# Properties

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

## Group action invariants

 Degree $n$ : $26$ Transitive number $t$ : $39$ Group : $\PSL(3,3)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) 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) $|\Aut(F/K)|$: $2$

## 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

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

## Group invariants

 Order: $5616=2^{4} \cdot 3^{3} \cdot 13$ Cyclic: No Abelian: No Solvable: No GAP id: Data not available
 Character table:  2 4 . 4 3 3 3 1 1 . . . . 3 3 2 1 . . . 3 1 . . . . 13 1 . . . . . . . 1 1 1 1 1a 3a 2a 4a 8a 8b 3b 6a 13a 13b 13c 13d 2P 1a 3a 1a 2a 4a 4a 3b 3b 13d 13a 13b 13c 3P 1a 1a 2a 4a 8a 8b 1a 2a 13a 13b 13c 13d 5P 1a 3a 2a 4a 8b 8a 3b 6a 13d 13a 13b 13c 7P 1a 3a 2a 4a 8b 8a 3b 6a 13b 13c 13d 13a 11P 1a 3a 2a 4a 8a 8b 3b 6a 13b 13c 13d 13a 13P 1a 3a 2a 4a 8b 8a 3b 6a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 12 . 4 . . . 3 1 -1 -1 -1 -1 X.3 13 1 -3 1 -1 -1 4 . . . . . X.4 16 1 . . . . -2 . B /C /B C X.5 16 1 . . . . -2 . /B C B /C X.6 16 1 . . . . -2 . C B /C /B X.7 16 1 . . . . -2 . /C /B C B X.8 26 -1 2 2 . . -1 -1 . . . . X.9 26 -1 -2 . A -A -1 1 . . . . X.10 26 -1 -2 . -A A -1 1 . . . . X.11 27 . 3 -1 -1 -1 . . 1 1 1 1 X.12 39 . -1 -1 1 1 3 -1 . . . . A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 B = E(13)^2+E(13)^5+E(13)^6 C = E(13)^4+E(13)^10+E(13)^12