# Properties

 Label 28T32 Degree $28$ Order $168$ Cyclic no Abelian no Solvable no Primitive no $p$-group no Group: $\PSL(2,7)$

## Group action invariants

 Degree $n$: $28$ Transitive number $t$: $32$ Group: $\PSL(2,7)$ Parity: $1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,13,28,18,22,6,10)(2,14,26,17,24,7,12)(3,16,25,19,23,8,11)(4,15,27,20,21,5,9), (1,14)(2,15)(3,16)(4,13)(5,28)(6,26)(7,25)(8,27)(11,12)(17,18)(19,20)(21,24)

## Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 4: None

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

Degree 14: None

## Low degree siblings

7T5 x 2, 8T37, 14T10 x 2, 21T14, 24T284, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1$ $21$ $2$ $( 3, 4)( 5,26)( 6,28)( 7,25)( 8,27)( 9,22)(10,23)(11,24)(12,21)(13,16)(17,19) (18,20)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 1$ $56$ $3$ $( 2, 3, 4)( 5,13,25)( 6,15,28)( 7,16,26)( 8,14,27)( 9,24,20)(10,21,19) (11,22,18)(12,23,17)$ $4, 4, 4, 4, 4, 4, 2, 2$ $42$ $4$ $( 1, 2, 3, 4)( 5,16,24,18)( 6,14,23,17)( 7,13,22,20)( 8,15,21,19)( 9,26) (10,28,12,27)(11,25)$ $7, 7, 7, 7$ $24$ $7$ $( 1, 5,13,17,25,23,12)( 2, 6,14,18,26,21, 9)( 3, 7,15,19,27,24,11) ( 4, 8,16,20,28,22,10)$ $7, 7, 7, 7$ $24$ $7$ $( 1, 5,15,22,11,28,19)( 2, 7,13,21,12,27,18)( 3, 6,14,24,10,25,20) ( 4, 8,16,23, 9,26,17)$

## 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 3a 4a 7a 7b 2P 1a 1a 3a 2a 7a 7b 3P 1a 2a 1a 4a 7b 7a 5P 1a 2a 3a 4a 7b 7a 7P 1a 2a 3a 4a 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