# Properties

 Label 21T14 Degree $21$ Order $168$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $\PSL(2,7)$

# Related objects

## Group action invariants

 Degree $n$: $21$ Transitive number $t$: $14$ Group: $\PSL(2,7)$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,4)(2,6)(3,5)(7,17)(8,16)(9,18)(13,14)(19,20), (1,5,9,12,13,17,19)(2,4,8,11,15,16,20)(3,6,7,10,14,18,21)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 3: None

Degree 7: $\GL(3,2)$ x 2

## Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1$ $21$ $2$ $( 5, 6)( 7,19)( 8,20)( 9,21)(11,12)(13,16)(14,17)(15,18)$ $4, 4, 4, 4, 2, 2, 1$ $42$ $4$ $( 2, 3)( 4,10)( 5,11, 6,12)( 7,16,19,13)( 8,18,20,15)( 9,17,21,14)$ $3, 3, 3, 3, 3, 3, 3$ $56$ $3$ $( 1, 2, 3)( 4, 7,13)( 5, 8,14)( 6, 9,15)(10,19,16)(11,21,17)(12,20,18)$ $7, 7, 7$ $24$ $7$ $( 1, 5, 9,12,13,17,19)( 2, 4, 8,11,15,16,20)( 3, 6, 7,10,14,18,21)$ $7, 7, 7$ $24$ $7$ $( 1, 5,13,17,21,11, 7)( 2, 4,14,16,20,12, 9)( 3, 6,15,18,19,10, 8)$

## 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 2a 4a 3a 7a 7b 2P 1a 1a 2a 3a 7a 7b 3P 1a 2a 4a 1a 7b 7a 5P 1a 2a 4a 3a 7b 7a 7P 1a 2a 4a 3a 1a 1a X.1 1 1 1 1 1 1 X.2 3 -1 1 . A /A X.3 3 -1 1 . /A A X.4 6 2 . . -1 -1 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