# Properties

 Label 13T7 Order $$5616$$ n $$13$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(3,3)$

# Related objects

## Group action invariants

 Degree $n$ : $13$ Transitive number $t$ : $7$ Group : $\PSL(3,3)$ CHM label : $L(13)=PSL(3,3)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,3,4,5,6,7,8,9,10,11,12,13), (2,12)(4,11)(5,6)(7,10) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

13T7, 26T39 x 2, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $13$ $432$ $13$ $( 1, 4, 5, 6, 3, 2, 7, 8,12,11, 9,10,13)$ $13$ $432$ $13$ $( 1,12, 6,10, 7, 4,11, 3,13, 8, 5, 9, 2)$ $13$ $432$ $13$ $( 1,13,10, 9,11,12, 8, 7, 2, 3, 6, 5, 4)$ $13$ $432$ $13$ $( 1, 2, 9, 5, 8,13, 3,11, 4, 7,10, 6,12)$ $2, 2, 2, 2, 1, 1, 1, 1, 1$ $117$ $2$ $( 1, 9)( 4, 7)( 6,10)( 8,12)$ $4, 4, 2, 2, 1$ $702$ $4$ $( 1, 7, 9, 4)( 2,11)( 3, 5)( 6, 8,10,12)$ $8, 4, 1$ $702$ $8$ $( 1,12, 4,10, 9, 8, 7, 6)( 2, 5,11, 3)$ $8, 4, 1$ $702$ $8$ $( 1, 6, 7, 8, 9,10, 4,12)( 2, 3,11, 5)$ $3, 3, 3, 1, 1, 1, 1$ $104$ $3$ $( 1, 6, 5)( 2,12,13)( 3, 7,10)$ $6, 3, 2, 1, 1$ $936$ $6$ $( 1, 3, 6, 7, 5,10)( 2,13,12)( 4, 9)$ $3, 3, 3, 3, 1$ $624$ $3$ $( 1, 4,11)( 2, 8,12)( 3, 5, 6)( 7,10, 9)$

## 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 1 1 . 3 3 3 . . . . 3 3 1 3 1 2 . . . . . . . 13 1 . . . . . . . 1 1 1 1 1a 2a 3a 6a 3b 4a 8a 8b 13a 13b 13c 13d 2P 1a 1a 3a 3a 3b 2a 4a 4a 13d 13a 13b 13c 3P 1a 2a 1a 2a 1a 4a 8a 8b 13a 13b 13c 13d 5P 1a 2a 3a 6a 3b 4a 8b 8a 13d 13a 13b 13c 7P 1a 2a 3a 6a 3b 4a 8b 8a 13b 13c 13d 13a 11P 1a 2a 3a 6a 3b 4a 8a 8b 13b 13c 13d 13a 13P 1a 2a 3a 6a 3b 4a 8b 8a 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 -3 4 . 1 1 -1 -1 . . . . X.4 16 . -2 . 1 . . . B /C /B C X.5 16 . -2 . 1 . . . /B C B /C X.6 16 . -2 . 1 . . . C B /C /B X.7 16 . -2 . 1 . . . /C /B C B X.8 26 2 -1 -1 -1 2 . . . . . . X.9 26 -2 -1 1 -1 . A -A . . . . X.10 26 -2 -1 1 -1 . -A A . . . . X.11 27 3 . . . -1 -1 -1 1 1 1 1 X.12 39 -1 3 -1 . -1 1 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