Properties

Label 42T3
Order \(42\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_3\times D_7$

Learn more about

Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $3$
Group :  $C_3\times D_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,32,4,34,6,36)(2,31,3,33,5,35)(7,25,9,27,11,29)(8,26,10,28,12,30)(13,24,16,19,17,22)(14,23,15,20,18,21)(37,42,40,38,41,39), (1,11,4,7,6,9)(2,12,3,8,5,10)(13,41,16,37,17,40)(14,42,15,38,18,39)(19,34,22,36,24,32)(20,33,21,35,23,31)(25,30,27,26,29,28)
$|\Aut(F/K)|$:  $42$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
14:  $D_{7}$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $D_{7}$

Degree 14: $D_{7}$

Degree 21: 21T3

Low degree siblings

There are no siblings 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 $ $7$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,41)( 8,42)( 9,37)(10,38)(11,40)(12,39)(13,35)(14,36) (15,32)(16,31)(17,33)(18,34)(19,30)(20,29)(21,25)(22,26)(23,27)(24,28)$
$ 6, 6, 6, 6, 6, 6, 6 $ $7$ $6$ $( 1, 3, 6, 2, 4, 5)( 7,37,11,41, 9,40)( 8,38,12,42,10,39)(13,31,17,35,16,33) (14,32,18,36,15,34)(19,26,24,30,22,28)(20,25,23,29,21,27)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 4, 6)( 2, 3, 5)( 7, 9,11)( 8,10,12)(13,16,17)(14,15,18)(19,22,24) (20,21,23)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,40,41)(38,39,42)$
$ 6, 6, 6, 6, 6, 6, 6 $ $7$ $6$ $( 1, 5, 4, 2, 6, 3)( 7,40, 9,41,11,37)( 8,39,10,42,12,38)(13,33,16,35,17,31) (14,34,15,36,18,32)(19,28,22,30,24,26)(20,27,21,29,23,25)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $1$ $3$ $( 1, 6, 4)( 2, 5, 3)( 7,11, 9)( 8,12,10)(13,17,16)(14,18,15)(19,24,22) (20,23,21)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,40)(38,42,39)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1, 8,18,24,27,33,41)( 2, 7,17,23,28,34,42)( 3, 9,13,20,30,36,38) ( 4,10,14,19,29,35,37)( 5,11,16,21,26,32,39)( 6,12,15,22,25,31,40)$
$ 21, 21 $ $2$ $21$ $( 1,10,15,24,29,31,41, 4,12,18,19,25,33,37, 6, 8,14,22,27,35,40) ( 2, 9,16,23,30,32,42, 3,11,17,20,26,34,38, 5, 7,13,21,28,36,39)$
$ 21, 21 $ $2$ $21$ $( 1,12,14,24,25,35,41, 6,10,18,22,29,33,40, 4, 8,15,19,27,31,37) ( 2,11,13,23,26,36,42, 5, 9,17,21,30,34,39, 3, 7,16,20,28,32,38)$
$ 21, 21 $ $2$ $21$ $( 1,14,25,41,10,22,33, 4,15,27,37,12,24,35, 6,18,29,40, 8,19,31) ( 2,13,26,42, 9,21,34, 3,16,28,38,11,23,36, 5,17,30,39, 7,20,32)$
$ 21, 21 $ $2$ $21$ $( 1,15,29,41,12,19,33, 6,14,27,40,10,24,31, 4,18,25,37, 8,22,35) ( 2,16,30,42,11,20,34, 5,13,28,39, 9,23,32, 3,17,26,38, 7,21,36)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,18,27,41, 8,24,33)( 2,17,28,42, 7,23,34)( 3,13,30,38, 9,20,36) ( 4,14,29,37,10,19,35)( 5,16,26,39,11,21,32)( 6,15,25,40,12,22,31)$
$ 21, 21 $ $2$ $21$ $( 1,19,40,18,35,12,27, 4,22,41,14,31, 8,29, 6,24,37,15,33,10,25) ( 2,20,39,17,36,11,28, 3,21,42,13,32, 7,30, 5,23,38,16,34, 9,26)$
$ 21, 21 $ $2$ $21$ $( 1,22,37,18,31,10,27, 6,19,41,15,35, 8,25, 4,24,40,14,33,12,29) ( 2,21,38,17,32, 9,28, 5,20,42,16,36, 7,26, 3,23,39,13,34,11,30)$
$ 7, 7, 7, 7, 7, 7 $ $2$ $7$ $( 1,24,41,18,33, 8,27)( 2,23,42,17,34, 7,28)( 3,20,38,13,36, 9,30) ( 4,19,37,14,35,10,29)( 5,21,39,16,32,11,26)( 6,22,40,15,31,12,25)$

