2a 2b 7a 4a 8b 8a
    5P 1a 3a 6a 2a 2b 7a 4a 8b 8a
    7P 1a 3a 6a 2a 2b 1a 4a 8a 8b

X.1     1  1  1  1  1  1  1  1  1
X.2     1  1 -1 -1  1  1  1 -1 -1
X.3     6  .  .  . -2 -1  2  .  .
X.4     6  .  .  .  2 -1  .  A -A
X.5     6  .  .  .  2 -1  . -A  A
X.6     7  1 -1 -1 -1  . -1  1  1
X.7     7  1  1  1 -1  . -1 -1 -1
X.8     8 -1 -1  2  .  1  .  .  .
X.9     8 -1  1 -2  .  1  .  .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2

magma: CharacterTable(G);