Properties

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

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 $\leq 47$

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

 Label Cycle Type Size Order Representative 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 $7$ $−1$ $1$ $0$ $0$ $0$ $−ζ9−1−ζ9$ $−ζ9−2−ζ92$ $−ζ9−4−ζ94$ 504.156.7b2 R $7$ $−1$ $1$ $0$ $0$ $0$ $−ζ9−2−ζ92$ $−ζ9−4−ζ94$ $−ζ9−1−ζ9$ 504.156.7b3 R $7$ $−1$ $1$ $0$ $0$ $0$ $−ζ9−4−ζ94$ $−ζ9−1−ζ9$ $−ζ9−2−ζ92$ 504.156.8a R $8$ $0$ $−1$ $1$ $1$ $1$ $−1$ $−1$ $−1$ 504.156.9a1 R $9$ $1$ $0$ $ζ7−3+ζ73$ $ζ7−1+ζ7$ $ζ7−2+ζ72$ $0$ $0$ $0$ 504.156.9a2 R $9$ $1$ $0$ $ζ7−2+ζ72$ $ζ7−3+ζ73$ $ζ7−1+ζ7$ $0$ $0$ $0$ 504.156.9a3 R $9$ $1$ $0$ $ζ7−1+ζ7$ $ζ7−2+ζ72$ $ζ7−3+ζ73$ $0$ $0$ $0$

magma: CharacterTable(G);