Properties

Label 42T31
Order \(168\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_7:A_4$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
6:  $C_6$
12:  $A_4$
24:  $A_4\times C_2$
42:  $F_7$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 7: $F_7$

Degree 14: None

Degree 21: 21T4

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

Group invariants

Order:  $168=2^{3} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [168, 49]
Character table:   
      2  3  3  3  3  1   1   1  1  2   2   2   2
      3  1  1  .  .  1   1   1  1  .   .   .   .
      7  1  .  1  .  .   .   .  .  1   1   1   1

        1a 2a 2b 2c 3a  6a  6b 3b 7a 14a 14b 14c
     2P 1a 1a 1a 1a 3b  3b  3a 3a 7a  7a  7a  7a
     3P 1a 2a 2b 2c 1a  2a  2a 1a 7a 14b 14c 14a
     5P 1a 2a 2b 2c 3b  6b  6a 3a 7a 14c 14a 14b
     7P 1a 2a 2b 2c 3a  6a  6b 3b 1a  2b  2b  2b
    11P 1a 2a 2b 2c 3b  6b  6a 3a 7a 14b 14c 14a
    13P 1a 2a 2b 2c 3a  6a  6b 3b 7a 14a 14b 14c

X.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
X.3      1 -1  1 -1  A  -A -/A /A  1   1   1   1
X.4      1 -1  1 -1 /A -/A  -A  A  1   1   1   1
X.5      1  1  1  1  A   A  /A /A  1   1   1   1
X.6      1  1  1  1 /A  /A   A  A  1   1   1   1
X.7      3 -3 -1  1  .   .   .  .  3  -1  -1  -1
X.8      3  3 -1 -1  .   .   .  .  3  -1  -1  -1
X.9      6  .  6  .  .   .   .  . -1  -1  -1  -1
X.10     6  . -2  .  .   .   .  . -1   B   C   D
X.11     6  . -2  .  .   .   .  . -1   C   D   B
X.12     6  . -2  .  .   .   .  . -1   D   B   C

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)-E(7)^2-E(7)^3-E(7)^4-E(7)^5+E(7)^6
C = -E(7)-E(7)^2+E(7)^3+E(7)^4-E(7)^5-E(7)^6
D = -E(7)+E(7)^2-E(7)^3-E(7)^4+E(7)^5-E(7)^6