Properties

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

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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 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, 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