# Properties

 Label 9T27 Degree $9$ 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(9, 27);

## Group action invariants

 Degree $n$: $9$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $27$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSL(2,8)$ CHM label: $L(9)=PSL(2,8)$ Parity: $1$ magma: IsEven(G); Primitive: yes magma: IsPrimitive(G); Nilpotency class: $-1$ (not nilpotent) magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $1$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (2,5)(3,6)(4,7)(8,9), (1,9)(2,3)(4,5)(6,7), (1,2,4,3,6,7,5) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 3: None

## Low degree siblings

28T70, 36T712

Siblings are shown with degree $\leq 47$

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

## Conjugacy classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $7, 1, 1$ $72$ $7$ $(3,4,6,5,7,9,8)$ $7, 1, 1$ $72$ $7$ $(3,5,8,6,9,4,7)$ $7, 1, 1$ $72$ $7$ $(3,6,7,8,4,5,9)$ $2, 2, 2, 2, 1$ $63$ $2$ $(2,3)(4,9)(5,6)(7,8)$ $3, 3, 3$ $56$ $3$ $(1,2,3)(4,6,7)(5,9,8)$ $9$ $56$ $9$ $(1,2,3,4,8,9,7,6,5)$ $9$ $56$ $9$ $(1,2,3,6,9,5,4,7,8)$ $9$ $56$ $9$ $(1,2,3,7,5,8,6,4,9)$

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); Label: 504.156 magma: IdentifyGroup(G);
 Character table:  2 3 . . . 3 . . . . 3 2 . . . . 2 2 2 2 7 1 1 1 1 . . . . . 1a 7a 7b 7c 2a 3a 9a 9b 9c 2P 1a 7c 7a 7b 1a 3a 9b 9c 9a 3P 1a 7b 7c 7a 2a 1a 3a 3a 3a 5P 1a 7c 7a 7b 2a 3a 9c 9a 9b 7P 1a 1a 1a 1a 2a 3a 9b 9c 9a X.1 1 1 1 1 1 1 1 1 1 X.2 7 . . . -1 -2 1 1 1 X.3 7 . . . -1 1 D F E X.4 7 . . . -1 1 E D F X.5 7 . . . -1 1 F E D X.6 8 1 1 1 . -1 -1 -1 -1 X.7 9 A B C 1 . . . . X.8 9 B C A 1 . . . . X.9 9 C A B 1 . . . . A = E(7)^3+E(7)^4 B = E(7)^2+E(7)^5 C = E(7)+E(7)^6 D = -E(9)^4-E(9)^5 E = -E(9)^2-E(9)^7 F = E(9)^2+E(9)^4+E(9)^5+E(9)^7 

magma: CharacterTable(G);