Properties

Label 28T14
Order \(84\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times C_7:C_3$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $14$
Group :  $C_2^2\times C_7:C_3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,15)(2,16)(3,6,24,17,19,10)(4,5,23,18,20,9)(7,14,11,22,27,25)(8,13,12,21,28,26), (1,25,21,17,13,10,5,2,26,22,18,14,9,6)(3,28,24,20,16,12,7,4,27,23,19,15,11,8)
$|\Aut(F/K)|$:  $4$

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
12:  $C_6\times C_2$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $C_7:C_3$

Degree 14: $(C_7:C_3) \times C_2$ x 3

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,19,24)( 4,20,23)( 5, 9,18)( 6,10,17)( 7,27,11)( 8,28,12)(13,26,21) (14,25,22)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $7$ $3$ $( 3,24,19)( 4,23,20)( 5,18, 9)( 6,17,10)( 7,11,27)( 8,12,28)(13,21,26) (14,22,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,20,24, 4,19,23)( 5,10,18, 6, 9,17)( 7,28,11, 8,27,12) (13,25,21,14,26,22)(15,16)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 2)( 3,23,19, 4,24,20)( 5,17, 9, 6,18,10)( 7,12,27, 8,11,28) (13,22,26,14,21,25)(15,16)$
$ 14, 14 $ $3$ $14$ $( 1, 3, 5, 7, 9,11,13,16,18,19,21,24,26,27)( 2, 4, 6, 8,10,12,14,15,17,20,22, 23,25,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 3,21,16,18, 7)( 2, 4,22,15,17, 8)( 5,11, 9,19,26,24)( 6,12,10,20,25,23) (13,27)(14,28)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 3,26,16,18,11)( 2, 4,25,15,17,12)( 5,19)( 6,20)( 7,13,24,21,27, 9) ( 8,14,23,22,28,10)$
$ 14, 14 $ $3$ $14$ $( 1, 4, 5, 8, 9,12,13,15,18,20,21,23,26,28)( 2, 3, 6, 7,10,11,14,16,17,19,22, 24,25,27)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 4,21,15,18, 8)( 2, 3,22,16,17, 7)( 5,12, 9,20,26,23)( 6,11,10,19,25,24) (13,28)(14,27)$
$ 6, 6, 6, 6, 2, 2 $ $7$ $6$ $( 1, 4,26,15,18,12)( 2, 3,25,16,17,11)( 5,20)( 6,19)( 7,14,24,22,27,10) ( 8,13,23,21,28, 9)$
$ 7, 7, 7, 7 $ $3$ $7$ $( 1, 5, 9,13,18,21,26)( 2, 6,10,14,17,22,25)( 3, 7,11,16,19,24,27) ( 4, 8,12,15,20,23,28)$
$ 14, 14 $ $3$ $14$ $( 1, 6, 9,14,18,22,26, 2, 5,10,13,17,21,25)( 3, 8,11,15,19,23,27, 4, 7,12,16, 20,24,28)$
$ 14, 14 $ $3$ $14$ $( 1, 7,13,19,26, 3, 9,16,21,27, 5,11,18,24)( 2, 8,14,20,25, 4,10,15,22,28, 6, 12,17,23)$
$ 14, 14 $ $3$ $14$ $( 1, 8,13,20,26, 4, 9,15,21,28, 5,12,18,23)( 2, 7,14,19,25, 3,10,16,22,27, 6, 11,17,24)$
$ 7, 7, 7, 7 $ $3$ $7$ $( 1,13,26, 9,21, 5,18)( 2,14,25,10,22, 6,17)( 3,16,27,11,24, 7,19) ( 4,15,28,12,23, 8,20)$
$ 14, 14 $ $3$ $14$ $( 1,14,26,10,21, 6,18, 2,13,25, 9,22, 5,17)( 3,15,27,12,24, 8,19, 4,16,28,11, 23, 7,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,15)( 2,16)( 3,17)( 4,18)( 5,20)( 6,19)( 7,22)( 8,21)( 9,23)(10,24)(11,25) (12,26)(13,28)(14,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3,18)( 4,17)( 5,19)( 6,20)( 7,21)( 8,22)( 9,24)(10,23)(11,26) (12,25)(13,27)(14,28)$

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [84, 9]
Character table:   
      2  2  2  2  2   2   2   2   2   2   2   2   2   2   2   2   2   2   2  2
      3  1  1  1  1   1   1   .   1   1   .   1   1   .   .   .   .   .   .  1
      7  1  .  .  1   .   .   1   .   .   1   .   .   1   1   1   1   1   1  1

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

X.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
X.3      1  1  1 -1  -1  -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  -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
X.6      1 /A  A -1 -/A  -A  -1 -/A  -A   1  /A   A   1  -1  -1   1   1  -1  1
X.7      1  A /A -1  -A -/A   1   A  /A  -1  -A -/A   1  -1   1  -1   1  -1 -1
X.8      1 /A  A -1 -/A  -A   1  /A   A  -1 -/A  -A   1  -1   1  -1   1  -1 -1
X.9      1  A /A  1   A  /A  -1  -A -/A  -1  -A -/A   1   1  -1  -1   1   1 -1
X.10     1 /A  A  1  /A   A  -1 -/A  -A  -1 -/A  -A   1   1  -1  -1   1   1 -1
X.11     1  A /A  1   A  /A   1   A  /A   1   A  /A   1   1   1   1   1   1  1
X.12     1 /A  A  1  /A   A   1  /A   A   1  /A   A   1   1   1   1   1   1  1
X.13     3  .  . -3   .   .   B   .   .  -B   .   .  -B   B  /B -/B -/B  /B  3
X.14     3  .  . -3   .   .  /B   .   . -/B   .   . -/B  /B   B  -B  -B   B  3
X.15     3  .  . -3   .   . -/B   .   .  /B   .   . -/B  /B  -B   B  -B   B -3
X.16     3  .  . -3   .   .  -B   .   .   B   .   .  -B   B -/B  /B -/B  /B -3
X.17     3  .  .  3   .   .   B   .   .   B   .   .  -B  -B  /B  /B -/B -/B -3
X.18     3  .  .  3   .   .  /B   .   .  /B   .   . -/B -/B   B   B  -B  -B -3
X.19     3  .  .  3   .   . -/B   .   . -/B   .   . -/B -/B  -B  -B  -B  -B  3
X.20     3  .  .  3   .   .  -B   .   .  -B   .   .  -B  -B -/B -/B -/B -/B  3

      2  2
      3  1
      7  1

        2c
     2P 1a
     3P 2c
     5P 2c
     7P 2c
    11P 2c
    13P 2c

X.1      1
X.2     -1
X.3      1
X.4     -1
X.5     -1
X.6     -1
X.7      1
X.8      1
X.9     -1
X.10    -1
X.11     1
X.12     1
X.13    -3
X.14    -3
X.15     3
X.16     3
X.17    -3
X.18    -3
X.19     3
X.20     3

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(7)-E(7)^2-E(7)^4
  = (1-Sqrt(-7))/2 = -b7