↑

Properties

Label 28T42
Degree $28$
Order $336$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PGL(2,7)$

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(28, 42);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $42$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PGL(2,7)$
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:  (1,14,21,9,15,6,4)(2,13,22,10,16,5,3)(7,11,26,17,20,28,23)(8,12,25,18,19,27,24), (1,18,13,28,22,11,5,26)(2,17,14,27,21,12,6,25)(3,19,4,20)(7,9,23,16,8,10,24,15)
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: $C_2$

Degree 4: None

Degree 7: None

Degree 14: 14T16

Low degree siblings

8T43, 14T16, 16T713, 21T20, 24T707, 28T46, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $21$ $2$ $( 3, 4)( 5,15)( 6,16)( 9,14)(10,13)(11,12)(17,25)(18,26)(19,24)(20,23)(21,22) (27,28)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $56$ $3$ $( 3, 5,10)( 4, 6, 9)( 7,28,11)( 8,27,12)(13,22,16)(14,21,15)(17,20,23) (18,19,24)$
$ 4, 4, 4, 4, 4, 4, 2, 2 $ $42$ $4$ $( 1, 2)( 3, 5,21,16)( 4, 6,22,15)( 7,28, 8,27)( 9,14,10,13)(11,12) (17,26,20,24)(18,25,19,23)$
$ 7, 7, 7, 7 $ $48$ $7$ $( 1, 3, 6,22,13,10,15)( 2, 4, 5,21,14, 9,16)( 7,23,18,26,12,19,28) ( 8,24,17,25,11,20,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $28$ $2$ $( 1, 7)( 2, 8)( 3,11)( 4,12)( 5,17)( 6,18)( 9,23)(10,24)(13,19)(14,20)(15,25) (16,26)(21,27)(22,28)$
$ 8, 8, 8, 4 $ $42$ $8$ $( 1, 7,22,26,16,19,10,11)( 2, 8,21,25,15,20, 9,12)( 3,17,14,27, 4,18,13,28) ( 5,24, 6,23)$
$ 6, 6, 6, 6, 2, 2 $ $56$ $6$ $( 1, 7,22,25,14,12)( 2, 8,21,26,13,11)( 3,18,16,23, 9,27)( 4,17,15,24,10,28) ( 5,20)( 6,19)$
$ 8, 8, 8, 4 $ $42$ $8$ $( 1, 7, 2, 8)( 3,27,22,12, 4,28,21,11)( 5,26,10,23,15,18,13,20) ( 6,25, 9,24,16,17,14,19)$

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  4  1  3  .  2  3  1  3
     3  1  .  1  .  .  1  .  1  .
     7  1  .  .  .  1  .  .  .  .

       1a 2a 3a 4a 7a 2b 8a 6a 8b
    2P 1a 1a 3a 2a 7a 1a 4a 3a 4a
    3P 1a 2a 1a 4a 7a 2b 8b 2b 8a
    5P 1a 2a 3a 4a 7a 2b 8b 6a 8a
    7P 1a 2a 3a 4a 1a 2b 8a 6a 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  .  2 -1  .  .  .  .
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);