# Properties

 Label 26T39 Degree $26$ Order $5616$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $\PSL(3,3)$

Show commands: Magma

magma: G := TransitiveGroup(26, 39);

## Group action invariants

 Degree $n$: $26$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $39$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $\PSL(3,3)$ 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: (1,2)(3,23)(4,24)(5,6)(7,21)(8,22)(9,11)(10,12)(13,19)(14,20)(17,18)(25,26), (1,4,5,8,9,11,13,16,18,20,21,24,25)(2,3,6,7,10,12,14,15,17,19,22,23,26) magma: Generators(G);

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 13: $\PSL(3,3)$

## Low degree siblings

13T7 x 2, 26T39, 39T43 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

## Conjugacy classes

 Label Cycle Type Size Order Representative 1A $1^{26}$ $1$ $1$ $()$ 2A $2^{12},1^{2}$ $117$ $2$ $( 1,19)( 2,20)( 3, 4)( 5,24)( 6,23)( 9,26)(10,25)(11,12)(13,14)(15,22)(16,21)(17,18)$ 3A $3^{6},1^{8}$ $104$ $3$ $( 1,15,10)( 2,16, 9)( 3,17,14)( 4,18,13)(19,22,25)(20,21,26)$ 3B $3^{8},1^{2}$ $624$ $3$ $( 1,16,11)( 2,15,12)( 3,25,19)( 4,26,20)( 5,14,17)( 6,13,18)( 7,24,22)( 8,23,21)$ 4A $4^{6},1^{2}$ $702$ $4$ $( 1, 5,22,11)( 2, 6,21,12)( 3,20,16,18)( 4,19,15,17)( 7,14, 8,13)( 9,26,10,25)$ 6A $6^{3},2^{3},1^{2}$ $936$ $6$ $( 1,25,15,19,10,22)( 2,26,16,20, 9,21)( 3,13,17, 4,14,18)( 5,24)( 6,23)(11,12)$ 8A1 $8^{3},2$ $702$ $8$ $( 1, 4, 5,19,22,15,11,17)( 2, 3, 6,20,21,16,12,18)( 7,10,14,25, 8, 9,13,26)(23,24)$ 8A-1 $8^{3},2$ $702$ $8$ $( 1,15, 5,17,22, 4,11,19)( 2,16, 6,18,21, 3,12,20)( 7, 9,14,26, 8,10,13,25)(23,24)$ 13A1 $13^{2}$ $432$ $13$ $( 1,22, 6,16,23,10,14, 3,19,26,17, 7,11)( 2,21, 5,15,24, 9,13, 4,20,25,18, 8,12)$ 13A-1 $13^{2}$ $432$ $13$ $( 1, 3,22,19, 6,26,16,17,23, 7,10,11,14)( 2, 4,21,20, 5,25,15,18,24, 8, 9,12,13)$ 13A2 $13^{2}$ $432$ $13$ $( 1,10,17, 6, 3,11,23,26,22,14, 7,16,19)( 2, 9,18, 5, 4,12,24,25,21,13, 8,15,20)$ 13A-2 $13^{2}$ $432$ $13$ $( 1,17, 3,23,22, 7,19,10, 6,11,26,14,16)( 2,18, 4,24,21, 8,20, 9, 5,12,25,13,15)$

magma: ConjugacyClasses(G);

## Group invariants

 Order: $5616=2^{4} \cdot 3^{3} \cdot 13$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: no magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 5616.a magma: IdentifyGroup(G); Character table:

 1A 2A 3A 3B 4A 6A 8A1 8A-1 13A1 13A-1 13A2 13A-2 Size 1 117 104 624 702 936 702 702 432 432 432 432 2 P 1A 1A 3A 3B 2A 3A 4A 4A 13A1 13A-2 13A2 13A-1 3 P 1A 2A 1A 1A 4A 2A 8A1 8A-1 13A-2 13A-1 13A1 13A2 13 P 1A 2A 3A 3B 4A 6A 8A-1 8A1 1A 1A 1A 1A Type

magma: CharacterTable(G);