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Properties

Label 42T4
Order \(42\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $4$
Group :  $F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,31,30,12,40)(2,19,32,29,11,39)(3,24,33,28,8,42)(4,23,34,27,7,41)(5,21,36,25,10,38)(6,22,35,26,9,37)(13,15,18,14,16,17), (1,27,33)(2,28,34)(3,26,36)(4,25,35)(5,29,31)(6,30,32)(7,40,17)(8,39,18)(9,42,15)(10,41,16)(11,38,14)(12,37,13)(19,23,22)(20,24,21)
$|\Aut(F/K)|$:  $42$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $F_7$

Degree 14: $F_7$

Degree 21: 21T4

Low degree siblings

7T4

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

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [42, 1]
Character table:    
     2  1  1  1   1  1   1  .
     3  1  1  1   1  1   1  .
     7  1  .  .   .  .   .  1

       1a 2a 3a  6a 3b  6b 7a
    2P 1a 1a 3b  3b 3a  3a 7a
    3P 1a 2a 1a  2a 1a  2a 7a
    5P 1a 2a 3b  6b 3a  6a 7a
    7P 1a 2a 3a  6a 3b  6b 1a

X.1     1  1  1   1  1   1  1
X.2     1 -1  1  -1  1  -1  1
X.3     1 -1  A  -A /A -/A  1
X.4     1 -1 /A -/A  A  -A  1
X.5     1  1  A   A /A  /A  1
X.6     1  1 /A  /A  A   A  1
X.7     6  .  .   .  .   . -1

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3