# Properties

 Label 42T26 Order $$168$$ n $$42$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $F_8:C_3$

## Group action invariants

 Degree $n$ : $42$ Transitive number $t$ : $26$ Group : $F_8:C_3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,25,39,2,26,40)(3,28,42)(4,27,41)(5,29,38)(6,30,37)(7,11,10,8,12,9)(13,34,24,14,33,23)(15,36,19,16,35,20)(17,32,21)(18,31,22), (1,13,27,38,8,22,36)(2,14,28,37,7,21,35)(3,16,29,39,10,23,31)(4,15,30,40,9,24,32)(5,17,26,42,12,19,33)(6,18,25,41,11,20,34) $|\Aut(F/K)|$: $6$

## Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 10$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: None

Degree 7: $C_7:C_3$

Degree 14: 14T11

Degree 21: 21T2

## Low degree siblings

8T36

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

## Group invariants

 Order: $168=2^{3} \cdot 3 \cdot 7$ Cyclic: No Abelian: No Solvable: Yes GAP id: [168, 43]
 Character table:  2 3 3 1 1 1 1 . . 3 1 1 1 1 1 1 . . 7 1 . . . . . 1 1 1a 2a 3a 6a 3b 6b 7a 7b 2P 1a 1a 3b 3b 3a 3a 7a 7b 3P 1a 2a 1a 2a 1a 2a 7b 7a 5P 1a 2a 3b 6b 3a 6a 7b 7a 7P 1a 2a 3a 6a 3b 6b 1a 1a X.1 1 1 1 1 1 1 1 1 X.2 1 1 A A /A /A 1 1 X.3 1 1 /A /A A A 1 1 X.4 3 3 . . . . B /B X.5 3 3 . . . . /B B X.6 7 -1 1 -1 1 -1 . . X.7 7 -1 A -A /A -/A . . X.8 7 -1 /A -/A A -A . . A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = E(7)^3+E(7)^5+E(7)^6 = (-1-Sqrt(-7))/2 = -1-b7