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Properties

Label 42T37
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 $t$ :  $37$
Group :  $\PSL(2,7)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,10,30,22,14,40,31)(2,9,29,21,13,39,32)(3,11,25,19,16,41,36)(4,12,26,20,15,42,35)(5,7,28,23,17,37,33)(6,8,27,24,18,38,34), (1,28,33)(2,27,34)(3,30,35)(4,29,36)(5,25,31)(6,26,32)(7,10,11)(8,9,12)(13,41,21)(14,42,22)(15,40,19)(16,39,20)(17,38,24)(18,37,23)
$|\Aut(F/K)|$:  $2$

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: None

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

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