# Properties

 Label 42T38 Order $$168$$ n $$42$$ Cyclic No Abelian No Solvable No Primitive No $p$-group No Group: $\PSL(2,7)$

## Group action invariants

 Degree $n$ : $42$ Transitive number $t$ : $38$ Group : $\PSL(2,7)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,24,40,9,25,35,14)(2,23,39,10,26,36,13)(3,19,42,11,28,31,18)(4,20,41,12,27,32,17)(5,21,38,8,30,34,15)(6,22,37,7,29,33,16), (1,32,39,24)(2,31,40,23)(3,35,37,21)(4,36,38,22)(5,33,42,19)(6,34,41,20)(7,25,11,29)(8,26,12,30)(9,28)(10,27)(13,17,14,18)(15,16) $|\Aut(F/K)|$: $6$

## Low degree resolvents

None

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 3: None

Degree 6: None

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

Degree 14: $\PSL(2,7)$

Degree 21: $\PSL(2,7)$

## Low degree siblings

7T5 x 2, 8T37

Siblings are shown with degree $\leq 10$

Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

## Conjugacy Classes

 Cycle Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1$ $21$ $2$ $( 3, 4)( 5, 6)( 9,12)(10,11)(13,37)(14,38)(15,39)(16,40)(17,42)(18,41)(21,24) (22,23)(25,31)(26,32)(27,33)(28,34)(29,36)(30,35)$ $4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2$ $42$ $4$ $( 1, 2)( 3, 5, 4, 6)( 7,19)( 8,20)( 9,21,12,24)(10,22,11,23)(13,31,37,25) (14,32,38,26)(15,35,39,30)(16,36,40,29)(17,34,42,28)(18,33,41,27)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $56$ $3$ $( 1, 3, 5)( 2, 4, 6)( 7,38,26)( 8,37,25)( 9,40,30)(10,39,29)(11,41,28) (12,42,27)(13,31,20)(14,32,19)(15,34,24)(16,33,23)(17,35,22)(18,36,21)$ $7, 7, 7, 7, 7, 7$ $24$ $7$ $( 1, 9,18,24,26,34,37)( 2,10,17,23,25,33,38)( 3, 8,15,22,30,32,40) ( 4, 7,16,21,29,31,39)( 5,11,14,20,27,36,41)( 6,12,13,19,28,35,42)$ $7, 7, 7, 7, 7, 7$ $24$ $7$ $( 1, 9,14,40,35,24,25)( 2,10,13,39,36,23,26)( 3,11,18,42,31,19,28) ( 4,12,17,41,32,20,27)( 5, 8,15,38,34,21,30)( 6, 7,16,37,33,22,29)$

## 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