# Properties

 Label 8T37 Degree $8$ Order $168$ Cyclic no Abelian no Solvable no Primitive yes $p$-group no Group: $\PSL(2,7)$

# Related objects

## Group action invariants

 Degree $n$: $8$ Transitive number $t$: $37$ Group: $\PSL(2,7)$ CHM label: $L(8)=PSL(2,7)$ Parity: $1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,2,4)(3,6,5), (1,2,3,4,5,6,8), (1,6)(2,3)(4,5)(7,8)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Degree 2: None

Degree 4: None

## Low degree siblings

7T5 x 2, 14T10 x 2, 21T14, 24T284, 28T32, 42T37, 42T38 x 2

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 $1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1, 1$ $56$ $3$ $(3,7,8)(4,5,6)$ $7, 1$ $24$ $7$ $(2,3,7,6,8,5,4)$ $7, 1$ $24$ $7$ $(2,4,5,8,6,7,3)$ $2, 2, 2, 2$ $21$ $2$ $(1,2)(3,4)(5,8)(6,7)$ $4, 4$ $42$ $4$ $(1,2,3,7)(4,6,5,8)$

## Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: no GAP id: [168, 42]
 Character table:  2 3 . . . 3 2 3 1 1 . . . . 7 1 . 1 1 . . 1a 3a 7a 7b 2a 4a 2P 1a 3a 7a 7b 1a 2a 3P 1a 1a 7b 7a 2a 4a 5P 1a 3a 7b 7a 2a 4a 7P 1a 3a 1a 1a 2a 4a X.1 1 1 1 1 1 1 X.2 3 . A /A -1 1 X.3 3 . /A A -1 1 X.4 6 . -1 -1 2 . X.5 7 1 . . -1 -1 X.6 8 -1 1 1 . . A = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7