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Properties

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

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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 TypeSizeOrderRepresentative
$ 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