Properties

Label 28T15
Order \(84\)
n \(28\)
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 $n$ :  $28$
Transitive number $t$ :  $15$
Group :  $C_2\times F_7$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2)(3,8,19,27,23,12)(4,7,20,28,24,11)(5,14,9,26,17,22)(6,13,10,25,18,21)(15,16), (1,12,18,16,26,4)(2,11,17,15,25,3)(5,19)(6,20)(7,9,28,21,23,13)(8,10,27,22,24,14)
$|\Aut(F/K)|$:  $4$

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:  $C_6\times C_2$
42:  $F_7$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 7: $F_7$

Degree 14: $F_7$, $F_7 \times C_2$ x 2

Low degree siblings

14T7 x 2, 42T10 x 2

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

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   1   2   2  1  2
      3  1  1  1   1   1  1  1   1   1   .   1   1  .  1
      7  1  .  .   .   .  .  .   .   .   1   .   .  1  1

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

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

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