Properties

Label 28T70
Degree $28$
Order $504$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $\PSL(2,8)$

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Show commands: Magma

magma: G := TransitiveGroup(28, 70);
 

Group action invariants

Degree $n$:  $28$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $70$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PSL(2,8)$
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:  (1,21,27,15,25,10,24,7,20)(2,19,8,26,17,16,6,5,3)(9,28,18,13,11,22,14,23,12), (1,28,21,4,25,18,10)(2,26,19,17,12,7,11)(3,24,20,16,13,14,9)(5,6,23,15,8,27,22)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 7: None

Degree 14: None

Low degree siblings

9T27, 36T712

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
1A $1^{28}$ $1$ $1$ $()$
2A $2^{12},1^{4}$ $63$ $2$ $( 1, 2)( 3, 7)( 4, 6)( 8,23)( 9,18)(10,20)(11,19)(12,21)(13,15)(17,24)(25,26)(27,28)$
3A $3^{9},1$ $56$ $3$ $( 1,21,17)( 2,26,27)( 3,15,25)( 4, 8,19)( 5,10,12)( 6, 9,18)( 7,11,22)(13,20,16)(14,23,24)$
7A1 $7^{4}$ $72$ $7$ $( 1,10,26,23,14,22,19)( 2, 7,17,25,13, 9,21)( 3, 5, 4, 6,16,24,20)( 8,11,15,28,18,27,12)$
7A2 $7^{4}$ $72$ $7$ $( 1,26,14,19,10,23,22)( 2,17,13,21, 7,25, 9)( 3, 4,16,20, 5, 6,24)( 8,15,18,12,11,28,27)$
7A3 $7^{4}$ $72$ $7$ $( 1,23,19,26,22,10,14)( 2,25,21,17, 9, 7,13)( 3, 6,20, 4,24, 5,16)( 8,28,12,15,27,11,18)$
9A1 $9^{3},1$ $56$ $9$ $( 1,18, 3,17, 9,25,21, 6,15)( 2, 7,14,27,22,24,26,11,23)( 4,10,13,19, 5,16, 8,12,20)$
9A2 $9^{3},1$ $56$ $9$ $( 1, 3, 9,21,15,18,17,25, 6)( 2,14,22,26,23, 7,27,24,11)( 4,13, 5, 8,20,10,19,16,12)$
9A4 $9^{3},1$ $56$ $9$ $( 1,25,18,21, 3, 6,17,15, 9)( 2,24, 7,26,14,11,27,23,22)( 4,16,10, 8,13,12,19,20, 5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $504=2^{3} \cdot 3^{2} \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  504.156
magma: IdentifyGroup(G);
 
Character table:

1A 2A 3A 7A1 7A2 7A3 9A1 9A2 9A4
Size 1 63 56 72 72 72 56 56 56
2 P 1A 1A 3A 7A2 7A3 7A1 9A1 9A2 9A4
3 P 1A 2A 1A 7A3 7A1 7A2 3A 3A 3A
7 P 1A 2A 3A 1A 1A 1A 9A1 9A2 9A4
Type
504.156.1a R 1 1 1 1 1 1 1 1 1
504.156.7a R 7 1 2 0 0 0 1 1 1
504.156.7b1 R 7 1 1 0 0 0 ζ91ζ9 ζ92ζ92 ζ94ζ94
504.156.7b2 R 7 1 1 0 0 0 ζ92ζ92 ζ94ζ94 ζ91ζ9
504.156.7b3 R 7 1 1 0 0 0 ζ94ζ94 ζ91ζ9 ζ92ζ92
504.156.8a R 8 0 1 1 1 1 1 1 1
504.156.9a1 R 9 1 0 ζ73+ζ73 ζ71+ζ7 ζ72+ζ72 0 0 0
504.156.9a2 R 9 1 0 ζ72+ζ72 ζ73+ζ73 ζ71+ζ7 0 0 0
504.156.9a3 R 9 1 0 ζ71+ζ7 ζ72+ζ72 ζ73+ζ73 0 0 0

magma: CharacterTable(G);