Properties

Label 28T13
Order \(84\)
n \(28\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times C_7:C_3$

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Group action invariants

Degree $n$ :  $28$
Transitive number $t$ :  $13$
Group :  $C_4\times C_7:C_3$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,17,15,21,3,2,8,18,16,22,4)(5,24,25,20,9,11,6,23,26,19,10,12)(13,27,14,28), (1,23,17,11,5,28,22,16,9,4,25,19,13,8,2,24,18,12,6,27,21,15,10,3,26,20,14,7)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
3:  $C_3$
4:  $C_4$
6:  $C_6$
12:  $C_{12}$
21:  $C_7:C_3$
42:  $(C_7:C_3) \times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 7: $C_7:C_3$

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

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

Group invariants

Order:  $84=2^{2} \cdot 3 \cdot 7$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [84, 2]
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 28a 12a 12b 28b 12c 12d  7a 14a 28c 28d  7b 14b 4a
     2P 1a 3b 3a 1a  3b  3a 14a  6b  6a 14a  6b  6a  7a  7a 14b 14b  7b  7b 2a
     3P 1a 1a 1a 2a  2a  2a 28d  4a  4a 28c  4b  4b  7b 14b 28b 28a  7a 14a 4b
     5P 1a 3b 3a 2a  6b  6a 28c 12b 12a 28d 12d 12c  7b 14b 28a 28b  7a 14a 4a
     7P 1a 3a 3b 2a  6a  6b  4a 12c 12d  4b 12a 12b  1a  2a  4a  4b  1a  2a 4b
    11P 1a 3b 3a 2a  6b  6a 28b 12d 12c 28a 12b 12a  7a 14a 28d 28c  7b 14b 4b
    13P 1a 3a 3b 2a  6a  6b 28c 12a 12b 28d 12c 12d  7b 14b 28a 28b  7a 14a 4a
    17P 1a 3b 3a 2a  6b  6a 28c 12b 12a 28d 12d 12c  7b 14b 28a 28b  7a 14a 4a
    19P 1a 3a 3b 2a  6a  6b 28d 12c 12d 28c 12a 12b  7b 14b 28b 28a  7a 14a 4b
    23P 1a 3b 3a 2a  6b  6a 28b 12d 12c 28a 12b 12a  7a 14a 28d 28c  7b 14b 4b

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   B   B   B  -B  -B  -B   1  -1   B  -B   1  -1 -B
X.4      1  1  1 -1  -1  -1  -B  -B  -B   B   B   B   1  -1  -B   B   1  -1  B
X.5      1  A /A -1  -A -/A   B   E -/E  -B  -E  /E   1  -1   B  -B   1  -1 -B
X.6      1  A /A -1  -A -/A  -B  -E  /E   B   E -/E   1  -1  -B   B   1  -1  B
X.7      1 /A  A -1 -/A  -A   B -/E   E  -B  /E  -E   1  -1   B  -B   1  -1 -B
X.8      1 /A  A -1 -/A  -A  -B  /E  -E   B -/E   E   1  -1  -B   B   1  -1  B
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   .   .   C   .   .  -C   .   . -/D  /D -/C  /C  -D   D  F
X.14     3  .  . -3   .   . -/C   .   .  /C   .   .  -D   D   C  -C -/D  /D  F
X.15     3  .  . -3   .   .  /C   .   . -/C   .   .  -D   D  -C   C -/D  /D -F
X.16     3  .  . -3   .   .  -C   .   .   C   .   . -/D  /D  /C -/C  -D   D -F
X.17     3  .  .  3   .   .   D   .   .   D   .   .  -D  -D  /D  /D -/D -/D -3
X.18     3  .  .  3   .   .  /D   .   .  /D   .   . -/D -/D   D   D  -D  -D -3
X.19     3  .  .  3   .   . -/D   .   . -/D   .   . -/D -/D  -D  -D  -D  -D  3
X.20     3  .  .  3   .   .  -D   .   .  -D   .   .  -D  -D -/D -/D -/D -/D  3

      2  2
      3  1
      7  1

        4b
     2P 2a
     3P 4a
     5P 4b
     7P 4a
    11P 4a
    13P 4b
    17P 4b
    19P 4a
    23P 4a

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

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = -E(4)
  = -Sqrt(-1) = -i
C = -E(28)^3-E(28)^19-E(28)^27
D = -E(7)-E(7)^2-E(7)^4
  = (1-Sqrt(-7))/2 = -b7
E = -E(12)^11
F = 3*E(4)
  = 3*Sqrt(-1) = 3i