Group invariants

Order:  $42=2 \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [42, 4]
Character table:   
      2  1  1   1   1   1   1  .   .   .   .   .  .   .   .  .
      3  1  1   1   1   1   1  1   1   1   1   1  1   1   1  1
      7  1  .   .   1   .   1  1   1   1   1   1  1   1   1  1

        1a 2a  6a  3a  6b  3b 7a 21a 21b 21c 21d 7b 21e 21f 7c
     2P 1a 1a  3b  3b  3a  3a 7b 21d 21c 21f 21e 7c 21b 21a 7a
     3P 1a 2a  2a  1a  2a  1a 7c  7c  7c  7a  7a 7a  7b  7b 7b
     5P 1a 2a  6b  3b  6a  3a 7b 21d 21c 21f 21e 7c 21b 21a 7a
     7P 1a 2a  6a  3a  6b  3b 1a  3a  3b  3a  3b 1a  3a  3b 1a
    11P 1a 2a  6b  3b  6a  3a 7c 21f 21e 21b 21a 7a 21d 21c 7b
    13P 1a 2a  6a  3a  6b  3b 7a 21a 21b 21c 21d 7b 21e 21f 7c
    17P 1a 2a  6b  3b  6a  3a 7c 21f 21e 21b 21a 7a 21d 21c 7b
    19P 1a 2a  6a  3a  6b  3b 7b 21c 21d 21e 21f 7c 21a 21b 7a

X.1      1  1   1   1   1   1  1   1   1   1   1  1   1   1  1
X.2      1 -1  -1   1  -1   1  1   1   1   1   1  1   1   1  1
X.3      1 -1   A  -A  /A -/A  1  -A -/A  -A -/A  1  -A -/A  1
X.4      1 -1  /A -/A   A  -A  1 -/A  -A -/A  -A  1 -/A  -A  1
X.5      1  1 -/A -/A  -A  -A  1 -/A  -A -/A  -A  1 -/A  -A  1
X.6      1  1  -A  -A -/A -/A  1  -A -/A  -A -/A  1  -A -/A  1
X.7      2  .   .   2   .   2  C   C   C   E   E  E   D   D  D
X.8      2  .   .   2   .   2  D   D   D   C   C  C   E   E  E
X.9      2  .   .   2   .   2  E   E   E   D   D  D   C   C  C
X.10     2  .   .   B   .  /B  C   F  /F   H  /H  E   G  /G  D
X.11     2  .   .  /B   .   B  C  /F   F  /H   H  E  /G   G  D
X.12     2  .   .   B   .  /B  D   G  /G   F  /F  C   H  /H  E
X.13     2  .   .  /B   .   B  D  /G   G  /F   F  C  /H   H  E
X.14     2  .   .   B   .  /B  E   H  /H   G  /G  D   F  /F  C
X.15     2  .   .  /B   .   B  E  /H   H  /G   G  D  /F   F  C

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3
B = 2*E(3)^2
  = -1-Sqrt(-3) = -1-i3
C = E(7)^2+E(7)^5
D = E(7)+E(7)^6
E = E(7)^3+E(7)^4
F = E(21)^8+E(21)^20
G = E(21)^11+E(21)^17
H = E(21)^2+E(21)^5