Properties

Label 42T8
Order \(84\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_7:A_4$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
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: 21T2

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

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [84, 11]
Character table:   
      2  2  2  .  .  2   2   2   2   2   2  2   2
      3  1  .  1  1  .   .   .   .   .   .  .   .
      7  1  1  .  .  1   1   1   1   1   1  1   1

        1a 2a 3a 3b 7a 14a 14b 14c 14d 14e 7b 14f
     2P 1a 1a 3b 3a 7a  7a  7a  7a  7b  7b 7b  7b
     3P 1a 2a 1a 1a 7b 14f 14d 14e 14a 14b 7a 14c
     5P 1a 2a 3b 3a 7b 14d 14e 14f 14b 14c 7a 14a
     7P 1a 2a 3a 3b 1a  2a  2a  2a  2a  2a 1a  2a
    11P 1a 2a 3b 3a 7a 14b 14c 14a 14e 14f 7b 14d
    13P 1a 2a 3a 3b 7b 14e 14f 14d 14c 14a 7a 14b

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7
C = -E(7)^3-E(7)^5+E(7)^6
D = E(7)^3-E(7)^5-E(7)^6
E = -E(7)^3+E(7)^5-E(7)^6