Properties

Label 42T10
Order \(84\)
n \(42\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times F_7$

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Group action invariants

Degree $n$ :  $42$
Transitive number $t$ :  $10$
Group :  $C_2\times F_7$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,5,4)(2,6,3)(7,19,13,37,26,32)(8,20,14,38,25,31)(9,23,17,40,29,33)(10,24,18,39,30,34)(11,22,15,42,27,36)(12,21,16,41,28,35), (1,18,9,35,20,28)(2,17,10,36,19,27)(3,15,12,31,24,25)(4,16,11,32,23,26)(5,13,8,34,22,30)(6,14,7,33,21,29)(37,42,39,38,41,40)
$|\Aut(F/K)|$:  $6$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
3:  $C_3$
4:  $C_2^2$
6:  $C_6$ x 3
42:  $F_7$

Resolvents shown for degrees $\leq 10$

Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 6: $C_6$

Degree 7: $F_7$

Degree 14: $F_7 \times C_2$

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

Group invariants

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

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

X.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
X.3      1 -1  1 -1   1  -1   1  -1  -1   1  -1   1   1  1
X.4      1  1 -1 -1  -1  -1   1   1   1   1  -1  -1  -1  1
X.5      1 -1 -1  1   A  -A  -A   A  /A -/A -/A  /A  -1  1
X.6      1 -1 -1  1  /A -/A -/A  /A   A  -A  -A   A  -1  1
X.7      1 -1  1 -1 -/A  /A -/A  /A   A  -A   A  -A   1  1
X.8      1 -1  1 -1  -A   A  -A   A  /A -/A  /A -/A   1  1
X.9      1  1 -1 -1   A   A  -A  -A -/A -/A  /A  /A  -1  1
X.10     1  1 -1 -1  /A  /A -/A -/A  -A  -A   A   A  -1  1
X.11     1  1  1  1 -/A -/A -/A -/A  -A  -A  -A  -A   1  1
X.12     1  1  1  1  -A  -A  -A  -A -/A -/A -/A -/A   1  1
X.13     6  . -6  .   .   .   .   .   .   .   .   .   1 -1
X.14     6  .  6  .   .   .   .   .   .   .   .   .  -1 -1

A = -E(3)
  = (1-Sqrt(-3))/2 = -b3