Properties

Label 42T39
Order \(252\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times C_7:C_3$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $39$
Group :  $A_4\times C_7:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,14,25,39,9,23,36,3,16,28,41,12,20,32,5,18,30,38,7,21,33)(2,13,26,40,10,24,35,4,15,27,42,11,19,31,6,17,29,37,8,22,34), (1,21,15)(2,22,16)(3,24,18)(4,23,17)(5,19,13)(6,20,14)(7,32,37)(8,31,38)(9,34,39)(10,33,40)(11,36,41)(12,35,42)(25,27,29)(26,28,30)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$ x 4
9:  $C_3^2$
12:  $A_4$
21:  $C_7:C_3$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 7: $C_7:C_3$

Degree 14: None

Degree 21: 21T7

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

Group invariants

Order:  $252=2^{2} \cdot 3^{2} \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [252, 27]
Character table:   
      2  2  2  2  2   2   2  .  .  .  .  .  .  2   2   .   .   2  2   .   .
      3  2  2  2  1   1   1  2  2  2  2  2  2  1   .   1   1   .  1   1   1
      7  1  .  .  1   .   .  1  .  .  1  .  .  1   1   1   1   1  1   1   1

        1a 3a 3b 2a  6a  6b 3c 3d 3e 3f 3g 3h 7a 14a 21a 21b 14b 7b 21c 21d
     2P 1a 3b 3a 1a  3b  3a 3f 3h 3g 3c 3e 3d 7a  7a 21b 21a  7b 7b 21d 21c
     3P 1a 1a 1a 2a  2a  2a 1a 1a 1a 1a 1a 1a 7b 14b  7b  7b 14a 7a  7a  7a
     5P 1a 3b 3a 2a  6b  6a 3f 3h 3g 3c 3e 3d 7b 14b 21d 21c 14a 7a 21b 21a
     7P 1a 3a 3b 2a  6a  6b 3c 3d 3e 3f 3g 3h 1a  2a  3c  3f  2a 1a  3c  3f
    11P 1a 3b 3a 2a  6b  6a 3f 3h 3g 3c 3e 3d 7a 14a 21b 21a 14b 7b 21d 21c
    13P 1a 3a 3b 2a  6a  6b 3c 3d 3e 3f 3g 3h 7b 14b 21c 21d 14a 7a 21a 21b
    17P 1a 3b 3a 2a  6b  6a 3f 3h 3g 3c 3e 3d 7b 14b 21d 21c 14a 7a 21b 21a
    19P 1a 3a 3b 2a  6a  6b 3c 3d 3e 3f 3g 3h 7b 14b 21c 21d 14a 7a 21a 21b

X.1      1  1  1  1   1   1  1  1  1  1  1  1  1   1   1   1   1  1   1   1
X.2      1  1  1  1   1   1  A  A  A /A /A /A  1   1   A  /A   1  1   A  /A
X.3      1  1  1  1   1   1 /A /A /A  A  A  A  1   1  /A   A   1  1  /A   A
X.4      1  A /A  1   A  /A  1  A /A  1  A /A  1   1   1   1   1  1   1   1
X.5      1 /A  A  1  /A   A  1 /A  A  1 /A  A  1   1   1   1   1  1   1   1
X.6      1  A /A  1   A  /A  A /A  1 /A  1  A  1   1   A  /A   1  1   A  /A
X.7      1 /A  A  1  /A   A /A  A  1  A  1 /A  1   1  /A   A   1  1  /A   A
X.8      1  A /A  1   A  /A /A  1  A  A /A  1  1   1  /A   A   1  1  /A   A
X.9      1 /A  A  1  /A   A  A  1 /A /A  A  1  1   1   A  /A   1  1   A  /A
X.10     3  3  3 -1  -1  -1  .  .  .  .  .  .  3  -1   .   .  -1  3   .   .
X.11     3  .  .  3   .   .  3  .  .  3  .  .  C   C   C   C  /C /C  /C  /C
X.12     3  .  .  3   .   .  3  .  .  3  .  . /C  /C  /C  /C   C  C   C   C
X.13     3  .  .  3   .   .  B  .  . /B  .  .  C   C   E   F  /C /C  /F  /E
X.14     3  .  .  3   .   . /B  .  .  B  .  .  C   C   F   E  /C /C  /E  /F
X.15     3  .  .  3   .   .  B  .  . /B  .  . /C  /C  /F  /E   C  C   E   F
X.16     3  .  .  3   .   . /B  .  .  B  .  . /C  /C  /E  /F   C  C   F   E
X.17     3  B /B -1  -A -/A  .  .  .  .  .  .  3  -1   .   .  -1  3   .   .
X.18     3 /B  B -1 -/A  -A  .  .  .  .  .  .  3  -1   .   .  -1  3   .   .
X.19     9  .  . -3   .   .  .  .  .  .  .  .  D  -C   .   . -/C /D   .   .
X.20     9  .  . -3   .   .  .  .  .  .  .  . /D -/C   .   .  -C  D   .   .

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = 3*E(3)^2
  = (-3-3*Sqrt(-3))/2 = -3-3b3
C = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
D = 3*E(7)^3+3*E(7)^5+3*E(7)^6
  = (-3-3*Sqrt(-7))/2 = -3-3b7
E = E(21)^2+E(21)^8+E(21)^11
F = E(21)+E(21)^4+E(21)^16