# Properties

 Label 10T26 Order $$360$$ n $$10$$ Cyclic No Abelian No Solvable No Primitive Yes $p$-group No Group: $\PSL(2,9)$

# Related objects

## Group action invariants

 Degree $n$ : $10$ Transitive number $t$ : $26$ Group : $\PSL(2,9)$ CHM label : $L(10)=PSL(2,9)$ Parity: $1$ Primitive: Yes Nilpotency class: $-1$ (not nilpotent) Generators: (1,2)(4,7)(5,8)(9,10), (1,2,10)(3,4,5)(6,7,8), (1,3,2,6)(4,5,8,7) $|\Aut(F/K)|$: $1$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 5: None

## Low degree siblings

6T15 x 2, 15T20 x 2, 20T89, 30T88 x 2, 36T555, 40T304, 45T49

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 1, 1$ $45$ $2$ $( 3, 6)( 4, 5)( 7, 8)( 9,10)$ $4, 4, 1, 1$ $90$ $4$ $( 3, 9, 6,10)( 4, 8, 5, 7)$ $3, 3, 3, 1$ $40$ $3$ $( 2, 3, 6)( 4, 9, 7)( 5, 8,10)$ $3, 3, 3, 1$ $40$ $3$ $( 2, 4, 5)( 3, 9, 8)( 6, 7,10)$ $5, 5$ $72$ $5$ $( 1, 2, 3, 4, 9)( 5, 7,10, 6, 8)$ $5, 5$ $72$ $5$ $( 1, 2, 3, 8,10)( 4, 7, 5, 9, 6)$

## Group invariants

 Order: $360=2^{3} \cdot 3^{2} \cdot 5$ Cyclic: No Abelian: No Solvable: No GAP id: [360, 118]
 Character table:  2 3 3 2 . . . . 3 2 . . 2 2 . . 5 1 . . . . 1 1 1a 2a 4a 3a 3b 5a 5b 2P 1a 1a 2a 3a 3b 5b 5a 3P 1a 2a 4a 1a 1a 5b 5a 5P 1a 2a 4a 3a 3b 1a 1a X.1 1 1 1 1 1 1 1 X.2 5 1 -1 2 -1 . . X.3 5 1 -1 -1 2 . . X.4 8 . . -1 -1 A *A X.5 8 . . -1 -1 *A A X.6 9 1 1 . . -1 -1 X.7 10 -2 . 1 1 . . A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5