# Properties

 Label 36T712 Degree $36$ Order $504$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,8)$

Show commands: Magma

magma: G := TransitiveGroup(36, 712);

## Group action invariants

 Degree $n$: $36$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $712$ 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: (1,2,6,17,19,8,4)(3,10,23,12,28,25,13)(5,15,9,22,20,14,18)(7,16,31,30,29,21,32)(11,26,34,35,27,33,24), (1,4)(2,8)(3,11)(5,14)(6,19)(7,21)(9,22)(10,24)(12,27)(13,26)(15,20)(16,29)(23,33)(25,34)(28,35)(30,31), (1,3)(2,7)(4,13)(5,12)(6,18)(8,9)(10,14)(11,26)(16,23)(17,31)(20,32)(21,22)(24,27)(25,29)(33,34)(35,36) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Degree 9: None

Degree 12: None

Degree 18: None

## Low degree siblings

9T27, 28T70

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 $1^{36}$ $1$ $1$ $()$ $2^{16},1^{4}$ $63$ $2$ $( 2, 4)( 3,26)( 5,18)( 6, 8)( 7,32)( 9,20)(10,11)(12,33)(13,34)(14,15)(16,21) (17,19)(23,24)(25,35)(27,28)(29,31)$ $7^{5},1$ $72$ $7$ $( 2,15,27,25,31,18,21)( 3,26,12,19,30,17,33)( 4,16, 5,29,35,28,14) ( 6,24,36,23, 8,20, 9)( 7,11,13,22,34,10,32)$ $7^{5},1$ $72$ $7$ $( 2,18,25,15,21,31,27)( 3,17,19,26,33,30,12)( 4,28,29,16,14,35, 5) ( 6,20,23,24, 9, 8,36)( 7,10,22,11,32,34,13)$ $7^{5},1$ $72$ $7$ $( 2,25,21,27,18,15,31)( 3,19,33,12,17,26,30)( 4,29,14, 5,28,16,35) ( 6,23, 9,36,20,24, 8)( 7,22,32,13,10,11,34)$ $3^{12}$ $56$ $3$ $( 1, 2, 5)( 3,19,34)( 4,25, 9)( 6,14,27)( 7,23,11)( 8,36,17)(10,20,12) (13,16,18)(15,32,35)(21,30,28)(22,26,24)(29,31,33)$ $9^{4}$ $56$ $9$ $( 1, 2, 9,17,24,36,12, 8,14)( 3,34,23, 4,31, 5,21, 6,11)( 7,26,28,18,32,16,15, 33,10)(13,20,22,35,25,30,19,29,27)$ $9^{4}$ $56$ $9$ $( 1, 2,13,23,22, 7,20,11,35)( 3, 8,17,34,16,27, 5,25,32)( 4,21,33,28,31, 6,12, 19,24)( 9,36,14,15,30,10,26,29,18)$ $9^{4}$ $56$ $9$ $( 1, 2,30,34,26,19,10, 3,29)( 4,27,32,20,36,22,16,25, 6)( 5,18,33, 8,11,23,17, 28,15)( 7,35,31, 9,12,24,14,21,13)$

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);