Properties

Label 42T29
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $A_4\times D_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $29$
Group :  $A_4\times D_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,11,14,24,26,36,41,6,9,18,21,29,33,39,3,7,16,20,27,31,38)(2,12,13,23,25,35,42,5,10,17,22,30,34,40,4,8,15,19,28,32,37), (1,31,3,34,5,35)(2,32,4,33,6,36)(7,26,9,28,12,30)(8,25,10,27,11,29)(13,24,16,20,17,22)(14,23,15,19,18,21)(37,41,39,38,42,40)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
14:  $D_{7}$
24:  $A_4\times C_2$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4\times C_2$

Degree 7: $D_{7}$

Degree 14: None

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 48]
Character table:   
      2  3  3  3  3  1   1  1   1  2   2   .   .   .   .   2  2   .   .   2  2
      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   1   1   1  1

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

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-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 = 3*E(7)^2+3*E(7)^5
G = 3*E(7)+3*E(7)^6
H = 3*E(7)^3+3*E(7)^4
I = E(21)^8+E(21)^20
J = E(21)^11+E(21)^17
K = E(21)^2+E(21)^5